# Coincidences between shtuka spaces

Adam Logan and Jared Weinstein, with an appendix by Masato Kuwata

# Higher modularity of elliptic curves over function fields

Adam Logan and Jared Weinstein, with an appendix by Masato Kuwata

###### Abstract

We investigate a notion of “higher modularity” for elliptic curves over
function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we
say that $E$ is $r$-modular when there is an algebraic correspondence between
a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as
an elliptic surface. The (known) case $r=1$ is analogous to the notion of
modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if
$E/\mathbf{F}_{q}(t)$ is a nonisotrivial elliptic curve whose conductor has
degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3
surfaces. Along the way we prove a result of independent interest: A K3
surface admits a finite morphism to a Kummer surface attached to a product of
elliptic curves if and only if its Picard lattice is rationally isometric to
the Picard lattice of such a Kummer surface.

###### Contents

  1. 1 Introduction
    1. 1.1 Analytic modularity, geometric modularity
    2. 1.2 Stacks of shtukas, and the definition of higher modularity
    3. 1.3 Relation to the Tate conjecture
    4. 1.4 Strategy of proof of 2-modularity
    5. 1.5 Application: Heegner-Drinfeld cycles on $\mathcal{E}^{r}$
    6. 1.6 Acknowledgments
  2. 2 The stacks of shtukas for $\operatorname{PGL}_{2}$
    1. 2.1 Vector bundles of rank 2, fractional twists, Atkin-Lehner automorphisms, and passage to $G=\operatorname{PGL}_{2}$
    2. 2.2 Stacks of $G$-shtukas
    3. 2.3 Cohomology of stacks of shtukas
    4. 2.4 Vector bundles with level structures on $\mathbf{P}^{1}$
    5. 2.5 Explicit presentations of shtuka spaces with $\Gamma_{0}(N)$-structure
    6. 2.6 The coincidence map
    7. 2.7 Explicit equations for the spaces of $G$-coincidences
  3. 3 Isogenies between K3 surfaces
    1. 3.1 Motivation: $2$-modularity of extremal rational elliptic fibrations
    2. 3.2 Elliptic fibrations on K3 surfaces
      1. 3.2.1 Generalities on elliptic surfaces
      2. 3.2.2 K3 elliptic surfaces
    3. 3.3 Kummer surfaces $\operatorname{Km}(E_{1}\times E_{2})$ of Picard rank 18
    4. 3.4 Extremal rational elliptic fibrations, and associated K3 surfaces
      1. 3.4.1 The $\mathrm{I}_{2}^{\ast}\mathrm{I}_{2}\mathrm{I}_{2}$ fibration, with Mordell-Weil group $(\mathbf{Z}/2\mathbf{Z})^{2}$
    5. 3.5 Appendix by Masato Kuwata: Kummer surfaces and Inose surfaces
      1. 3.5.1 The $\mathrm{II}^{*}\mathrm{I}_{1}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $0$
      2. 3.5.2 The $\mathrm{III}^{*}\mathrm{I}_{2}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/2\mathbf{Z}$
      3. 3.5.3 The $\mathrm{IV}^{*}\mathrm{I}_{3}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/3\mathbf{Z}$
    6. 3.6 The semistable fibrations
      1. 3.6.1 The $\mathrm{I}_{9}\mathrm{I}_{1}\mathrm{I}_{1}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/3\mathbf{Z}$
      2. 3.6.2 The $\mathrm{I}_{8}\mathrm{I}_{2}\mathrm{I}_{1}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/4\mathbf{Z}$
      3. 3.6.3 The $\mathrm{I}_{5}\mathrm{I}_{5}\mathrm{I}_{1}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/5\mathbf{Z}$
      4. 3.6.4 The $\mathrm{I}_{6}\mathrm{I}_{3}\mathrm{I}_{2}\mathrm{I}_{1}$ fibration, with Mordell-Weil group $\mathbf{Z}/6\mathbf{Z}$
  4. 4 Calabi-Yau threefolds and 3-modularity
    1. 4.1 Fiber products of two elliptic fibrations
    2. 4.2 The Kummer threefold
    3. 4.3 A birational map between a Kummer threefold and a fiber product of elliptic fibrations

##  1 Introduction

Let $K$ be a function field over a finite field $\mathbf{F}_{q}$, and let
$E/K$ be an elliptic curve. In this article we investigate a notion of “higher
modularity” for $E/K$.

###  1.1 Analytic modularity, geometric modularity

Let us briefly recall the meaning of modularity for an elliptic curve $E$ over
the rational numbers $\mathbf{Q}$. Let $N$ be the conductor of $E$. Following
[Ulm03], we might distinguish between two notions of modularity for
$E/\mathbf{Q}$:

  1. 1.

(Analytic modularity) There exists a cuspidal holomorphic form $f$ of weight
$2$ and level $\Gamma_{0}(N)$ such that $L(f,\chi,s)=L(E,\chi,s)$ for all
Dirichlet characters $\chi$.

  2. 2.

(Geometric/motivic modularity) There exists a non-constant morphism
$X_{0}(N)\to E$ defined over $\mathbf{Q}$, where $X_{0}(N)$ is the modular
curve of level $\Gamma_{0}(N)$.

The two notions are equivalent. If $E$ is geometrically modular, the required
cusp form $f$ may be found by pulling back a nonzero holomorphic differential
form on $E$ through $X_{0}(N)\to E$. Conversely, if $E$ is analytically
modular, there exists an elliptic curve quotient $\operatorname{{\rm
Jac}}X_{0}(N)\to E^{\prime}$ corresponding to its cusp form $f$. The equality
of $L$-functions implies that the representations of
$\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ on the rational
$\ell$-adic Tate modules of $E$ and $E^{\prime}$ are isomorphic. A theorem of
Faltings [Fal83] implies that $E$ and $E^{\prime}$ are isogenous, from which
we deduce that $E$ is geometrically modular.

In any case, the conjecture of Shimura-Tanayama is a theorem [BCDT01]; every
elliptic curve $E/\mathbf{Q}$ is analytically modular, hence geometrically
modular. An important consequence is that the Heegner point on $X_{0}(N)$ for
an imaginary quadratic field $F$ satisfying the Heegner hypothesis relative to
$N$, gives rise to a Heegner point $P_{F}\in E(F)$. The celebrated theorem of
Gross-Zagier [GZ86] relates the height of $P_{F}$ to the derivative
$L^{\prime}(E,F,1)$. The theorem of Kolyvagin [Kol90] shows that if $P_{F}$ is
not torsion, then it generates a finite-index subgroup of $E(F)$, and
furthermore one gets finiteness of the Shafarevich-Tate group.

Let us now return to the setting of an elliptic curve $E$ over a function
field $K$ with field of scalars $\mathbf{F}_{q}$. We assume that $E$ is
nonisotrivial, meaning that $j(E)\not\in\mathbf{F}_{q}$. Then $L(E,\chi,s)$ is
entire for every Hecke character $\chi$. (In fact $L(E,\chi,s)$ is a
polynomial in $q^{-s}$.) The analogue of analytic modularity for $E$ is the
condition that there exists a cuspidal automorphic representation $\pi$ of
$\operatorname{GL}_{2}$ over $K$, such that $L(\pi,\chi,s)=L(E,\chi,s)$ for
all $\chi$. The analytic modularity of $E$ follows from a theorem of Deligne
together with Weil’s converse theorem (see [Ulm03] for a discussion).

The analogue of geometric modularity for $E$ is the subject of this paper. The
interest here hinges on the correct analogue for the modular curve. Let
$X/\mathbf{F}_{q}$ be the nonsingular projective curve with function field
$K$. Let $N$ be the conductor of $E$, with support $\Sigma\subset\left\lvert
X\right\rvert$. As a natural starting point, we may consider the Drinfeld
modular curve $\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)$. Here
$\infty\in\left\lvert X\right\rvert$ is a place of split multiplicative
reduction for $E$; thus $N=N_{f}+(\infty)$ for a divisor $N_{f}$ which is
prime to $\infty$. If $A$ is the ring of functions on $X$ regular away from
$\infty$, then $\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)$ parametrizes
Drinfeld $A$-modules of rank 2 endowed with $\Gamma_{0}(N_{f})$-structure.
There is a morphism $\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)\to
X\smallsetminus\Sigma$ (sending a Drinfeld module to its “characteristic”),
whose fibers are smooth curves. Let
$\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)_{K}$ be the generic fiber of
this morphism, and let
$\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)_{\overline{K}}$ be its base
change to an algebraic closure $\overline{K}/K$.

Drinfeld’s computation [Dri74] of the cohomology of Drinfeld modular curves
shows that $H^{1}(E_{\overline{K}})$ appears as a summand of
$H^{1}(\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)_{\overline{K}})$. (We
mean étale cohomology with $\mathbf{Q}_{\ell}$-coefficients, considered as a
representation of the Galois group of $K$.) We may once again apply Faltings’
theorem to conclude that there is a nonconstant morphism
$\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)_{K}\to E$. If $K^{\prime}/K$
is a separable quadratic extension satisfying the Heegner hypothesis relative
to $E$, there is a Heegner point $P_{K^{\prime}}\in E(K^{\prime})$, whose
height is related to the derivative $L^{\prime}(E/K^{\prime},1)$, see [YZ19,
Remark 1.5]. If $L^{\prime}(E/K^{\prime},1)\neq 0$, then $P_{K^{\prime}}$ is
not a torsion point. But also we have that the rank of $E(K^{\prime})$ is
$\leq 1$, since the “easy inequality” of the Birch and Swinnerton-Dyer
conjecture is a theorem for function fields. We conclude immediately that the
rank is exactly 1, and so the conjecture of Birch and Swinnerton-Dyer holds
for $E^{\prime}/K$.

###  1.2 Stacks of shtukas, and the definition of higher modularity

The notion of geometric modularity in the function field setting may be
generalized far beyond the Drinfeld modular curve. Namely, for $r=1,2,\dots$
we have the moduli stack of $r$-legged shtukas of rank 2, which lies over the
$r$-fold product $X^{r}$ over $\mathbf{F}_{q}$. These were introduced by
Drinfeld [Dri80] in the case $r=2$ to establish the Langlands correspondence
for $\operatorname{GL}_{2}$. For general $r$, these are examples of the spaces
used by V. Lafforgue [Laf18] to prove the automorphic-to-Galois direction of
the Langlands correspondence for general reductive groups.

The spaces of shtukas relevant to us are relative to the group
$G=\operatorname{PGL}_{2}$. To define them, we need an effective divisor $N$
of $X$, with support $\Sigma$. We also need a subset
$\Sigma_{\infty}\subset\Sigma$, such that each $v\in\Sigma_{\infty}$ appears
with multiplicity 1 in $N$. Finally, we need an integer $r\geq 1$ satisfying
the parity condition $\\#\Sigma_{\infty}\equiv r\pmod{2}$. Following [YZ19] we
let

$\lambda_{r}\colon\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\to(X\smallsetminus\Sigma)^{r}$

denote the moduli stack of $G$-shtukas with $\Gamma_{0}(N)$-level structure,
with $r$ “moving legs” in $X\smallsetminus\Sigma$, and “fixed legs” at the
places in $\Sigma_{\infty}$ (these are analogous to the archimedean places in
the number field setting). These legs are “minuscule”; i.e., they are the type
considered by Drinfeld in his original definition. We save the precise
definition of $\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ for
§2. It is a Deligne-Mumford stack, generically smooth of relative dimension
$r$ over $(X\smallsetminus\Sigma)^{r}$.

For example, when $r=0$, the set $\Sigma_{\infty}$ is the ramification locus
of a projective quaternionic group $G^{\prime}$, and then
$\operatorname{Sht}^{0}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ is the discrete
set $G^{\prime}(K)\backslash G^{\prime}(\mathbf{A}_{K})/\Gamma^{\prime}$, for
an appropriate level subgroup $\Gamma^{\prime}\subset
G^{\prime}(\mathbf{A}_{K})$. When $r=1$ and
$\Sigma_{\infty}=\left\\{\infty\right\\}$ is a singleton, one of the connected
components of
$\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\left\\{\infty\right\\})$ is
isomorphic over $X\smallsetminus\Sigma$ to a quotient of
$\operatorname{DrMod}(\Gamma_{0}(N_{f});\infty)$; this is the shtuka
correspondence [Mum78].

We can now start to describe our notion of “higher modularity” for our
elliptic curve $E/K$. Let $U=X\smallsetminus\Sigma$, and let $\mathcal{E}\to
U$ be the family of elliptic curves with generic fiber $E$. The idea goes like
this: Consider the family $h^{1}(\mathcal{E})$ of motives over $U$. Let
$h^{1}(\mathcal{E})^{\boxtimes r}$ be its $r$th external tensor power, meaning
the family of motives over $U^{r}$ whose fiber over $(s_{1},\dots,s_{r})$ is
$h^{1}(\mathcal{E}_{s_{1}})\otimes\cdots\otimes h^{1}(\mathcal{E}_{s_{r}})$.
For $E$ to be $r$-modular should mean that $h^{1}(\mathcal{E})^{\boxtimes r}$
is a quotient of the (compactly supported) cohomology of a stack of $r$-legged
shtukas.

The precise condition involves the existence of a degree 0 algebraic
correspondence between
$\operatorname{Sht}^{r}:=\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$
and $\mathcal{E}^{r}$. By a degree 0 algebraic correspondence, we mean a
$\mathbf{Q}$-linear combination of irreducible closed substacks

$Z\subset\operatorname{Sht}^{r}\times_{U^{r}}\mathcal{E}^{r}$

such that each geometric fiber of $Z\to U^{r}$ has dimension $r$.

Let $s=(s_{1},\dots,s_{r})$ be a geometric point of $U^{r}$, and let $\ell$ be
a prime not dividing $q$. The $\ell$-adic cycle class of $Z_{s}$ gives rise to
a map

$p_{Z}(s)\colon
H^{r}_{c}(\operatorname{Sht}^{r}_{s},\mathbf{Q}_{\ell})\to\bigotimes_{i=1}^{r}H^{1}(\mathcal{E}_{s_{i}},\mathbf{Q}_{\ell})$
(1.2.1)

(Explanation: the $\ell$-adic cycle class of $Z_{s}$ belongs to
$H^{2r}(\operatorname{Sht}^{r}_{s}\times\mathcal{E}^{r}_{s},\mathbf{Q}_{\ell})(r)$,
which maps via the Künneth isomorphism to
$H^{r}(\operatorname{Sht}^{r}_{s},\mathbf{Q}_{\ell})(r)\otimes\bigotimes_{i=1}^{r}H^{1}(\mathcal{E}_{s_{i}},\mathbf{Q}_{\ell})$.
Poincaré duality identifies the latter with the space of maps as in (1.2.1).)
For $Z=\sum_{j}a_{j}Z_{j}$ a formal $\mathbf{Q}$-linear combination of
correspondences $Z_{j}$ as above, we define
$p_{Z}(s)=\sum_{j}a_{j}p_{Z_{j}}(s)$.

######  Definition 1.2.1.

Let $K/\mathbf{F}_{q}$ be a function field, and let $E/K$ be a nonisotrivial
elliptic curve whose conductor $N$ has support $\Sigma$. Let
$U=X\smallsetminus\Sigma$, and let $\mathcal{E}\to U$ be the family of
elliptic curves with generic fiber $E$. Let $\Sigma_{\infty}\subset\Sigma$ be
a subset of places appearing with multiplicity 1 in $N$, satisfying
$\\#\Sigma_{\infty}\equiv r\pmod{2}$. We say that $E/K$ is
$(r,\Sigma_{\infty})$-modular if there exists a degree 0 algebraic
correspondence $Z$ between
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ and
$\mathcal{E}^{r}$ such that for all geometric points $s=(s_{1},\dots,s_{r})$
of $U^{r}$, the map $p_{Z}(s)$ of (1.2.1) is surjective. We say that $E/K$ is
$r$-modular if it is $(r,\Sigma_{\infty})$-modular for all $\Sigma_{\infty}$
satisfying the hypotheses above.

######  Remark 1.2.2.

When $r$ is odd, we must assume that $E/K$ has at least one place of
multiplicative reduction in order to have an interesting notion of
$r$-modularity.

######  Conjecture 1.2.3.

A nonisotrivial elliptic curve $E/K$ is $r$-modular for all $r\geq 1$.

The main goal of this article is to give the first examples of 2-modular and
3-modular elliptic curves. Our results apply to what is arguably the simplest
class of elliptic fibrations, namely the extremal rational elliptic fibrations
$\mathcal{E}\to X$. Here “rational” means that $\mathcal{E}$ is birational to
$\mathbf{P}^{2}$ (and consequently $X\cong\mathbf{P}^{1}$), and “extremal”
means that $\mathcal{E}\to X$ has Mordell-Weil rank 0. (The generic rational
elliptic fibration has Mordell-Weil rank 8.) For such a fibration we
necessarily have $\deg N=4$ and $L(E,s)=1$ identically. Conversely, if an
elliptic fibration $\mathcal{E}\to\mathbf{P}^{1}$ has degree 4 conductor, then
it is extremal rational.

The extremal rational elliptic fibrations $\mathcal{E}\to X$ are classified in
[MP86]. The number of singular geometric fibers of a extremal rational
elliptic fibration is either two, three, or four, with two occurring only if
$\mathcal{E}\to X$ is isotrivial or if it has wild fibers (which happens only
in characteristics 2 and 3). A nonisotrivial extremal rational elliptic
fibration always has at least one multiplicative fiber, and so there is always
a nontrivial notion of higher modularity for $\mathcal{E}$.

######  Theorem 1.2.4.

Let $\mathcal{E}\to X$ be a nonisotrivial extremal rational elliptic fibration
with generic fiber $E$. Then $E$ is 2-modular.

In the next two subsections we will (a) explain why one should believe
Conjecture 1.2.3, and (b) sketch the proof of Theorem 1.2.4.

###  1.3 Relation to the Tate conjecture

We attempt to motivate the definition of higher modularity. Using analytic
modularity (which is a theorem) combined with prior results on the cohomology
of stacks of shtukas, we will reduce Conjecture 1.2.3 to a sufficiently strong
version of the Tate conjecture. Let $\eta=\operatorname{Spec}K$ be the generic
point of $X$, and let $\overline{\eta}=\operatorname{Spec}\overline{K}$ be a
geometric generic point.

We use the normalization of the global Langlands correspondence found in
[Dri80], adapted to the group $G=\operatorname{PGL}_{2}$. This is a bijection
between isomorphism classes:

  * •

Cuspidal automorphic representations $\pi$ of $G$ with coefficients in
$\overline{\mathbf{Q}}_{\ell}$, and

  * •

Irreducible representations
$\sigma\colon\operatorname{Gal}(\overline{K}/K)\to\operatorname{GL}_{2}\overline{\mathbf{Q}}_{\ell}$,
such that $\det\sigma=\overline{\mathbf{Q}}_{\ell}(-1)$.

Suppose that $\pi$ and $\sigma$ correspond. Let $N$ be the conductor of $\pi$,
in the sense that $\dim\pi^{\Gamma_{0}(N)}=1$, and let $\Sigma$ be the support
of $N$. Let $U=X\smallsetminus\Sigma$. Then $\sigma$ factors through a
representation of $\pi_{1}(U,\overline{\eta})$ of conductor $N$. The
correspondence is characterized by the property that for all $v\in U$ with
residue field $\mathbf{F}_{q_{v}}$ and Frobenius $\operatorname{Frob}_{v}$, we
have
$L(s-1/2,\pi_{v})=\det(1-\sigma(\operatorname{Frob}_{v})q_{v}^{-s})^{-1}$.

######  Remark 1.3.1.

We have chosen this normalization of the global Langlands correspondence so
that the $H^{1}$ of a nonisotrivial elliptic curve over $K$ corresponds to a
cuspidal automorphic representation of $G$. Under the “usual” normalization of
the correspondence, the Langlands parameter of an automorphic representation
of $G$ is a homomorphism
$\operatorname{Gal}(\overline{K}/K)\to\operatorname{SL}_{2}\overline{\mathbf{Q}}_{\ell}$.
This is largely an aesthetic choice; working with $G$ rather than
$\operatorname{GL}_{2}$ saves us certain notational headaches having to do
with the center of $\operatorname{GL}_{2}$.

We now reference some facts about the relation between the cohomology of
stacks of shtukas and the global Langlands correspondence. Let $r\geq 1$,
write $\eta^{r}$ for the $r$-fold product of $\eta$ over $\mathbf{F}_{q}$, and
write $\eta_{r}$ for the generic point of $X^{r}$. Let
$\overline{\eta}_{r}\to\eta_{r}$ be an algebraic closure lying over
$\overline{\eta}^{r}\to\eta^{r}$. Then there is a homomorphism

$\pi_{1}(U^{r},\overline{\eta}_{r})\to\pi_{1}(U,\overline{\eta})^{r}$ (1.3.1)

The cohomology
$H^{r}_{c}(\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\overline{\eta}_{r}},\mathbf{Q}_{\ell})$
admits commuting actions of $\pi_{1}(U^{r},\overline{\eta}_{r})$ and the Hecke
algebra $T_{0}(N)$ for $G$ with level $\Gamma_{0}(N)$. The following
proposition is along the lines of the Kottwitz conjecture for Shimura
varieties. It appears later as Proposition 2.3.1. We warn that it is
conditional on the extension of the main results of Xue in [Xue20a] and
[Xue20b] to the situation $\Sigma_{\infty}\neq\emptyset$.

######  Proposition 1.3.2.

Let
$\sigma\colon\pi_{1}(U,\overline{\eta})\to\operatorname{GL}_{2}\overline{\mathbf{Q}}_{\ell}$
be an irreducible representation of conductor $N$ with
$\det\sigma=\overline{\mathbf{Q}}_{\ell}(-1)$. Then as representations of
$\pi_{1}(U^{r},\overline{\eta}_{r})$, the external tensor power
$\sigma^{\boxtimes r}$ appears as a direct summand of
$H^{r}_{c}(\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\overline{\eta}_{r}},\overline{\mathbf{Q}}_{\ell})$.
(Here $\pi_{1}(U^{r},\overline{\eta}_{r})$ acts on $\sigma^{\boxtimes r}$ by
means of the homomorphism (1.3.1).)

Now suppose $E/K$ is a nonisotrivial elliptic curve of conductor $N$. Let
$\sigma$ be the the representation of $\pi_{1}(U,\overline{\eta})$ on
$H^{1}(E_{\overline{K}},\mathbf{Q}_{\ell})$. Then $\sigma$ is irreducible, and
$\det\sigma=\mathbf{Q}_{\ell}(-1)$. By Proposition 1.3.2, $\sigma^{\boxtimes
r}$ appears as a quotient of
$H^{r}_{c}(\operatorname{Sht}^{r}_{\overline{\eta}_{r}},\mathbf{Q}_{\ell})$,
where
$\operatorname{Sht}^{r}=\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$.
By Poincaré duality, this means there exists a nonzero
$\pi_{1}(U^{r},\overline{\eta}_{r})$-invariant vector $w$ lying in

$H^{r}(\operatorname{Sht}^{r}_{\overline{\eta}_{r}},\mathbf{Q}_{\ell})\otimes
H^{1}(\mathcal{E}_{\overline{\eta}},\mathbf{Q}_{\ell})^{\boxtimes r}(r)\subset
H^{2r}(\operatorname{Sht}^{r}_{\overline{\eta}_{r}}\times\mathcal{E}^{r}_{\overline{\eta}_{r}},\mathbf{Q}_{\ell}(r))$

If the stack of shtukas were a projective variety, the Tate conjecture would
predict that $w$ is the class of an algebraic correspondence $Z$. This is why
we have claimed that a sufficiently strong version of the Tate conjecture
predicts the algebraic correspondence $Z$ appearing in Definition 1.2.1.

###  1.4 Strategy of proof of 2-modularity

Let $N$ be an effective divisor of $X=\mathbf{P}^{1}_{\mathbf{F}_{q}}$ of
degree 4 with support $\Sigma$. As before, let $U=X\smallsetminus\Sigma$. Let
$\Sigma_{\infty}\subset\Sigma$ be a set of places appearing with multiplicity
1 in $N$, such that $\\#\Sigma_{\infty}$ is even. The stack of shtukas in this
situation can be described with simple equations, at least up to birational
equivalence. More precisely, each fiber of

$\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\to U^{2}$

is birational to an elliptic surface (Theorem 2.5.1). For $q$ large these
elliptic surfaces are of general type, which makes it hard to imagine finding
the desired algebraic correspondence directly. Instead, we found an unexpected
relation between shtuka spaces with different numbers of legs, which we have
called the “coincidence map”. The coincidence map is a rational map

$c\colon\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\dashrightarrow\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime}).$

between stacks of shtukas with 2 and 1 legs, respectively; its domain of
definition meets every fiber over $U^{2}$. Here
$\Sigma^{\prime}_{\infty}\subset\Sigma_{\infty}$ is another set of places
appearing with multiplicity 1 in $N$, such that $\\#\Sigma_{\infty}^{\prime}$
is odd. The coincidence map fits into a cartesian diagram of varieties with
rational maps:

$\textstyle{\operatorname{Sht}_{G}^{2}(\Gamma_{0}(N);\Sigma_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c\times\lambda_{2}}$$\textstyle{\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})\times
U^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{1}\times\operatorname{id}}$$\textstyle{\operatorname{Coinc}^{3}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{3}}$$\textstyle{U^{3}}$
(1.4.1)

Here,
$\operatorname{Coinc}^{3}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime})$
is a moduli space of what we have called “3-legged coincidences”: these are
certain modifications of vector bundles on the projective line, but Frobenius
is not involved (and indeed the space of coincidences can be defined in any
characteristic). The leg map $\lambda_{3}$ in (1.4.1) is a double cover of
$U^{3}$, branched over a divisor of degree $(2,2,2)$.

Now suppose $E/K$ is a nonisotrivial elliptic curve of conductor $N$,
corresponding to a family of elliptic curves $\mathcal{E}\to U$. We apply the
fact that $E$ is 1-modular, which is to say there is a finite morphism

$\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})\to\mathcal{E}\\\
$ (1.4.2)

commuting with the maps to $U$. Combining (1.4.1) with (1.4.2), we obtain a
dominant rational map:

$\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})\dashrightarrow{\mathcal{Z}}^{2}(\mathcal{E})$
(1.4.3)

where ${\mathcal{Z}}^{2}(\mathcal{E})$ is defined as the cartesian product:

$\textstyle{{\mathcal{Z}}^{2}(\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\times
U^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Coinc}^{3}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U^{3}}$
(1.4.4)

We have a map ${\mathcal{Z}}^{2}(\mathcal{E})\to U^{2}$ obtained by composing
the upper row of (1.4.4) with the projection; with respect to this, the
morphism (1.4.3) commutes with the maps to $U^{2}$. By replacing
$\mathcal{E}\to U$ with the complete elliptic fibration over $\mathbf{P}^{1}$
in (1.4.4), it is possible to redefine ${\mathcal{Z}}^{2}(\mathcal{E})$ so
that the morphism ${\mathcal{Z}}^{2}(\mathcal{E})\to U^{2}$ is proper and
smooth; we do this. Then each fiber of ${\mathcal{Z}}^{2}(\mathcal{E})\to
U^{2}$ is the base change of a rational elliptic fibration by a double cover
of $\mathbf{P}^{1}$ ramified at 2 points. Generically, such a base change is a
K3 surface of Picard rank 18. Our plan of attack now shifts to the study of K3
surfaces.

Recall that the Kummer surface $\operatorname{Km}(A)$ of an abelian surface
$A$ is the K3 surface obtained by resolving singularities on the quotient
$A/[-1]$. If $A$ is the product of generic nonisogenous elliptic curves, then
$\operatorname{Km}(A)$ has Picard rank 18.

We prove the following theorem:

######  Theorem 1.4.1.

Let $S$ be a K3 surface over an algebraically closed field. Assume there is an
isometry
$\operatorname{Pic}S\otimes\mathbf{Q}\cong\operatorname{Pic}K\otimes\mathbf{Q}$,
where $K$ is a Kummer surface of the form $\operatorname{Km}(E_{1}\times
E_{2})$, where $E_{1},E_{2}$ are nonisogenous elliptic curves. Then there
exists a finite morphism from $S$ to a Kummer surface of this form.

######  Remark 1.4.2.

Theorem 1.4.1 is in the spirit of a result of Buskin [Bus19, Theorem 1.1]
which states that if $S$ and $S^{\prime}$ are two complex K3 surfaces, then a
Hodge isometry $H^{2}(S,\mathbf{Q})\cong H^{2}(S^{\prime},\mathbf{Q})$ is
necessarily induced from an algebraic correspondence between $S$ and
$S^{\prime}$.

Recall our family of K3 surfaces ${\mathcal{Z}}^{2}(\mathcal{E})\to U^{2}$.
The idea now is to apply Theorem 1.4.1 to every fiber. We verify the existence
of two families $\mathcal{E}_{1},\mathcal{E}_{2}\to U$ of elliptic curves and
a finite morphism
${\mathcal{Z}}^{2}(\mathcal{E})\to\operatorname{Km}(\mathcal{E}_{1}\times\mathcal{E}_{2})$.
Theorem 1.4.1 can be made effective, in the sense that the product of elliptic
curves $\mathcal{E}_{1}\times\mathcal{E}_{2}$ can be computed. In fact our
computations show that $\mathcal{E}_{1}\times\mathcal{E}_{2}$ is isogenous
over $U^{2}$ to $\mathcal{E}\times\mathcal{E}$, at least after extending the
scalar field $\mathbf{F}_{q}$. These computations appear in §3.4. Combining
(1.4.3) with the morphism
${\mathcal{Z}}^{2}(\mathcal{E})\to\operatorname{Km}(\mathcal{E}^{2})$ gives a
dominant rational map
$\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\dashrightarrow\operatorname{Km}(\mathcal{E}^{2})$
whose domain of definition meets every fiber over $U^{2}$. Pulling back along
the rational map
$\mathcal{E}^{2}\dashrightarrow\operatorname{Km}(\mathcal{E}^{2})$, we obtain
a correspondence $Z$ between
$\operatorname{Sht}^{2}(\Gamma_{0}(N);\Sigma_{\infty})$ and $\mathcal{E}^{2}$.
(A priori the correspondence is defined only over dense open subsets; to
define $Z$, we pass to the Zariski closure.)

It remains to check that for each geometric point $s$ of $U^{r}$, the induced
map $p_{Z}(s)$ on cohomology is surjective. This surjectivity reduces to the
following fact: if $S$ is a nonsingular quasi-projective surface over an
algebraically closed field, and $S\to S^{\prime}$ is a dominant morphism to a
projective surface $S^{\prime}$, then the image of the induced map
$H^{2}_{c}(S,\mathbf{Q}_{\ell})\to H^{2}(S^{\prime},\mathbf{Q}_{\ell})$
contains the transcendental part of $H^{2}(S^{\prime},\mathbf{Q}_{\ell})$.

The phenomenon of the coincidence map applies to spaces of shtukas with
arbitrarily many legs, and opens up an avenue of attack to prove
$r$-modularity for any $r$.

######  Theorem 1.4.3.

Assume $q$ is odd. Let $E/\mathbf{F}_{q}(t)$ be the Legendre elliptic curve,
with Weierstrass equation $y^{2}=x(x-1)(x-t)$. Then $E$ is 3-modular.

This $E$ has conductor $N=(0)+(1)+2(\infty)$. Theorem 1.4.3 was proved by
finding a birational map between two families of Calabi-Yau threefolds
${\mathcal{Z}}^{3}(\mathcal{E})$ and $\operatorname{Km}(\mathcal{E}^{3})$ over
$U^{3}$, where $U=\mathbf{P}^{1}\smallsetminus\left\\{0,1,\infty\right\\}$.
Here ${\mathcal{Z}}^{3}(\mathcal{E})$ is defined as a fiber product analogous
to (1.4.4), and $\operatorname{Km}(\mathcal{E}^{3})$ is a generalized Kummer
variety.

###  1.5 Application: Heegner-Drinfeld cycles on $\mathcal{E}^{r}$

This article was inspired by the theorem of Yun-Zhang [YZ19] relating Heegner-
Drinfeld cycles on stacks of shtukas, to the Taylor expansion of $L$-functions
of automorphic forms. We explain here the contact between higher modularity
and the conjecture of Birch and Swinnerton-Dyer for elliptic curves over
function fields.

As before, we let $X$ be a curve over $\mathbf{F}_{q}$ with function field
$K$. Let $f\colon\mathcal{E}\to X$ be an elliptic fibration with generic fiber
$E/K$. Let $\ell$ be a prime not dividing $q$. Recall [Tat95] that the
inequality $\operatorname{ord}_{s=1}L(E/K,s)\geq\operatorname{rank}E(K)$ is
known unconditionally, and that the following are equivalent:

  1. 1.

The conjecture of Birch and Swinnerton-Dyer (BSD) holds for $E/K$; i.e.,
$\operatorname{ord}_{s=1}L(E/K,s)=\operatorname{rank}E(K)$.

  2. 2.

The Tate conjecture holds for the surface $\mathcal{E}/\mathbf{F}_{q}$. By
this, we mean that
$H^{2}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))^{\operatorname{Fr}_{q}=1}$
is spanned by classes of divisors.

  3. 3.

The Tate-Shafarevich group $\Sha(E/K)$ is finite.

We briefly review the connection between these statements. Let
$\operatorname{NS}(\mathcal{E})$ be the Néron-Severi group of $\mathcal{E}$
(divisors modulo algebraic equivalence). Define a decreasing two-step
filtration $\operatorname{Fil}^{i}\operatorname{NS}(\mathcal{E})$, with
$\operatorname{Fil}^{1}\operatorname{NS}(\mathcal{E})$ the subgroup of
divisors whose intersection pairing with a fiber of $f$ is zero, and with
$\operatorname{Fil}^{2}\operatorname{NS}(\mathcal{E})$ the subgroup generated
by irreducible components of fibers. Then
$\operatorname{NS}(\mathcal{E})/\operatorname{Fil}^{1}\operatorname{NS}(\mathcal{E})\cong\mathbf{Z}$
and
$\operatorname{Fil}^{1}\operatorname{NS}(\mathcal{E})/\operatorname{Fil}^{2}\operatorname{NS}(\mathcal{E})\cong
E(K)$, the Mordell-Weil group. The identity section splits off the first
filtrant:
$\operatorname{NS}(\mathcal{E})\cong\operatorname{Fil}^{1}\operatorname{NS}(\mathcal{E})\oplus\mathbf{Z}$.

Meanwhile, the Leray spectral sequence for the composition $\mathcal{E}\to
X\to\operatorname{Spec}\mathbf{F}_{q}$ degenerates on the second page. As a
result there is a decreasing two-step filtration
$\operatorname{Fil}^{i}H^{2}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))$
with graded pieces
$H^{2}(X_{{\overline{\mathbf{F}}_{q}}},f_{*}\mathbf{Q}_{\ell}(1))\cong\mathbf{Q}_{\ell}$,
$H^{1}(X_{\overline{\mathbf{F}}_{q}},R^{1}f_{*}\mathbf{Q}_{\ell}(1))$,
$H^{0}(X_{{\overline{\mathbf{F}}_{q}}},R^{2}f_{*}\mathbf{Q}_{\ell}(1))$. This
filtration is preserved by the action of Frobenius $\operatorname{Fr}_{q}$.
Once again, the identity section splits off the first filtrant:
$H^{2}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))\cong\operatorname{Fil}^{1}H^{2}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))\oplus\mathbf{Q}_{\ell}$.

The cycle class map $\operatorname{NS}(\mathcal{E})\otimes\mathbf{Q}_{\ell}\to
H^{2}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))$ respects
the filtrations on either side, and induces an isomorphism on the first and
third graded pieces. On the second graded pieces, we get an injective map

$E(K)\otimes\mathbf{Q}_{\ell}\to
V:=H^{1}(X_{\overline{\mathbf{F}}_{q}},R^{1}f_{*}\mathbf{Q}_{\ell}(1))$

which lands in the Frobenius-fixed part $V^{\operatorname{Fr}_{q}=1}$. The
Tate conjecture for $\mathcal{E}/\mathbf{F}_{q}$ reduces to the statement that
$E(K)\otimes\mathbf{Q}_{\ell}\to V^{\operatorname{Fr}_{q}=1}$ is an
isomorphism. Note that $V^{\operatorname{Fr}_{q}=1}$ is the Selmer group for
$E/K$ (with $\mathbf{Q}_{\ell}$-coefficients), and the statement that
$E(K)\otimes\mathbf{Q}_{\ell}\to V^{\operatorname{Fr}_{q}=1}$ is an
isomorphism is equivalent to the statement that the $\ell$-primary part of
$\Sha(E/K)$ is finite.

The vector space $V$ is related to the $L$-function via:

$L(E/K,s)=\det(1-q^{1-s}\operatorname{Fr}_{q}|V).$

Thus $\operatorname{ord}_{s=1}L(E/K,s)$ is the dimension of the generalized
$1$-eigenspace for $\operatorname{Fr}_{q}$ acting on $V$, which a priori may
be larger than $V^{\operatorname{Fr}_{q}=1}$. Thus we have inequalities:

$\operatorname{rank}E(K)\leq\dim
V^{\operatorname{Fr}_{q}=1}\leq\operatorname{ord}_{s=1}L(E/K,s)$

If BSD holds for $E/K$, then the leftmost and rightmost quantities are equal,
and consequently the Tate conjecture holds for $\mathcal{E}$, and as a bonus
we also find that the generalized $1$-eigenspace for $\operatorname{Fr}_{q}$
on $V$ coincides with the eigenspace. Conversely, if the Tate conjecture holds
for $\mathcal{E}$, Tate shows that the generalized $1$-eigenspace for
$\operatorname{Fr}_{q}$ on $V$ coincides with the eigenspace, and therefore
BSD holds for $E/K$. Note that the statement that the generalized
$1$-eigenspace for $\operatorname{Fr}_{q}$ on $V$ coincides with the
eigenspace is a consequence of the general conjecture that
$\operatorname{Fr}_{q}$ acts semisimply on all
$H^{i}(\mathcal{E}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell})$; this
statement is sometimes packaged along with the Tate conjecture.

Let $N$ be the conductor of $E$, and let $\Sigma$ be the support of $N$.
Assume that $N$ is multiplicity-free. Let $\Sigma_{\infty}\subset\Sigma$ be a
subset. For each $r\geq 0$ with the same parity as $\\#\Sigma_{\infty}$, we
have a space of $r$-legged shtukas
$\lambda_{r}\colon\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\to
X^{r}$. (The map is really defined over all of $X^{r}$, without removing
$\Sigma$. In the following discussion we are working with the iterated shtukas
as defined in [YZ19]. In particular
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ is a smooth
Deligne-Mumford stack.)

Let $K^{\prime}/K$ be a quadratic extension corresponding to a (possibly
branched) double cover $X^{\prime}\to X$. We assume that $X^{\prime}$ is
geometrically connected over $\mathbf{F}_{q}$, and that the branch points of
$X^{\prime}\to X$ are disjoint from $\Sigma$. Let us write
$\Sigma=\Sigma_{f}\cup\Sigma_{\infty}$. Assume the Heegner hypothesis: all
places in $\Sigma_{f}$ are split in $X^{\prime}$, and all places in
$\Sigma_{\infty}$ are inert in $X^{\prime}$. Let
$\mathcal{E}^{\prime}=\mathcal{E}\times_{X}X^{\prime}$. Then (under an
assumption on the expected behavior of the cohomology of the stacks of
shtukas, see [YZ19, Equation 1.9]), Yun-Zhang construct a Heegner-Drinfeld
class

$HD_{E,r}\in((V^{\prime})^{\otimes r})^{\operatorname{Fr}_{q}=1}$

where $V^{\prime}$ is the analogue of $V$ for $E^{\prime}\to X^{\prime}$.

Forthcoming work of Yun-Zhang (see [YZ19, Section 1.3.1]) will show:

######  Theorem 1.5.1.

Let $r_{0}\geq 0$ be the smallest integer $r$ such that $HD_{E,r}\neq 0$. Then
the Selmer rank $\dim(V^{\prime})^{\operatorname{Fr}_{q}=1}$ is $r_{0}$, and
$HD_{E,r_{0}}$ spans the line
$\bigwedge^{r_{0}}(V^{\prime})^{\operatorname{Fr}_{q}=1}$.

On the other hand, the main result [YZ19, Corollary 1.4] applied to this
situation reads:

$(HD_{E,r},HD_{E,r})\dot{=}L^{(r)}(E/K^{\prime},1),$ (1.5.1)

where $(\;,\;)\colon V\times V\to\mathbf{Q}_{\ell}$ is the natural symmetric
pairing on $V$, and $\dot{=}$ means equality up to an explicit nonzero
constant. If one knew definiteness of the restriction of the pairing to the
Heegner-Drinfeld classes, one could conclude that the $r_{0}$ of Theorem 1.5.1
is the analytic rank of $E/K^{\prime}$, and therefore that Selmer rank =
analytic rank. (This is true unconditionally in the case of analytic rank 3,
using the Hodge index theorem for surfaces.) The upshot is that Heegner-
Drinfeld classes can be used to construct a basis for the determinant of the
Selmer group of $E/K^{\prime}$, but (unless the analytic rank is 1) one cannot
use them to construct elements of the Mordell-Weil group.

Under the assumption that $E$ is $(r,\Sigma_{\infty})$-modular, the Heegner-
Drinfeld classes actually arise from algebraic cycles on
$(\mathcal{E}^{\prime})^{r}$, in the following sense. Define the base change

$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\prime}=\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\times_{X^{r}}X^{\prime}.$

The Heegner-Drinfeld cycle is a codimension $r$ cycle with proper support:

$\mathcal{HD}_{r}\in
Z^{r}_{c}(\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\prime},\mathbf{Q}).$

It is obtained as the locus of rank 2 $X$-shtukas which arise from rank 1
$X^{\prime}$-shtukas. (To define $\mathcal{HD}_{r}$ one needs to make some
auxiliary choices, see [YZ19, 1.1.3]. In particular one has to choose a place
of $X^{\prime}$ above each $v\in\Sigma_{f}$.)

Let $Z$ be an algebraic correspondence between
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ and
$\mathcal{E}^{r}$ as in Definition 1.2.1 (extended over all of $X^{r}$). Such
a correspondence induces a map on algebraic cycles:

$p_{Z}\colon
Z^{r}_{c}(\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\prime},\mathbf{Q})\to
Z^{r}((\mathcal{E}^{\prime})^{r},\mathbf{Q}).$

Define

$\mathcal{HD}_{E,r}:=p_{r}(\mathcal{HD}_{r})\in
Z^{r}((\mathcal{E}^{\prime})^{r},\mathbf{Q}).$

Its cohomology class is an element

$\operatorname{cl}(\mathcal{HD}_{E,r})\in
H^{2r}((\mathcal{E}^{\prime})^{r}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(r))$

which is $\operatorname{Fr}_{q}$-invariant.

Owing to the Künneth formula and the splitting of
$H^{2}(\mathcal{E}^{\prime}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(1))$
noted above, we have a surjective map

$H^{2r}((\mathcal{E}^{\prime})^{r}_{\overline{\mathbf{F}}_{q}},\mathbf{Q}_{\ell}(r))\to(V^{\prime})^{\otimes
r}$ (1.5.2)

We expect that the image of $\operatorname{cl}(\mathcal{HD}_{E,r})$ in
$(V^{\prime})^{\otimes r}$ is $HD_{E,r}$. The moral of our story is this:
suppose $r$ is the Selmer rank of $E/K^{\prime}$. Under the assumption of
$r$-modularity, one still doesn’t know that the Selmer group
$(V^{\prime})^{\operatorname{Fr}_{q}=1}$ is spanned by classes of algebraic
cycles in $\mathcal{E}^{\prime}$, but one does know that the determinant of
the Selmer group is spanned by the class of an algebraic cycle in
$(\mathcal{E}^{\prime})^{r}$.

###  1.6 Acknowledgments

The authors would like to thank Noam Elkies, María Inés de Frutos Fernández,
Colin Ingalls, Masato Kuwata, Will Sawin, Ari Schnidman, John Voight, Ziquan
Yang, Wei Zhang, and Xinwen Zhu for many helpful discussions. A.L. would like
to thank the Tutte Institute for its support of his external research. J.W.
was supported by NSF grant DMS-1902148 and an award from the Simons
Foundation.

##  2 The stacks of shtukas for $\operatorname{PGL}_{2}$

In this section we review the construction of the stack of $r$-legged shtukas
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N),\Sigma_{\infty})$ for the group
$G=\operatorname{PGL}_{2}$. We have tried to keep our notation consistent with
[YZ19].

From §2.4 on we specialize to the case that the base curve $X$ is
$\mathbf{P}^{1}_{\mathbf{F}_{q}}$. In that case the shtuka space
$\operatorname{Sht}^{r}(\Gamma;\Sigma_{\infty})$ is amenable to concrete
calculations.

###  2.1 Vector bundles of rank 2, fractional twists, Atkin-Lehner
automorphisms, and passage to $G=\operatorname{PGL}_{2}$

Let $F$ be a field, and let $X/F$ be a smooth projective curve with fraction
field $K$. For each closed point $x\in\left\lvert X\right\rvert$, let $K_{x}$
be the completion of $K$ at $x$. We have the stack $\operatorname{Bun}_{2}$,
which assigns to an $F$-scheme $S$ the groupoid of rank 2 vector bundles
${\mathcal{F}}$ on $X\times S$. If $D\in\operatorname{Div}X$, we have the
twist ${\mathcal{F}}(D):={\mathcal{F}}\otimes\mathcal{O}_{X}(D)$. If $N\subset
X$ is an effective divisor and ${\mathcal{F}}$ is a rank 2 vector bundle on
$X\times S$, then a $\Gamma_{0}(N)$-structure on ${\mathcal{F}}$ is a rank 1
subbundle $\mathcal{L}_{N}\subset{\mathcal{F}}|_{N\times S}$.

In the special case that $N$ is multiplicity-free with support $\Sigma$, a
$\Gamma_{0}(N)$-structure may alternatively be described as a system of
fractional twists ${\mathcal{F}}(\textstyle\frac{1}{2}P)$ for each
$P\in\Sigma$, which lies in between ${\mathcal{F}}$ and ${\mathcal{F}}(P)$:

${\mathcal{F}}\subset{\mathcal{F}}(\textstyle\frac{1}{2}P)\subset{\mathcal{F}}(P)$

The quotients ${\mathcal{F}}(\textstyle\frac{1}{2}P)/{\mathcal{F}}$ and
${\mathcal{F}}(P)/{\mathcal{F}}(\textstyle\frac{1}{2}P)$ are required to be
rank 1 vector bundles on $P\times S$. Given the data of the
${\mathcal{F}}(\textstyle\frac{1}{2}P)$, one can define for each $D\leq N$ the
twist ${\mathcal{F}}(\textstyle\frac{1}{2}D)$, defined as the subbundle of
${\mathcal{F}}(D)$ generated by ${\mathcal{F}}$ and the
${\mathcal{F}}(\textstyle\frac{1}{2}P)$ for each $P\in\operatorname{supp}D$.
The line bundle $\mathcal{L}_{N}$ can be recovered as the kernel of
${\mathcal{F}}|_{N\times S}\to{\mathcal{F}}(\textstyle\frac{1}{2}N)|_{N\times
S}$.

We can go further than this and define ${\mathcal{F}}(D)$ for any $D$
belonging to $\textstyle\frac{1}{2}\mathbf{Z}\Sigma$, in such a way that
${\mathcal{F}}(D+D^{\prime})={\mathcal{F}}(D)(D^{\prime})$ for all
$D,D^{\prime}\in\textstyle\frac{1}{2}\mathbf{Z}N$. Then whenever $D\leq
D^{\prime}$ for $D,D^{\prime}\in\textstyle\frac{1}{2}\mathbf{Z}N$, we have an
inclusion ${\mathcal{F}}(D)\subset{\mathcal{F}}(D^{\prime})$ with cokernel
supported on $D^{\prime}-D$.

Keeping the hypothesis that $N$ is multiplicity-free, we may write objects of
$\operatorname{Bun}_{2}(\Gamma_{0}(N))$ as pairs
${\mathcal{F}}^{\dagger}=({\mathcal{F}},\left\\{{\mathcal{F}}(D)\right\\})$,
where $\left\\{{\mathcal{F}}(D)\right\\}$ is a system of fractional twists for
$D\in\textstyle\frac{1}{2}\mathbf{Z}\Sigma$. For an object
${\mathcal{F}}^{\dagger}$ of $\operatorname{Bun}_{2}(\Gamma_{0}(N))$ and
$D^{\prime}\in\textstyle\frac{1}{2}\mathbf{Z}\Sigma$ we can form the
fractional twist ${\mathcal{F}}^{\dagger}(D^{\prime})$, simply by replacing
each ${\mathcal{F}}(D)$ with ${\mathcal{F}}(D+D^{\prime})$. The resulting
automorphism of $\operatorname{Bun}_{2}(\Gamma_{0}(N))$ is called the Atkin-
Lehner automorphism $\operatorname{AL}(D^{\prime})$. These satisfy
$\operatorname{AL}(D)\circ\operatorname{AL}(D^{\prime})=\operatorname{AL}(D+D^{\prime})$.111To
be pedantic: a certain diagram of stacks 2-commutes.

A rank 2 vector bundle ${\mathcal{F}}$ on a curve is semistable if
$2\deg\mathcal{L}\leq\deg{\mathcal{F}}$ for all rank 1 subbundles
$\mathcal{L}\subset{\mathcal{F}}$. The index of instability of a vector bundle
is

$\operatorname{inst}({\mathcal{F}})=\max_{{\mathcal{L}}}(2\deg\mathcal{L}-\deg{\mathcal{F}})$

where ${\mathcal{L}}$ runs through rank 1 subbundles of ${\mathcal{F}}$. Note
that $\operatorname{inst}({\mathcal{F}})$ is invariant under (integral) twists
by line bundles. This index allows us to define a stratification on
$\operatorname{Bun}_{2}(\Gamma_{0}(N))$ For $i\geq 0$, let

$\operatorname{Bun}_{2}(\Gamma_{0}(N))^{\leq
i}\subset\operatorname{Bun}_{2}(\Gamma_{0}(N))$

denote the open substack determined by the condition that
$\operatorname{inst}{\mathcal{F}}\leq i$ on all geometric points. Similarly
define $\operatorname{Bun}_{2}(\Gamma_{0}(N))^{d,\leq i}$ as the degree $d$
component. Then each $\operatorname{Bun}_{2}(\Gamma_{0}(N))^{d,\leq i}$ is an
Artin stack of finite type.

As in [YZ17], our main focus is on shtukas for the group

$G=\operatorname{PGL}_{2}.$

Let $\operatorname{Bun}_{G}$ be the stack which assigns to an $F$-scheme $S$
the groupoid of $G$-torsors ${\mathcal{F}}$ on $X\times S$. Then

$\operatorname{Bun}_{G}\cong\operatorname{Bun}_{2}/\operatorname{Pic}_{X},$

where the Picard stack $\operatorname{Pic}_{X}$ acts on
$\operatorname{Bun}_{2}$ by twisting. We may also define the stack
$\operatorname{Bun}_{G}(\Gamma_{0}(N))$, classifing $G$-torsors
${\mathcal{F}}$ over $X\times S$ together with a reduction of
${\mathcal{F}}|_{N\times S}$ to the Borel subgroup of $G$.

The degree of an object of $\operatorname{Bun}_{G}$ is valued in
$\mathbf{Z}/2\mathbf{Z}$, and we write
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{d}$ ($d=0,1$) for the appropriate
component. Similarly as above we have
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{d,\leq i}$, an Artin stack of finite
type. The Atkin-Lehner automorphisms described above descend to
$\operatorname{Bun}_{G}(\Gamma_{0}(N))$: for any subset
$\Sigma_{0}\subset\Sigma$ supported on places appearing with multiplicity 1 in
$N$, we have the Atkin-Lehner automorphism
$\operatorname{AL}(\textstyle\frac{1}{2}\Sigma_{0})$, which is in fact an
involution. For two such subsets
$\Sigma_{0},\Sigma_{0}^{\prime}\subset\Sigma$, we have the relation
$\operatorname{AL}(\textstyle\frac{1}{2}\Sigma_{0})\circ\operatorname{AL}(\textstyle\frac{1}{2}\Sigma_{0}^{\prime})=\operatorname{AL}(\textstyle\frac{1}{2}\Sigma_{0}^{\prime\prime})$,
where $\Sigma_{0}^{\prime\prime}\subset\Sigma$ is the symmetric difference of
$\Sigma_{0}$ and $\Sigma_{0}^{\prime}$.

###  2.2 Stacks of $G$-shtukas

Now let $\mathbf{F}_{q}$ be a finite field, and let $X/\mathbf{F}_{q}$ be a
curve. The definition of shtukas we use coincides with that appearing in
Varshavsky [Var04] for general reductive groups $G$, except that our shtukas
feature a set of “archimedean places” $\Sigma_{\infty}$, as in [YZ19].

######  Definition 2.2.1.

Let $S$ be an $\mathbf{F}_{q}$-scheme. Let $G$ be $\operatorname{GL}_{2}$ or
$\operatorname{PGL}_{2}$. Suppose we are given the following data:

  * •

An effective divisor $N\subset X$, with support $\Sigma$.

  * •

A subset $\Sigma_{\infty}\subset\Sigma$ of degree 1 places222The assumption
that all places of $\Sigma_{\infty}$ are degree 1 is just for notational
convenience. appearing in $N$ with multiplicity 1.

  * •

An integer $r\geq 0$.

  * •

Morphisms $x_{i}\colon S\to X\smallsetminus\Sigma$ for $i=1,\dots,r$. Let
$\gamma_{i}\subset X\times_{\mathbf{F}_{q}}S$ be the graph of $x_{i}$; this is
a divisor of $X\times_{\mathbf{F}_{q}}S$.

A $G$-shtuka with legs $x_{1},\dots,x_{r}$, archimedean places
$\Sigma_{\infty}$, and $\Gamma_{0}(N)$-structure is an isomorphism of
$G$-torsors with $\Gamma_{0}(N)$-structure:

$f\colon{\mathcal{F}}^{\dagger}|_{(X\times_{\mathbf{F}_{q}}S)\smallsetminus\bigcup_{i=1}^{r}\gamma_{i}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\operatorname{Fr}_{S}^{*}{\mathcal{F}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})|_{(X\times_{\mathbf{F}_{q}}S)\smallsetminus\bigcup_{i=1}^{r}\gamma_{i}}.$

Here the “$\operatorname{Fr}_{S}$” is shorthand for the endomorphism
$\operatorname{id}_{X}\times\operatorname{Fr}_{q}$ on $X\times S$, where
$\operatorname{Fr}_{q}\colon S\to S$ is the $q$th power Frobenius morphism.

One can measure the failure of $f$ to be an isomorphism along
$\gamma_{1},\dots,\gamma_{r}$ by means of an algebraic representation $W$ of
the $r$-fold product $\hat{G}^{r}$, where $\hat{G}$ is the Langlands dual
group. We write $\operatorname{Sht}^{W}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$
for the moduli stack of $G$-shtukas with $\Gamma_{0}(N)$-structure and
archimedean places $\Sigma_{\infty}$ which are bounded by $W$.

We explain here in explicit terms the condition that a shtuka as above is
“bounded by $W$”, in the case that $G=\operatorname{GL}_{2}$ (so that
$\hat{G}=\operatorname{GL}_{2}$ as well) and $W$ is the representation

$W_{\mu}=\mathrm{std}^{\mu_{1}}\boxtimes\cdots\boxtimes\mathrm{std}^{\mu_{r}}$

of $\hat{G}^{r}$, where $\mathrm{std}$ is the standard representation of
$\operatorname{GL}_{2}$ and
$\mu=(\mu_{1},\dots,\mu_{r})\in\left\\{1,-1\right\\}^{r}$ is a vector of signs
(with $\mathrm{std}^{-1}$ meaning the dual of $\mathrm{std}$). Consider the
following data:

  * •

Vector bundles ${\mathcal{G}}_{0},{\mathcal{G}}_{1},\cdots,{\mathcal{G}}_{r}$
on $X\times_{\mathbf{F}_{q}}S$,

  * •

For each $i=1,\dots,r$, an isomorphism
$f_{i}\colon{\mathcal{G}}_{i-1}|_{(X\times_{\mathbf{F}_{q}}S)\backslash\gamma_{i}}\to{\mathcal{G}}_{i}|_{(X\times_{\mathbf{F}_{q}}S)\backslash\gamma_{i}}$.
If $\mu_{i}=1$, then $f_{i}$ is required to extend to a morphism of vector
bundles ${\mathcal{G}}_{i-1}\to{\mathcal{G}}_{i}$ whose cokernel is length 1
supported on $\gamma_{i}$. If $\mu_{i}=-1$, then $f_{i}^{-1}$ is required to
have this property,

An isomorphism between ${\mathcal{G}}_{0}$ and ${\mathcal{G}}_{r}$ away from
$\bigcup_{i}^{r}\gamma_{i}$ is bounded by $W_{\mu}$ if it factors into
$f_{r}\circ\cdots\circ f_{1}$, where $f_{1},\dots,f_{r}$ are as above.
Comparing degrees of vector bundles, we find that such an isomorphism can
exist only if $\sum_{i}\mu_{i}=\deg{\mathcal{G}}_{r}-\deg{\mathcal{G}}_{0}$.

From now on we fix

$G=\operatorname{PGL}_{2},$

so that $\hat{G}=\operatorname{SL}_{2}$. Note that $\mathrm{std}$ is self-dual
when considered as a representation of $\hat{G}$. Let us simply write

$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$

for the moduli stack of $G$-shtukas with $\Gamma_{0}(N)$-structure and
archimedean places $\Sigma_{\infty}$ which are bounded by

$W_{r}:=\mathrm{std}^{\boxtimes r}.$

We have an isomorphism

$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})\cong\operatorname{Sht}_{\operatorname{GL}_{2}}^{W_{\mu}}(\Gamma_{0}(N);\Sigma_{\infty})/\operatorname{Pic}_{X}(\mathbf{F}_{q}),$

where $\mu$ is any vector of signs satisfying
$\sum_{i}\mu_{i}=\\#\Sigma_{\infty}$. Therefore a necessary condition for
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ to be nonempty is
the congruence

$r\equiv\\#\Sigma_{\infty}\pmod{2}.$

The stack $\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$ is a
Deligne-Mumford stack, and the morphism

$\lambda^{r}\colon\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})\to(X\smallsetminus\Sigma)^{r}$

sending a shtuka to its legs $x_{1},\dots,x_{r}$ is generically smooth of
relative dimension $r$.

######  Remark 2.2.2.

Keeping the intermediate ${\mathcal{G}}_{i}$ as above, one obtains the notion
of an iterated shtuka. The moduli space of iterated
$\operatorname{GL}_{2}$-shtukas bounded by $\mu=(1,-1)$ was originally
considered by Drinfeld [Dri80]. If we had worked with iterated shtukas, the
morphism $\lambda^{r}$ would be smooth over $(X\smallsetminus\Sigma)^{r}$. The
spaces of iterated and non-iterated shtukas coincide over the locus in $X^{r}$
where the legs are pairwise distinct.

######  Example 2.2.3 (Relation to Drinfeld modular curves).

In the special case that $r=1$ and $\Sigma_{\infty}=\left\\{\infty\right\\}$
is a singleton, the stack $\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\infty)$
is related to a Drinfeld modular curve. Namely, decompose $N$ as
$N=N_{f}+(\infty)$, and let $\mathrm{DrMod}(\Gamma_{0}(N_{f});\infty)$
classify Drinfeld $A$-modules, where $A$ is the ring of functions on $X$
regular away from $\infty$, equipped with a $\Gamma_{0}(N_{f})$-structure. The
precise relationship is that there is an isomorphism

$\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\infty)\cong\mathrm{DrMod}(\Gamma_{0}(N_{f});\infty)/\operatorname{Pic}^{\circ}_{X}(\mathbf{F}_{q}).$

For $r>1$ the stack
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$ is generally not
of finite type. To produce finite-type stacks, we need to pass to truncations.
There is a map

$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})\to\operatorname{Bun}_{G}(\Gamma_{0}(N))$

sending a shtuka as in Definition 2.2.1 to its $G$-torsor ${\mathcal{F}}$. For
each $i\geq 0$, let
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{\leq i}$ be the
pullback under this map of $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\leq i}$.
Then $\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{\leq i}$ is
of finite type.

###  2.3 Cohomology of stacks of shtukas

Here we prove Proposition 1.3.2, which states that irreducible Langlands
parameters for $G$ appear as direct summands in the cohomology of stacks of
shtukas. We will need the main results of [Xue20a] and [Xue20b], which we now
review.

Recall our conventions on generic points: $\eta$ (resp., $\eta_{r}$) is the
generic point of $X$ (resp, $X^{r}$), and $\overline{\eta}_{r}\to\eta_{r}$ is
an algebraic closure lying over $\overline{\eta}^{r}\to\eta^{r}$. Let $\Sigma$
be the support of $N$, and let $U=X\smallsetminus\Sigma$.

The truncated stack of shtukas
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{\leq i}$ is of
finite type over $U^{r}$, so it makes sense to define the cohomology with
compact supports

$H^{r}_{c}(\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{\leq
i}_{\overline{\eta}_{r}},\overline{\mathbf{Q}}_{\ell}),$

a finite-dimensional $\overline{\mathbf{Q}}_{\ell}$-vector space admitting an
action of $\pi_{1}(U^{r},\overline{\eta}_{r})$. Define

$H^{r}_{c}(\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})_{\overline{\eta}_{r}},\overline{\mathbf{Q}}_{\ell}):=\varinjlim_{i\geq
0}H^{r}_{c}(\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{\leq
i}_{\overline{\eta}_{r}},\overline{\mathbf{Q}}_{\ell}),$

and abbreviate

$H^{r}_{c}:=H^{r}_{c}(\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})_{\overline{\eta}_{r}},\overline{\mathbf{Q}}_{\ell}).$

Then $H^{r}_{c}$ admits an action of the Hecke algebra $T_{0}(N)$ which
commutes with the action of $\pi_{1}(U^{r},\overline{\eta}_{r})$. Let
$(H^{r}_{c})^{\text{cusp}}\subset H^{r}_{c}$ be the cuspidal part. Assuming
that $\Sigma_{\infty}=\emptyset$, a theorem of Xue [Xue20a, Theorem 5.0.1]
shows that $(H^{r}_{c})^{\text{cusp}}$ is finite-dimensional, and then
Drinfeld’s lemma extends the action of $\pi_{1}(U^{r},\overline{\eta}_{r})$ on
$(H^{r}_{c})^{\text{cusp}}$ to an action of $\pi_{1}(U,\overline{\eta})^{r}$.

######  Proposition 2.3.1.

Assume that Xue’s theorem [Xue20a, Theorem 5.0.1] extends to the case of
arbitrary $(r,\Sigma_{\infty})$. As a representation of
$T_{0}(N)\times\pi_{1}(U,\overline{\eta})^{r}$, there is a decomposition

$(H^{r}_{c})^{\mathrm{cusp}}=\bigoplus_{\pi}\pi^{\Gamma_{0}(N)}\otimes\sigma^{\boxtimes
r}_{\pi},$

where $\pi$ runs over cuspidal representations of $G$ containing a
$\Gamma_{0}(N)$-fixed vector, and $\sigma_{\pi}$ is the 2-dimensional
representation of $\pi_{1}(U,\overline{\eta})$ corresponding to $\pi$ under
the global Langlands correspondence.

######  Remark 2.3.2.

When $r=2$ and $\Sigma_{\infty}=\emptyset$, (2.3.1) is a special case of a
theorem of Drinfeld [Dri80].

###### Proof.

The formalism of excursion operators developed in [Laf18] furnishes a
decomposition of $T_{0}(N)\times\pi_{1}(U,\overline{\eta})^{r}$-modules
$(H^{r}_{c})^{\mathrm{cusp}}=\bigoplus_{\sigma}(H^{r}_{c})^{\mathrm{cusp}}_{\sigma}$
indexed by isomorphism classes of Langlands parameters
$\sigma\colon\pi_{1}(U,\overline{\eta})\to\operatorname{GL}_{2}\overline{\mathbf{Q}}_{\ell}$.

Let $\sigma$ be an irreducible Langlands parameter for $G$. In the case that
$\Sigma_{\infty}=\emptyset$, Corollaire 2.1 to [LZ18, Proposition 1.2]
identifies $(H^{r}_{c})^{\mathrm{cusp}}_{\sigma}$ with
$\pi^{\Gamma_{0}(N)}\otimes\sigma^{\boxtimes r}$, where $\pi$ corresponds to
$\sigma$ under the Langlands correspondence. (We remark here that our
$(H^{r}_{c})^{\mathrm{cusp}}$ is the space denoted $H^{\text{cusp}}_{I,W}$ in
[LZ18], where $I$ is a set of cardinality $r$ and $W=\mathrm{std}^{\boxtimes
I}$ is the external tensor product of $I$ copies of the standard
representation std of $\hat{G}=\operatorname{SL}_{2}$.) The proposition
follows.

For $r$ even and $\Sigma_{\infty}$ nonempty, let $G^{\prime}$ be the form of
$G$ corresponding to the quaternion algebra over $K$ which is ramified exactly
at $\Sigma_{\infty}$. Consider $G^{\prime}$ as a group scheme over $X$ which
is parahoric at the places in $\Sigma_{\infty}$ and reductive elsewhere. Then
our stack of shtukas
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$ is isomorphic to a
stack of shtukas $\operatorname{Sht}_{G^{\prime}}^{r}(\Gamma_{0}(N_{f}))$ for
the group $G^{\prime}$ (with no archimedean places). Here we have written
$N_{f}$ for the part of $N$ away from $\Sigma_{\infty}$. Under the Langlands
correspondence, $\sigma$ corresponds to a cuspidal representation
$\pi^{\prime}$ of $G^{\prime}$ of level $N_{f}$. We now apply [LZ18,
Proposition 1.2]: there exists a finite-dimensional $T_{0}(N)$-module
$A_{\sigma}$ such that for all even $r$, there is an isomorphism of
$T_{0}(N)\times\pi_{1}(U,\overline{\eta})^{r}$-modules:

$(H^{r}_{c})^{\mathrm{cusp}}_{\sigma}\cong A_{\sigma}\otimes\sigma^{\boxtimes
r}.$

(Note that the group $\mathcal{S}_{\sigma}$ in [LZ18, Proposition 1.2] is
trivial in this setting.) Setting $r=0$ we find that
$A_{\sigma}=(H^{0}_{c})^{\mathrm{cusp}}_{\sigma}=\pi^{\Gamma_{0}(N)}$ is
nonzero. Thus for any even $r$ we find a nonzero $\sigma$-isotypic component
in $(H^{r}_{c})^{\mathrm{cusp}}$.

For $r$ odd, let $\infty\in\Sigma_{\infty}$, and let $G^{\prime}$ be the form
of $G$ corresponding to the quaternion algebra $D/K$ ramified exactly at
$\Sigma_{\infty}\smallsetminus\left\\{\infty\right\\}$. Then
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})$ is isomorphic to a
stack of shtukas
$\operatorname{Sht}_{G^{\prime}}^{r}(\Gamma_{0}(N_{f});\left\\{\infty\right\\})$.
Let $\pi^{\prime}$ be the automorphic representation of $G^{\prime}$
corresponding to $\sigma$. The same technique as in [LZ18] applies
(conditionally on the extension of Xue’s theorem to this case) to give a
finite-dimensional $T_{0}(N)$-module $A_{\sigma}$ such that for all odd $r$,
there is an isomorphism of
$T_{0}(N)\times\pi_{1}(U,\overline{\eta})^{r}$-modules:

$(H^{r}_{c})^{\mathrm{cusp}}_{\sigma}\cong A_{\sigma}\otimes\sigma^{\boxtimes
r}.$

Setting $r=1$, we have that $(H^{1}_{c})^{\mathrm{cusp}}_{\sigma}$ is the
cohomology of either a Drinfeld modular curve (if
$\Sigma_{\infty}=\left\\{\infty\right\\}$) or of the (also 1-dimensional)
moduli space of D-elliptic sheaves in the sense of [LRS93] (otherwise). In
either case the decomposition of $H^{1}$ is known [LRS93, Theorem 14.9]:
$A_{\sigma}=\pi^{\Gamma_{0}(N)}$. We proceed as above. ∎

In the situation of Proposition 1.3.2, suppose $\sigma$ is an irreducible
Langlands parameter of conductor $N$. Proposition 2.3.1 shows that
$\sigma^{\boxtimes r}$ appears as a summand of $(H^{r}_{c})^{\mathrm{cusp}}$.
To prove that it is a summand of $H^{r}_{c}$, we will need to exhibit
$\sigma^{\boxtimes r}$ as a quotient of $H^{r}_{c}$, which requires further
argument because $H^{r}_{c}$ is generally infinite-dimensional.

To proceed, we require the result [Xue20b, Theorem 2]. We must assume that
this result applies to arbitrary pairs $(r,\Sigma_{\infty})$. The result is
that $H^{r}_{c}$ is finitely generated over the local spherical Hecke algebra
at any place of $U$. The appropriate adaptation of Drinfeld’s lemma extends
the action of $\pi_{1}(U^{r},\overline{\eta}_{r})$ on $H^{r}_{c}$ to an action
of $\mathrm{Weil}(\overline{\eta}/\eta)^{r}$, where
$\mathrm{Weil}(\overline{\eta}/\eta)$ is the Weil group.

Furthermore one gets once again a decomposition over Langlands parameters, as
explained in [Xue20b, Proposition 3.7.10]. Let $v\in U$ be a place, let
$\mathcal{H}_{G,v}$ be the local spherical Hecke algebra at a place $v\in U$.
Let $\mathcal{J}\subset\mathcal{H}_{G,v}$ be an ideal of finite codimension,
so that the quotient $H^{r}_{c}/\mathcal{J}H^{r}_{c}$ is finite-dimensional.
The formalism of excursion operators applies to give a decomposition into
$T_{0}(N)\times\mathrm{Weil}(\overline{\eta}/\eta)^{r}$-modules:

$H^{r}_{c}/\mathcal{J}H^{r}_{c}=\bigoplus_{\sigma}(H^{r}_{c}/\mathcal{J}H^{r}_{c})_{\sigma}$

The methods of [LZ18] show as before that as a
$\mathrm{Weil}(\overline{\eta}/\eta)^{r}$-module,
$(H^{r}_{c}/\mathcal{J}H^{r}_{c})_{\sigma}$ is a direct sum of copies of
$\eta^{\boxtimes r}$.

Now suppose that $\sigma$ is an irreducible representation of
$\operatorname{Gal}(\overline{\eta}/\eta)$ of conductor $N$, corresponding to
$\pi$, and that $\mathcal{J}$ is the ideal of $\mathcal{H}_{G,v}$
corresponding to $\pi_{v}$. To prove Proposition 1.3.2 for $\sigma$, it
suffices to show that $(H^{r}_{c}/\mathcal{J}H^{r}_{c})_{\sigma}$ is nonzero.
For this, note that the inclusion
$(H^{r}_{c})^{\mathrm{cusp}}/\mathcal{J}(H^{r}_{c})^{\mathrm{cusp}}\to
H^{r}_{c}/\mathcal{J}H^{r}_{c}$ commutes with excursion operators, and
therefore respects the decomposition into Langlands parameters. By Proposition
2.3.1,
$((H^{r}_{c})^{\mathrm{cusp}}/\mathcal{J}(H^{r}_{c})^{\mathrm{cusp}})_{\sigma}$
is nonzero, as it contains $(H^{r}_{c})^{\mathrm{cusp}}_{\sigma}\neq 0$.

###  2.4 Vector bundles with level structures on $\mathbf{P}^{1}$

At this point we specialize our study of shtuka spaces to the case that the
base curve $X$ is the projective line. We make heavy use of the fact that the
degree $d$ component $\operatorname{Bun}_{2}^{d}$ admits a dense open substack
containing only one isomorphism class, namely $\mathcal{O}(d/2)^{\oplus 2}$ or
$\mathcal{O}((d-1)/2)\oplus\mathcal{O}((d+1)/2)$ as $d$ is even or odd,
respectively; these are exactly the objects with instability index $\leq 1$.
As a result, it becomes possible to give explicit equations for dense open
substacks of the moduli stacks of shtukas.

The following lemmas, concerning $\operatorname{Bun}_{G}(\Gamma_{0}(N))$ for
divisors $N$ of degree 3 and 4, exemplify the simplifications that will occur.

######  Lemma 2.4.1.

Let $F$ be a field, and let $X=\mathbf{P}^{1}_{F}$. Let $N$ be an effective
divisor of degree 3. There is a dense substack
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))$, such that each connected component
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{d,\omega}$ ($d=0,1$) is isomorphic to
a single point $\operatorname{Spec}F$.

###### Proof.

We given an explicit description of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ in the case that
$N=P_{1}+P_{2}+P_{3}$ where each $P_{i}$ is $F$-rational, the other cases
being similar. It is the locus of ${\mathcal{F}}^{\dagger}$ where the
instability of each fractional twist ${\mathcal{F}}(\textstyle\frac{1}{2}D)$
is $\leq 1$, where $D$ runs over divisors supported on $N_{1},N_{2},N_{3}$.
One sees that each of its components
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{d,\omega}$ is a single point. In
degree 0, for instance, let
${\mathcal{F}}^{\dagger}\in\operatorname{Bun}_{G}^{0}(\Gamma_{0}(N))(F)$ be
the object with ${\mathcal{F}}$ the trivial $G$-torsor. The level structure,
represented by a triple of pairwise distinct elements of $\mathbf{P}^{1}(F)$,
determines an object of $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ if
and only if the three points in the triple are pairwise distinct. (Translated
into vector bundles, this is the observation that elementary modifications of
$\mathcal{O}^{\oplus 2}$ at points $P_{1},P_{2}$ along the lines
$L_{1},L_{2}\subset F^{2}$ produce $\mathcal{O}(1)\oplus\mathcal{O}(1)$
generically, but will produce $\mathcal{O}(2)\oplus\mathcal{O}$ if
$L_{1}=L_{2}$.) As $G$ acts simply transitively on such triples, we find an
isomorphism
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{0,\omega}\cong\operatorname{Spec}F$.

∎

######  Lemma 2.4.2.

Let $F$ be a field, and let $X=\mathbf{P}^{1}_{F}$.

  1. 1.

Let $N=P_{1}+P_{2}+P_{3}+P_{4}$ be a multiplicity-free divisor of degree 4.
There exists a dense open substack
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$, stable under the Atkin-
Lehner operators, each of whose connected components is isomorphic to
$\mathbf{P}^{1}_{F}\smallsetminus N$. With respect to these isomorphisms, the
Atkin-Lehner involution
$\operatorname{AL}(\textstyle\frac{1}{2}(P_{1}+P_{2})))$ (here $i\neq j$ are
distinct indices) is the unique involution which exchanges $P_{1}$ with
$P_{2}$ and $P_{3}$ with $P_{4}$.

  2. 2.

Let $N=P_{1}+P_{2}+2P_{3}$, with $P_{1},P_{2},P_{3}$ pairwise distinct. There
exists a dense open substack $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$,
stable under the Atkin-Lehner operators (supported only on $P_{1},P_{2}$),
each of whose connected components is isomorphic to $\mathbf{A}^{1}_{F}$.

###### Proof.

(1) Let $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ be the locus of
${\mathcal{F}}^{\dagger}$ where (once again) all Atkin-Lehner twists
${\mathcal{F}}(\textstyle\frac{1}{2}\Sigma)$ have instability index $\leq 1$.
This time, the $\Gamma_{0}(N)$-level structures ${\mathcal{F}}^{\dagger}$ on
the trivial $G$-torsor correspond to quadruples of points of $\mathbf{P}^{1}$.
One computes that ${\mathcal{F}}^{\dagger}$ lies in
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ if and only if the quadruple
is pairwise distinct and is not in the $G$-orbit of
$(P_{1},P_{2},P_{3},P_{4})$. The set of $G$-orbits of such quadruples is
$P^{1}_{F}\smallsetminus N$. (The claim about Atkin-Lehner involutions is not
used in the sequel and we omit its proof.)

(2) Let $\operatorname{Bun}_{2}(\Gamma_{0}(N))^{\omega}$ be the preimage of
the substack described in Lemma 2.4.1. The parameter space for
$\Gamma_{0}(2P_{3})$-structures lying over a given
$\Gamma_{0}(P_{3})$-structure is described by the fibers of
$\operatorname{Res}^{2P_{3}}_{P_{3}}\mathbf{P}^{1}_{F}\to\mathbf{P}^{1}_{F}$,
which are all $\mathbf{A}^{1}_{F}$.

∎

###  2.5 Explicit presentations of shtuka spaces with
$\Gamma_{0}(N)$-structure

The following theorem and its proof give an explicit presentation of the space
of shtukas with $\Gamma_{0}(N)$-structure, where $N$ is a divisor of degree 3
or 4. It is a special case of the main results of [dFF20]. Define
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\omega}$ to be the
dense open substack where the underlying $G$-bundle of the shtuka lies in the
substack $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{\omega}$ described by Lemmas
2.4.1 and 2.4.2.

######  Theorem 2.5.1.

Let $X=\mathbf{P}^{1}_{\mathbf{F}_{q}}$, let $r\geq 1$ be an integer, and let
$\Sigma_{\infty}\subset\left\lvert X\right\rvert$ be a finite set of degree 1
points satisfying $\\#\Sigma_{\infty}\equiv r\pmod{2}$. Let
$\eta_{r}=\operatorname{Spec}\mathbf{F}_{q}(t_{1},\dots,t_{r})$ be the generic
point of $X^{r}$.

  1. 1.

Let $N$ be an effective divisor of degree 3, such that each point of
$\Sigma_{\infty}$ has multiplicity 1 in $N$. The stack
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\omega}_{\eta_{r}}$
is a quasiprojective variety, each of whose connected components is rational;
more precisely, there is a birational morphism from each component to
$(\mathbf{P}^{1}_{\eta_{r}})^{r}$.

  2. 2.

Let $N$ be a multiplicity-free effective divisor of degree 4 whose support
contains $\Sigma_{\infty}$. The stack
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\omega}_{\eta_{r}}$
is a quasiprojective variety. Each of its connected components is birational
to a hypersurface in $(\mathbf{P}^{1}_{\eta})^{r+1}$ of degree
$(2,2,\dots,2,q+1)$.

  3. 3.

Let $N=P_{1}+P_{2}+2P_{3}$ be an effective divisor of degree 4, such that
$P_{1},P_{2},P_{3}$ are pairwise distinct and
$\Sigma_{\infty}\subset\left\\{P_{1},P_{2}\right\\}$. The stack
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{\omega}_{\eta_{r}}$
is a quasiprojective variety. Each of its connected components is birational
to a hypersurface in $(\mathbf{P}^{1}_{\eta})^{r+1}$ of degree
$(2,2,\dots,2,q)$.

######  Remark 2.5.2.

There are no cusp forms on $\operatorname{PGL}_{2}$ of conductor $N$ if $\deg
N=3$, whereas if $\deg N=4$ and $N$ is multiplicity-free, then the dimension
of the space of cusp forms of conductor $N$ is exactly $q$. (See [DF13, §7]
for these results and related formulas.) So it is not surprising that non-
rational varieties appear when passing from degree 3 to degree 4.

######  Remark 2.5.3.

It is possible to extend Theorem 2.5.1 to include the cases $N=2P_{1}+2P_{2}$,
$N=P_{1}+3P_{2}$, and $N=4P$. We refer the reader to [dFF20] for details.

###### Proof.

(1) By Lemma 2.4.1, $\operatorname{Bun}_{G}(\Gamma_{0}(N))^{d,\omega}$ is a
point, corresponding to an object ${\mathcal{F}}^{\dagger}$ of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))(\mathbf{F}_{q})$. Thus for a test
scheme $S$, the shtukas in
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{d,\omega}(S)$
classify isomorphisms of $G$-torsors with $\Gamma_{0}(N)$-structure:

$f\colon{\mathcal{F}}^{\dagger}|_{(X\times_{\mathbf{F}_{q}}S)\backslash\cup_{i}\gamma_{i}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}{\mathcal{F}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})|_{(X\times_{\mathbf{F}_{q}}S)\backslash\cup_{i}\gamma_{i}}$

which are bounded by $W_{r}$.

With Frobenius out of the picture, all that remains is linear algebra. Let
${\mathcal{F}}_{1}^{\dagger}$ and ${\mathcal{F}}_{2}^{\dagger}$ be any two
lifts of ${\mathcal{F}}$ and
${\mathcal{F}}(\textstyle\frac{1}{2}\Sigma_{\infty})$ to
$\operatorname{Bun}_{2}(\mathbf{F}_{q})$, such that
$\deg{\mathcal{F}}_{2}=\deg{\mathcal{F}}_{1}+r$. (This is possible due to the
parity condition on $r$.) Note that the dimension of
$\operatorname{Hom}({\mathcal{F}}_{1}^{\dagger},{\mathcal{F}}_{2}^{\dagger})$
(meaning, morphisms of vector bundles, not necessarily injective, which
preserve level structures) is $2r+1$. The stack of shtukas
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{d,\omega}$ is
isomorphic to the quotient by $\mathbf{G}_{m}$ of the moduli stack of
morphisms of vector bundles with $\Gamma_{0}(N)$-structure
${\mathcal{F}}_{1}^{\dagger}\to{\mathcal{F}}_{2}^{\dagger}$ which are bounded
by $\mathrm{std}^{\boxtimes r}$ at the legs. This is a quasi-projective
variety; indeed it is the locally closed subvariety of

$(\mathbf{P}^{1})^{r}\times\mathbf{P}\operatorname{Hom}({\mathcal{F}}_{1}^{\dagger},{\mathcal{F}}_{2}^{\dagger})\cong(\mathbf{P}^{1})^{r}\times\mathbf{P}^{2r},$

consisting of data $(x_{1},\dots,x_{r},f)$, where the divisor of $\det f$ is
$\gamma_{1}+\dots+\gamma_{r}$.

For each $i=1,\dots,r$, the local behavior of $f$ at $\gamma_{i}$ gives an
$S$-point of the affine Grassmannian $\mathrm{Gr}_{G}$: this is the period
morphism at the $i$th leg. If the legs are pairwise disjoint, all the period
morphisms land in the minuscule Schubert cell
$\mathrm{Gr}_{G}^{\mathrm{std}}=\mathbf{P}^{1}$. We claim that over the
generic point $\eta_{r}$, the product of the period morphisms

$\Pi\colon\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{d,\omega}_{\eta_{r}}\to(\mathbf{P}^{1}_{\eta_{r}})^{r}$

is a birational map. Indeed, let $x_{1},\dots,x_{r}$ be generic legs, and
consider the fiber of $\Pi$ over $(u_{1},\dots,u_{r})\in(\mathbf{P}^{1})^{r}$.
This is the subspace of
$f\in\mathbf{P}\operatorname{Hom}({\mathcal{F}}_{1}^{\dagger},{\mathcal{F}}_{2}^{\dagger})\cong\mathbf{P}^{2r}$
such that $f$ is generically an isomorphism, and such that $f(x_{i})$ has rank
1, with kernel $u_{i}$. This is $2r$ linear conditions; for generic
$(u_{1},\dots,u_{r})$ there is a unique solution.

A computation involving minors shows that the rational map
$\Pi^{-1}\colon(\mathbf{P}^{1}_{\eta_{r}})^{r}\to\mathbf{P}_{\eta_{r}}^{2r}$
has degree $(2,\dots,2)$.

(2) Now suppose $N$ is multiplicity-free of degree 4. Let $N=N_{0}+P$ for a
point $P$ disjoint from $N_{0}$. The quasi-projective variety
$\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{r}}$ can be
modeled on a subvariety of $\mathbf{P}^{2r}\times\mathbf{P}^{1}$, where the
$\mathbf{P}^{2r}$ is as in part (1) and the $\mathbf{P}^{1}$ records the
additional level structure at $P$. Namely, it is the space of pairs $(f,v)$,
where the divisor of $\det f$ is $\gamma_{1}+\dots+\gamma_{r}$, and $f$
preserves the level structure at $P$ in the sense that
$f(P)(v)=\operatorname{Fr}_{q}(v)$. Part (1) shows that this is birational via
$\Pi$ to the subvariety $Z$ of $(\mathbf{P}^{1})^{r}\times\mathbf{P}^{1}$
consisting of pairs $(u,v)$, where $f=\Pi^{-1}(u)$ satisfies
$f(P)(v)=\operatorname{Fr}_{q}(v)$. We saw that $\Pi^{-1}$ has degree
$(2,\dots,2)$, which is to say that $f(t)$ has coordinates which are degree
$(2,\dots,2)$ in $u=(u_{1},\dots,u_{r})$. Finally we observe that the equation
$f(P)(v)=\operatorname{Fr}_{q}(v)$ has degree $q+1$ in $v$.

(3) Finally, suppose $N=P_{1}+P_{2}+2P_{3}=N_{0}+P_{3}$. The calculation is
similar to (2), except that the additional level structure at $P_{3}$ has
parameter space $\mathbf{A}^{1}$, this being the fiber of
$\operatorname{Res}^{2P_{3}}_{P_{3}}\mathbf{P}^{1}\to\mathbf{P}^{1}$. The
operator $f(P)$ is an automorphism of this $\mathbf{A}^{1}$; i.e., an affine
transformation. Thus $f(P)(v)=\operatorname{Fr}_{q}(v)$ has degree $q$ in $v$.
∎

######  Remark 2.5.4.

Consider the case $r=2$. Let $N$ be a divisor of degree 4 as in part (2) or
(3) of Theorem 2.5.1. Let
$\eta_{2}\cong\operatorname{Spec}\mathbf{F}_{q}(t_{1},t_{2})$ be the generic
point of $X^{2}$. The theorem shows that each connected component of
$\operatorname{Sht}_{G}^{2}(\Gamma_{0}(N);\Sigma_{\infty})^{\omega}_{\eta_{2}}$
is birational to a hypersurface in $(\mathbf{P}^{1}_{\eta_{2}})^{3}$ of degree
$(2,2,q)$ or $(2,2,q+1)$. Since $(2,2)$-curves in $(\mathbf{P}^{1})^{2}$ are
genus 1, there is a desingularization, call it $Z$, which is a genus 1
fibration over $\mathbf{P}^{1}_{\eta_{2}}$. In fact
$Z\to\mathbf{P}^{1}_{\eta_{2}}$ admits a section, so that $Z$ is an elliptic
surface. The nature of the singular fibers of $Z\to\mathbf{P}^{1}_{\eta_{2}}$
was studied in [dFF20].

Specializing even further to the case $q=2$ and $N=(0)+(1)+2(\infty)$, one
obtains a hypersurface $Z$ defined over $\mathbf{F}_{2}(t_{1},t_{2})$ of
degree $(2,2,2)$, which is a K3 elliptic surface. This project originated in
2019, when the second author sent the equation for $Z$ to Noam Elkies, who (a)
recognized $Z$ as the universal K3 elliptic surface with a 6-torsion section,
and (b) computed a finite morphism from $Z$ to the Kummer surface
$\operatorname{Km}(\mathcal{E}_{\eta_{2}}^{2})$, where
$\mathcal{E}\to\mathbf{P}^{1}_{\mathbf{F}_{2}}$ is an elliptic fibration of
conductor $N$. (More precisely,
$\mathcal{E}\to\mathbf{P}^{1}_{\mathbf{F}_{2}}$ is the elliptic fibration with
$\mathrm{I}_{2},\mathrm{I}_{6},\mathrm{IV}$ fibers at $0,1,\infty$, with
Mordell-Weil group $\mathbf{Z}/6\mathbf{Z}$.) This $\mathcal{E}$ was the first
example of a elliptic fibration we knew to be 2-modular.

###  2.6 The coincidence map

Theorem 2.5.1 allows us to derive equations for shtuka spaces of level
$\Gamma_{0}(N)$, where $N$ is a degree 4 effective divisor of
$\mathbf{P}^{1}_{\mathbf{F}_{q}}$ supported on at least one place $\infty$ of
multiplicity 1. In an early phase of this line of research, we dealt directly
with those equations in an attempt to prove 2-modularity of some elliptic
fibrations of conductor $N$. To our surprise, a change of variables turned the
defining equation for
$\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N))^{d,\omega}_{\eta_{2}}$ into one in
which we could recognize an equation for the Drinfeld modular curve
$\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\infty)$. Eventually we found a
generalization of this phenomenon, in the form of the coincidence map
described in the introduction.

Let $r\geq 2$, and let
$\Sigma_{\infty},\Sigma^{\prime}_{\infty}\subset\operatorname{Supp}N$ be sets
of degree 1 places with multiplicity 1 in $N$, such that
$\\#\Sigma_{\infty}\equiv r\pmod{2}$ and $\\#\Sigma_{\infty}^{\prime}\equiv
1\pmod{2}$. (Under our hypotheses on $N$, such sets exist for any $r$.) Let
$\Sigma_{\infty}^{\prime\prime}$ be the symmetric difference of these sets, so
that $\\#\Sigma_{\infty}^{\prime\prime}\equiv r+1\pmod{2}$. Our goal is to
give a factorization of the variety of $r$-legged shtukas into a cartesian
square:

$\textstyle{\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c\times\lambda_{r}}$$\scriptstyle{\beta}$$\textstyle{\operatorname{Sht}^{1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})\times
U^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{1}\times\operatorname{id}}$$\textstyle{\operatorname{Coinc}^{r+1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda_{r+1}}$$\textstyle{U^{r+1}}$
(2.6.1)

We have called the variety
$\operatorname{Coinc}^{r+1}(\Gamma_{0}(N);\Sigma^{\prime\prime}_{\infty})$ the
moduli space of coincidences, and $c$ the coincidence map. The surprising
thing (to us) about this state of affairs was that the family of varieties
$\operatorname{Coinc}^{r+1}(\Gamma_{0}(N);\Sigma^{\prime\prime}_{\infty})\to
U^{r+1}$ was symmetric not just in the first $r$ copies of $U$ as we would
expect, but also in the final copy.

First we define the variety of coincidences. The “coincidences” classified by
these spaces are modifications of vector bundles, but Frobenius is not
involved and they may be defined in any characteristic. Therefore let $F$ be
any field, and let $N$ be an effective divisor on $\mathbf{P}^{1}$ of degree
4. Let $\Sigma_{\infty}\subset\operatorname{Supp}N$ be a set of places
appearing with multiplicity 1 in $N$, such that $\\#\Sigma_{\infty}\equiv
r\pmod{2}$.

Let ${\mathcal{H}}^{\dagger}\in\operatorname{Bun}_{G}(\Gamma_{0}(N))(F)$ be
the following “reducible object” of instability index 2. This will be the
image of an object of $\operatorname{Bun}_{2}(\Gamma_{0}(N))(F)$ represented
by the data ${\mathcal{H}}=\mathcal{O}(2)\oplus\mathcal{O}$ together with the
$\Gamma_{0}(N)$-level structure which is just $\mathcal{O}_{N}$ embedded into
${\mathcal{H}}|_{N}$ along the second factor. Note that
$\operatorname{Aut}{\mathcal{H}}^{\dagger}$ (considered as a group scheme) is
$\mathbf{T}$, the diagonal torus in $G$. Thus
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{0}$ contains a locally closed substack
isomorphic to $B\mathbf{T}$, whose $S$ points classify those
${\mathcal{H}}_{0}^{\dagger}\in\operatorname{Bun}_{G}(\Gamma_{0}(S))$ which
are locally isomorphic to ${\mathcal{H}}^{\dagger}$.

We need ${\mathcal{H}}^{\dagger}$ for the following property:

######  Lemma 2.6.1.

Consider the stack $\mathcal{U}$ whose $S$-points classify tuples
$({\mathcal{H}}^{\dagger}_{0},{\mathcal{F}}^{\dagger},\tau,h)$, where
${\mathcal{H}}^{\dagger}_{0}$ is an object of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{0}(S)$ which is locally isomorphic to
${\mathcal{H}}^{\dagger}$, where
${\mathcal{F}}^{\dagger}\in\operatorname{Bun}_{G}(\Gamma_{0}(N))^{1,\omega}(S)$,
where $\tau\in\mathbf{P}^{1}(S)$, and where

$h\colon{\mathcal{H}}_{0}^{\dagger}|_{\mathbf{P}^{1}\smallsetminus\gamma_{\tau}}\cong{\mathcal{F}}^{\dagger}|_{\mathbf{P}^{1}\smallsetminus\gamma_{\tau}}$

is an isomorphism bounded by $\mathrm{std}$ on $\tau$. Then the projection
$\mathcal{U}\mapsto\operatorname{Bun}_{G}(\Gamma_{0}(N))^{1,\omega}$ is an
isomorphism. That is, ${\mathcal{F}}^{\dagger}$ determines the data
$(\tau,{\mathcal{H}}^{\dagger},h)$.

###### Proof.

A restatement of the lemma is that, given ${\mathcal{F}}^{\dagger}$, the space
classifying isomorphisms $h$ between ${\mathcal{H}}^{\dagger}$ and
${\mathcal{F}}^{\dagger}$ away from an unspecified leg and bounded by
$\mathrm{std}$ there is a $\mathbf{T}$-torsor. The space of such $h$ can be
modeled in an ambient space
$\mathbf{P}\operatorname{Hom}(\mathcal{O}(2)\oplus\mathcal{O},\mathcal{O}(2)\oplus\mathcal{O}(1))\cong\mathbf{P}^{5}$.
The fact that $h$ preserves $\Gamma_{0}(N)$-level structures (recall that
$\deg N=4$) means that it belongs to a $\mathbf{P}^{1}$, but then two of the
points on the $\mathbf{P}^{1}$ correspond to $h$ with $\det h=0$. The
complement of these points is a $\mathbf{T}$-torsor. ∎

######  Definition 2.6.2 (The stack of $r$-legged coincidences).

A $G$-coincidence with legs $x_{i}\colon S\to\mathbf{P}^{1}$ (for
$i=1,\dots,r$), archimedean places $\Sigma_{\infty}$, and
$\Gamma_{0}(N)$-structure is an isomorphism of $G$-torsors with
$\Gamma_{0}(N)$-structure:

$f\colon{\mathcal{H}}_{1}^{\dagger}|_{\mathbf{P}^{1}_{S}\smallsetminus\bigcup_{i}\gamma_{i}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}{\mathcal{H}}_{2}^{\dagger}|_{\mathbf{P}^{1}_{S}\smallsetminus\bigcup_{i}\gamma_{i}}$

such that ${\mathcal{H}}_{1}^{\dagger}\cong{\mathcal{H}}^{\dagger}$ and
${\mathcal{H}}_{2}^{\dagger}\cong{\mathcal{H}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})$,
and such that $f$ is bounded by $\mathrm{std}^{\boxtimes r}$ at the legs. Let
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ be the stack
classifying such $G$-coincidences, and let

$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\to(\mathbf{P}^{1})^{r}$

be the morphism sending a $G$-coincidence to its legs.

In other words, $\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$
classifies such isomorphisms involving ${\mathcal{H}}^{\dagger}$ and
${\mathcal{H}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})$, modulo the
action of $\mathbf{T}\times\mathbf{T}$.

Now we can construct the coincidence map $c$. Return to the setting in the
overview, so that the base field is $\mathbf{F}_{q}$ and we have
$\Sigma_{\infty},\Sigma_{\infty}^{\prime},\Sigma_{\infty}^{\prime\prime}\subset\Sigma$,
whose parities are $r,1,r+1$, respectively. Consider the stack $\mathcal{V}$
whose $S$-points classifying the following data:

  * •

${\mathcal{F}}^{\dagger}$ is an object of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{0,\omega}(S)$,

  * •

${\mathcal{H}}_{1}^{\dagger},{\mathcal{H}}_{2}^{\dagger}$ are objects of
$\operatorname{Bun}_{G}(\Gamma_{0}(N))^{0}(S)$ locally isomorphic to
${\mathcal{H}}^{\dagger}$,

  * •

$f\colon{\mathcal{F}}^{\dagger}|_{X_{S}\smallsetminus\bigcup_{i=1}^{r}\gamma_{i}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\operatorname{Fr}^{\ast}{\mathcal{F}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})|_{X_{S}\smallsetminus\bigcup_{i=1}^{r}\gamma_{i}}$
is a shtuka bounded by $\mathrm{std}^{\boxtimes r}$ at the legs
$x_{1},\dots,x_{r}\in\mathbf{P}^{1}(S)$,

  * •

$f^{\prime}\colon{\mathcal{F}}^{\dagger}|_{X_{S}\smallsetminus\gamma_{r+1}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}\operatorname{Fr}^{\ast}(\textstyle\frac{1}{2}\Sigma_{\infty}^{\prime})|_{X_{S}\smallsetminus\gamma_{r+1}}$
is a shtuka bounded by $\mathrm{std}$ at the leg
$x_{r+1}\in\mathbf{P}^{1}(S)$,

  * •

For $i=1,2$,
$h_{i}\colon{\mathcal{H}}_{i}^{\dagger}|_{X_{S}\smallsetminus\gamma_{\tau}}\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}{\mathcal{F}}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty}^{\prime})|_{X_{S}\smallsetminus\gamma_{\tau}}$
is an isomorphism bounded by $\mathrm{std}$ at $\tau\in\mathbf{P}^{1}(S)$.

These are subject to a condition ($\ast$) that the composition

$g:=h_{2}(\textstyle\frac{1}{2}\Sigma_{\infty}^{\prime\prime})^{-1}\circ
f^{\prime}(\textstyle\frac{1}{2}\Sigma_{\infty})^{-1}\circ
f(\textstyle\frac{1}{2}\Sigma_{\infty}^{\prime})\circ h_{1},$ (2.6.2)

a priori an isomorphism from ${\mathcal{H}}_{1}^{\dagger}$ to
${\mathcal{H}}_{2}(\textstyle\frac{1}{2}\Sigma_{\infty})^{\prime\prime}$ away
from $x_{1},\dots,x_{r+1},\tau$ and bounded by $\mathrm{std}^{\boxtimes
r+1}\boxtimes\mathrm{std}^{\otimes 2}$, extends over $\tau$ (more precisely,
is bounded by the subrepresentation $\mathrm{std}^{\boxtimes
r+1}\boxtimes\mathrm{triv}$).

Now, by Lemma 2.6.1, ${\mathcal{F}}^{\dagger}$ completely determines the data
of $\tau$ and the $h_{i}$, so we see that $\mathcal{V}$ is a closed substack
of
$\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{0,\omega}\times\operatorname{Sht}_{G}^{1}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})^{0,\omega}$.
We claim that $\mathcal{V}$ is birational to the graph of a dominant rational
map
$c\colon\operatorname{Sht}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty})^{0}\dashrightarrow\operatorname{Sht}_{G}^{1}(\Gamma_{0}(N);\Sigma_{\infty})^{0}$.

Proof of claim: Given an $r$-legged shtuka $f$ involving
${\mathcal{F}}^{\dagger}$, the space of shtukas $f^{\prime}$ with one
indeterminate leg lives in an ambient space
$\mathbf{P}\operatorname{Hom}(\mathcal{O}^{2},\mathcal{O}(1)\oplus\mathcal{O})\cong\mathbf{P}^{5}$,
subject to 4 level conditions plus the additional linear condition $(\ast)$,
which determines $f^{\prime}$ uniquely, at least for generic $f$.

We have defined a rational map

$c\colon\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{0}\dashrightarrow\operatorname{Sht}_{G}^{1}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime})^{0}$

sending $f$ to $f^{\prime}$. There is also a morphism
$\mathcal{V}\to\operatorname{Coinc}_{G}^{r}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime})$
whose output is the composition $g$ above, inducing a rational map

$\beta\colon\operatorname{Sht}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})^{0,\omega}\dashrightarrow\operatorname{Coinc}^{r+1}_{G}(\Gamma_{0}(N);\Sigma_{\infty}^{\prime\prime}).$

We have defined a commutative diagram of rational maps (2.6.1) at least on the
degree 0 component. To see that it is cartesian, suppose $g$ and $f^{\prime}$
are given. By Lemma 2.6.1 we get an $h$, from which it is possible to solve
for $f$ in (2.6.2). Finally, we may extend $c$ and $\beta$ to the degree 1
component using an Atkin-Lehner involution.

###  2.7 Explicit equations for the spaces of $G$-coincidences

Let $F$ be a field, let $N\subset\mathbf{P}^{1}_{F}$ be an effective divisor
of degree 4 with support $\Sigma$, let
$U=\mathbf{P}^{1}_{F}\smallsetminus\Sigma$, and let
$\Sigma_{\infty}\subset\Sigma$ be a subset consisting of points appearing in
$N$ with multiplicity 1. Then we have defined the space of $G$-coincidences
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\to U^{r}$ in
Definition 2.6.2. The fiber of this morphism over the generic point
$\eta=\operatorname{Spec}F(t_{1},\dots,t_{r})$ is, as we will see, a quasi-
projective variety over $\eta$ of dimension $r-3$. We record here some
explicit equations for
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{r}}$. The
case

$N=(0)+(1)+2(\infty)$

being especially simple, we assume this. Fix $\Sigma_{\infty}$ in the
following way: $\emptyset$ if $r$ is even, and $\left\\{1\right\\}$ if $r$ is
odd. Let ${\mathcal{H}}_{0}^{\dagger}$ be the object of
$\mathcal{O}(2)\oplus\mathcal{O}$ of $\operatorname{Bun}_{2}(\Gamma_{0}(N))$,
where the $\Gamma_{0}(N)$-level structure is $\mathcal{O}_{N}$ embedded in the
second factor. Let ${\mathcal{H}}_{1}^{\dagger}$ be a twist of
${\mathcal{H}}_{0}^{\dagger}(\textstyle\frac{1}{2}\Sigma_{\infty})$ which has
degree $r+2$: thus ${\mathcal{H}}_{1}$ is isomorphic to
$\mathcal{O}((r+3)/2)\oplus\mathcal{O}((r+1)/2)$ if $r$ is odd and
$\mathcal{O}(r/2+2)\oplus\mathcal{O}(r/2)$ if $r$ is even. We can model
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})$ on the quotient
by $\mathbf{T}\times\mathbf{T}$ of the subvariety
$A\subset\mathbf{P}\operatorname{Hom}({\mathcal{H}}_{0}^{\dagger},{\mathcal{H}}^{\dagger}_{1})\cong\mathbf{P}^{2r-1}$
describing those $f$ for which $\det f$ is nonzero. Explicitly, $A$ classifies
matrices with polynomial entries

$f(T)=\begin{pmatrix}a(T)&b(T)\\\ c(T)&d(T)\end{pmatrix}$

taken up to nonzero scalar, where the entries $a,b,c,d$ have degrees bounded
by $r/2$,$r/2+2$,$r/2-2$,$r/2$ (for $r$ even) or
$(r-1)/2,(r-1)/2,(r-3)/2,(r+1)/2$ (for $r$ odd), such that $f(T)$ preserves
the $\Gamma_{0}(N)$-level structure, and such that $\det
f(T)=\prod_{i=1}^{r}(T-t_{i})$ (again up to nonzero scalar). The preservation
of level structure means:

  * •

$b(0)=b(1)=0$, and $\deg b(t)\leq r/2$, if $r$ even,

  * •

$b(0)=d(1)=0$, and $\deg b(t)\leq(r-3)/2$, if $r$ odd.

The action of $\mathbf{T}\times\mathbf{T}$ is by left and right translation on
$f(T)$. This is a quasi-projective variety over $\eta_{r}$ of dimension $r-3$.

######  Example 2.7.1.

[The case $r=3$.] In this case,
$\operatorname{Coinc}^{3}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{3}}\to\eta_{3}$
is a separable double cover of
$\eta_{3}\cong\operatorname{Spec}F(t_{1},t_{2},t_{3})$. For our values of $N$
and $\Sigma_{\infty}$, we have that
$\operatorname{Coinc}^{3}_{\circ}(\Gamma_{0}(N);\left\\{1\right\\})_{\eta_{3}}$
is isomorphic to the 0-dimensional subvariety of $\mathbf{A}^{3}$ in variables
$a,b,d$ over $\eta_{3}=\operatorname{Spec}F(t_{1},t_{2},t_{3})$, which
classifies matrices

$f(T)=\begin{pmatrix}(T-a)&bT\\\ 1&(T-1)(T-d)\end{pmatrix}$

which are singular at $t_{1},t_{2},t_{3}$. This translates into the equations

$(t_{i}-1)(t_{i}-a)(t_{i}-d)-bt_{i}=0,\;i=1,2,3.$

We can eliminate $b$ to obtain two equations for $a,d$, revealing that these
are the roots of the irreducible polynomial

$s^{2}-(t_{1}+t_{2}+t_{3}-1)s+t_{1}t_{2}t_{3}.$ (2.7.1)

Then (2.7.1) is the equation for the double cover
$\operatorname{Coinc}^{3}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{3}}\to\eta_{3}$.
If we think of $t_{1}$ and $t_{2}$ as scalars, then (2.7.1) defines a double
cover of the $t_{3}$-line with discriminant $16t_{1}t_{2}(t_{1}-1)(t_{2}-1)$.
Therefore the double cover can be spread out into a family of double covers
$\mathbf{P}^{1}_{s}\times U^{2}\to\mathbf{P}^{1}_{t_{3}}\times U^{2}$ defined
by the equation (2.7.1), which has the property that it is nonsplit over every
point of $U^{2}$.

######  Example 2.7.2.

[The case $r=4$.] $\operatorname{Coinc}^{4}_{G}(\Gamma_{0}(N))_{\eta_{4}}$ is
isomorphic to the 1-dimensional subvariety of $\mathbf{A}^{5}_{\eta_{4}}$ in
variables $a_{0},a_{1},b,d_{0},d_{1}$ over the base $\eta_{4}\cong
F(t_{1},t_{2},t_{3},t_{4})$, which classifies matrices

$f(T)=\begin{pmatrix}T^{2}+a_{1}T+a_{0}&bT(T-1)\\\
1&T^{2}+d_{1}T+d_{0}\end{pmatrix}$

which are singular at $t_{1},t_{2},t_{3},t_{4}$. The closure of
$\operatorname{Coinc}^{4}_{G}(\Gamma_{0}(N))_{\eta_{4}}$ in
$\mathbf{A}^{5}_{\eta_{4}}$ contains 6 obvious rational points $P_{ij}$, one
for each partition of $\left\\{1,2,3,4\right\\}$ into ${i,j}$ and
${i^{\prime},j^{\prime}}$, defined by

$f_{ij}(T)=\begin{pmatrix}(T-t_{i})(T-t_{j})&0\\\
1&(T-t_{i^{\prime}})(T-t_{j^{\prime}})\end{pmatrix}$

Each of these points is nonsingular. There is also an obvious involution of
$\operatorname{Coinc}^{4}_{G}(\Gamma_{0}(N))_{\eta_{4}}$ which exchanges
$(a_{0},a_{1})$ with $(d_{0},d_{1})$.

The completion of this curve, call it ${\mathcal{C}}$, has genus 1. (One way
to see this is that the projective closure in $\mathbf{P}^{5}$ has a unique
singular point, and projection from this point to $\mathbf{P}^{4}$ gives a
nonsingular curve of genus 1. Another way to see this is to observe that
projection onto $\mathbf{A}^{2}$ via the final coordinates $d_{0},d_{1}$ is an
isomorphism of $\operatorname{Coinc}^{4}_{G}(\Gamma_{0}(N))_{\eta_{4}}$ onto
an affine cubic, whose projective closure is nonsingular.) A Weierstrass
equation for ${\mathcal{C}}$ is

$y^{2}+e_{1}xy=x^{3}+(-e_{2}+e_{3}-2e_{4})x^{2}+(1-e_{1}+e_{2}-e_{3}+e_{4})e_{4}x,$
(2.7.2)

where $e_{1},\dots,e_{4}$ are the elementary symmetric polynomials in
$t_{1},\dots,t_{4}$. The points $P_{ij}$ generate the Mordell-Weil group of
${\mathcal{C}}$, which is isomorphic to
$\mathbf{Z}^{3}\oplus\mathbf{Z}/2\mathbf{Z}$. The points $P_{ij}$ have
coordinates

$P_{ij}:\;\;(t_{i}(t_{i}-1)t_{j}(t_{j}-1),t_{i}(t_{i}-1)t_{j}(t_{j}-1)(-t_{i}t_{j}+t_{i^{\prime}}t_{j^{\prime}}-t_{i^{\prime}}-t_{j^{\prime}}))$

in the model (2.7.2); we remark that $P_{12},P_{13},P_{14}$ are independent.
The involution noted above is translation by the 2-torsion point, $T$ which
has coordinates $(0,0)$ in (2.7.2).

It turns out that ${\mathcal{C}}$ is the universal elliptic curve with this
data, in the following sense.

######  Proposition 2.7.3.

Let ${\mathcal{M}}_{1,4}^{\prime}$ be the moduli space of curves over $k$ of
genus 1 together with 4 marked points and also an element of order 2 in
$\operatorname{Pic}^{0}$. There is an isomorphism of
$\eta=\operatorname{Spec}k(t_{1},t_{2},t_{3},t_{4})$ with the generic point of
${\mathcal{M}}_{1,4}^{\prime}$, such that the pullback of the universal object
is $({\mathcal{C}};O,P_{12},P_{13},P_{14},T)$.

###### Proof.

We assume $\operatorname{{\rm char}}k\neq 2$ for convenience. Let $V$ be the
group $\\{(\pm 1,\pm 1)\\}$ under multiplication, and let $V$ act on
${\overline{\mathcal{M}}}^{\prime}_{1,4}$ by
$(\sigma_{1},\sigma_{2})(E;P_{1},P_{2},P_{3},P_{4};t)=(E;P_{1},P_{2},\sigma_{1}(P_{3}),\sigma_{2}(P_{4});t)$,
where negation is with respect to the origin $P_{1}$. We claim that
${\overline{\mathcal{M}}}^{\prime}_{1,4}/V$ is birational to
$\mathbf{A}^{4}_{k}$. Indeed, an elliptic curve with a point of order $2$ has
a model $y^{2}=x(x^{2}+rx+s)$ which is unique up to replacing $(r,s)$ with
$(\lambda^{2}r,\lambda^{4}s)$. Since negation is an automorphism, $P_{2}$ is
only defined up to sign. Our birational map
${\overline{\mathcal{M}}}^{\prime}_{1,4}/V\dashrightarrow\mathbf{A}^{4}_{k}$
sends $(E;P_{1},P_{2},P_{3},P_{4};t)$ to
$(s/r^{2},x(P_{2}),x(P_{3}),x(P_{4}))\in\mathbf{A}_{k}^{4}$.

On the other hand, the data $({\mathcal{C}},O,P_{12},P_{13},P_{14},T)$ defines
a map $\eta\to{\overline{\mathcal{M}}}^{\prime}_{1,4}$. One calculates [LW]
that the composition
$\eta\to{\overline{\mathcal{M}}}^{\prime}_{1,4}\to{\overline{\mathcal{M}}}^{\prime}_{1,4}/V\dashrightarrow\mathbf{A}_{k}^{4}$
is dominant of degree 4. We conclude that
$\eta\to{\overline{\mathcal{M}}}^{\prime}_{1,4}$ is an isomorphism from $\eta$
onto the generic point of ${\overline{\mathcal{M}}}^{\prime}_{1,4}$. ∎

We can only speculate on the geometry of
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{r}}$ for
general $r$. Let $n_{1},n_{2}$ be the bounds on the degree of $c(T),d(T)$ in
the description we gave above: thus $n_{1}=r/2,(r-1)/2$ and
$n_{2}=r/2-2,(r-3)/2$ as $r$ is even or odd, respectively, and
$n_{1}+n_{2}=r-2$. Then projection onto the pair $(c(T),d(T))$ gives a
birational equivalence between
$\operatorname{Coinc}^{r}_{G}(\Gamma_{0}(N);\Sigma_{\infty})_{\eta_{r}}$ and a
hypersurface $C$ in $\mathbf{P}^{n_{1}}\times\mathbf{P}^{n_{2}}$ of degree
$(n_{1}+1,n_{2}+1)$. Such a hypersurface has trivial canonical bundle, which
led us to guess that $C$ is birational to a (nonsingular) Calabi-Yau variety.
For instance, this holds in the case $r=5$, in which the variety $C$ is
birational to an elliptic K3 surface.

##  3 Isogenies between K3 surfaces

###  3.1 Motivation: $2$-modularity of extremal rational elliptic fibrations

Let $E/\mathbf{F}_{q}(t)$ be a nonisotrivial elliptic curve with degree 4
conductor $N$, with corresponding rational elliptic fibration
$\mathcal{E}\to\mathbf{P}^{1}$. Let $\Sigma$ be the support of $N$, and let
$U=\mathbf{P}^{1}\smallsetminus\Sigma$. Let $\Sigma_{\infty}\subset\Sigma$ be
a subset of even cardinality consisting of places appearing in $N$ with
multiplicity 1. In the introduction we sketched a plan to prove that $E$ is
$(2,\Sigma_{\infty})$-modular. The first phase of the plan was to find a
rational map

$\operatorname{Sht}^{2}_{G}(\Gamma_{0}(N);\Sigma_{\infty})\dashrightarrow{\mathcal{Z}}^{2}(\mathcal{E}),$

where ${\mathcal{Z}}^{2}(\mathcal{E})$ is defined as a certain cartesian
product:

$\textstyle{{\mathcal{Z}}^{2}(\mathcal{E})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\times_{\mathbf{F}_{q}}U^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{P}^{1}\times
U^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{2:1}$$\textstyle{\mathbf{P}^{1}\times
U^{2}}$ (3.1.1)

The lower horizontal arrow is a certain family of nonsplit double covers of
$\mathbf{P}^{1}$ parametrized by $U$, derived from the moduli space of
3-legged coincidences. Now, the base change of a rational elliptic surface
$\mathcal{A}\to\mathbf{P}^{1}$ by a nonsplit double cover
$\mathbf{P}^{1}\to\mathbf{P}^{1}$ is a K3 surface. Thus
${\mathcal{Z}}^{2}(\mathcal{E})\to U^{2}$ is a family of K3 surfaces. As
explained in the introduction, in order to prove that $E$ is 2-modular, it
suffices to find a finite morphism
${\mathcal{Z}}^{2}(\mathcal{E})\to\operatorname{Km}(\mathcal{E}^{2})$
commuting with the maps to $U^{2}$. This is what we accomplish in this
section:

######  Theorem 3.1.1.

There exists a finite morphism
${\mathcal{Z}}^{2}(\mathcal{E})\to\operatorname{Km}(\mathcal{E}^{2})$
commuting with the maps to $U^{2}$.

######  Remark 3.1.2.

Theorem 3.1.1 can be stated in any characteristic, since the definition of the
space of coincidences is independent of characteristic. In the cases where
$\mathcal{E}\to\mathbf{P}^{1}$ is unstable our formulas confirm Theorem 3.1.1
in any characteristic.

Generally, it is an interesting problem to give sufficient conditions for two
K3 surfaces to be isogenous, see [Muk87], [Bus19], [Yan22]. A necessary
condition is some relation between the Picard lattices of the two surfaces.
Naturally one wants to know if this is sufficient:

######  Question 3.1.3.

Let $S$ be a K3 surface over an algebraically closed field. Let $\Lambda$ be a
lattice, such that $\operatorname{Pic}S\otimes\mathbf{Q}$ and
$\Lambda\otimes\mathbf{Q}$ are isometric. Does there exist a K3 surface
$S^{\prime}$ in correspondence with $S$ such that
$\operatorname{Pic}S^{\prime}\cong\Lambda$?

For many values of $\Lambda$, operations on elliptic fibrations can be used to
construct the correspondence. The main theorem of this section answers a
special case of Question 3.1.3.

######  Theorem 3.1.4.

Let $S$ be a K3 surface over an algebraically closed field. Suppose there is
an isometry
$\operatorname{Pic}S\otimes\mathbf{Q}\cong\operatorname{Pic}K\otimes\mathbf{Q}$,
where $K=\operatorname{Km}(E_{1}\times E_{2})$ for two non-isogenous elliptic
curves $E_{1}$ and $E_{2}$. Then there is a morphism of finite degree from $S$
to a Kummer surface of this form.

###  3.2 Elliptic fibrations on K3 surfaces

We begin with a few remarks on elliptic fibrations and especially elliptic K3
surfaces. Some standard references are [SS10] for elliptic fibrations and
[Huy16] for K3 surfaces. Unless stated otherwise, by “surface” we mean a
smooth projective surface over an algebraically closed field.

####  3.2.1 Generalities on elliptic surfaces

Let $S$ be a surface. A genus 1 fibration on $S$ is a morphism $S\to X$ to a
curve whose geometric generic fiber is a smooth integral curve of genus 1. If
$S\to X$ admits a section, we call it an elliptic fibration. In any case it
always has a multisection, meaning a curve $C\subset S$ such that $C\to X$ is
finite; the degree of the multisection is the degree of $C\to X$. An elliptic
surface is a surface admitting an elliptic fibration, such that no fiber
contains an exceptional curve (meaning a smooth rational curve of self-
intersection $-1$).

For a surface $S$, the Néron-Severi group $\operatorname{NS}(S)$ is the group
of divisors modulo algebraic equivalence. It is a finitely generated torsion-
free abelian group endowed with an intersection pairing, which we write as
$(v,w)$. This pairing is nondegenerate on the torsion-free part of
$\operatorname{NS}(S)$. We write $\rho(S)$ for the rank of the torsion-free
part of $\operatorname{NS}(S)$. If $S\to X$ is an elliptic fibration with
general fiber $F$ and section $O$, the trivial lattice
$T\subset\operatorname{NS}(S)$ is generated by $O$ and all irreducible
components of fibers of $S\to X$. Suppose the reducible fibers occur at
$R\subset X$. For $v\in R$, let
$x_{0},\dots,x_{m_{v}-1}\in\operatorname{NS}(S)$ be the irreducible components
of the fiber $S_{v}$, where $x_{0}$ is the unique such component which meets
$O$. As all fibers are equivalent to $F$, the trivial lattice $T$ is the
orthogonal direct sum of $\left<O,F\right>$ and
$\left<x_{1},\dots,x_{m_{v}-1}\right>$.

The isomorphism classes of singular fibers of an elliptic surface have a well-
known classification by Kodaira symbol. For each such fiber, the corresponding
summand of $T$ is a root lattice of type $A_{n}$, $D_{n}$ $(n\geq 4)$,
$E_{6}$, $E_{7}$, or $E_{8}$. Here our convention is that our root lattices
are negative definite. We will make use of the standard facts about each
Kodaira symbol without comment, e.g., the corresponding Dynkin diagram,
multiplicity of components, discriminant, etc.

There are two useful formulas relating the geometry of an elliptic surface $S$
to the singular fibers of an elliptic fibration on $S$. One is the Shioda-Tate
formula, and the other is a formula for the Euler number of $S$.

The Mordell-Weil group of an elliptic fibration $S\to X$ is the group of
sections $S(X)$. Equivalently it is the set of rational points of the generic
fiber of $S\to X$, which is an elliptic curve. There is an isomorphism

$NS(X)/T\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}S(X)$
(3.2.1)

sending a divisor to its generic fiber. In particular we have the Shioda-Tate
formula [SS10, Proposition 6.6]:

$\rho(S)=2+\sum_{v\in R}(m_{v}-1)+\operatorname{rank}S(X)/(\text{torsion})$
(3.2.2)

The smooth locus in each singular fiber $S_{v}$ of an elliptic fibration has a
group structure, with neutral component either $\mathbf{G}_{m}$ (type $I_{n}$)
or $\mathbf{G}_{a}$ (all other types). We call $S_{v}$ a multiplicative or
additive fiber, respectively. For additive fiber $v$, let $\delta_{v}$ be the
index of wild ramification, defined by the formula:

$\delta_{v}=v(\Delta)-1-m_{v}$

where $\Delta$ is the discriminant of a Weierstrass equation for $S$ over the
local ring $\mathcal{O}_{X,v}$. Then $\delta_{v}\neq 0$ only if the ground
field has characteristic 2 or 3. If $\delta_{v}=0$ for all $v$, we say that
$S\to X$ is tame.

For a projective variety $Y$, write $e(Y)$ for the Euler number (= topological
Euler characteristic = alternating sum of Betti numbers of $Y$). For a
singular fiber $S_{v}$ of an elliptic fibration $S\to X$, we have

$e(S_{v})=\begin{cases}m_{v},&\text{if $S_{v}$ is multiplicative}\\\
m_{v}+1,&\text{if $S_{v}$ is additive.}\end{cases}$ (3.2.3)

We have the following relations:

$\chi(S,\mathcal{O}_{S})=\frac{1}{12}e(S)$ (3.2.4)

(where $\chi$ = Euler characteristic), and

$e(S)=\sum_{v}(e(S_{v})+\delta_{v}),$ (3.2.5)

where the sum is over the singular fibers of $S\to X$.

####  3.2.2 K3 elliptic surfaces

A smooth projective surface $S$ is K3 if it has trivial canonical bundle and
$H^{1}(S,\mathcal{O}_{S})=0$.

Let $S$ be a K3 surface. Then on $S$, the notions of linear, algebraic, and
numerical equivalence of divisors agree, and
$\operatorname{Pic}S\cong\operatorname{NS}(S)$ is torsion-free. We refer to
this group as the Picard lattice. It has signature $(1,\rho(S)-1)$. By
Riemann-Roch, an irreducible curve $C\subset S$ of arithmetic genus $g$ has
$(C,C)=2g-2$. This implies that $\operatorname{Pic}S$ is an even lattice.

In the case that $k=\mathbf{C}$ is the field of complex numbers, then
$\operatorname{Pic}S$ is a primitive sublattice of $H^{2}(S,\mathbf{Z})$. For
its part, $H^{2}(S,\mathbf{Z})$ is isomorphic to the K3 lattice
$E_{8}^{2}\oplus U^{3}$, where $U$ is the hyperbolic plane. By the Lefschetz
principle, this fact about $\operatorname{Pic}S$ extends to any field of
characteristic 0. In positive characteristic, we can use the following lemma.

######  Proposition 3.2.1 ([Huy16, Proposition 5.8]).

Let $X$ be a K3 surface of finite height over a perfect field $k$ of
characteristic $p$. Then $X$ can be lifted to $W(k)$ in such a way that the
specialization map from the Néron-Severi group of the general to the special
fibre is an isomorphism.

We are interested in those K3 surfaces which admit genus 1 fibrations. If $S$
is K3 and $S\to X$ is a genus 1 fibration, then necessarily
$X\cong\mathbf{P}^{1}$. An interesting fact about elliptic K3 surfaces $S$ is,
if their Picard rank is not too small, they typically admit several genus 1
fibrations which are distinct modulo the action of $\operatorname{Aut}S$. It
will be important for us to recognize the genus 1 fibrations on $S$ purely by
examining the lattice $\operatorname{Pic}S$.

######  Proposition 3.2.2 ([Pvv71, §3, Theorem 1]).

Assume that the characteristic of $k$ is not 2 or 3. Let $S$ be a K3 surface.

  1. 1.

Suppose $F$ is a primitive class in $\operatorname{Pic}S$ with $(F,F)=0$. Then
there exists a genus 1 fibration $S\to\mathbf{P}^{1}$ whose fiber class is
$\O(\operatorname{Pic}S)$-equivalent to $F$.

  2. 2.

Continuing, let $d$ be the greatest common divisor of $(F,D)$ for all
$D\in\operatorname{Pic}S$. Then $S\to\mathbf{P}^{1}$ admits a multisection of
degree $d$.

  3. 3.

Continuing further, suppose $d=1$, so that $S\to\mathbf{P}^{1}$ is an elliptic
fibration with fiber $F$ and section $O$. Then $\left<F,O\right>$ is the
hyperbolic plane. The reducible fibers of $S\to\mathbf{P}^{1}$ correspond
exactly to the root lattice summands of the root sublattice of
$\left<F,O\right>^{\perp}$. (The root sublattice is the sublattice spanned by
vectors $v$ with $(v,v)=-2$.)

###### Proof.

(1) One can apply standard results on $\operatorname{Pic}S$ [Huy16, Corollary
8.2.9] to find a sequence of smooth rational curves $C_{1},\dots,C_{r}$ on $S$
such that $F^{\prime}=\pm\rho_{C_{r}}\circ\cdots\circ\rho_{C_{1}}(F)$ is
numerically effective (nef), where $\rho_{C}(v)=D-(2(v,C)/(C,C))C$ is the
reflection about the line spanned by a curve $C\subset S$. Since each $C_{i}$
is a smooth rational curve, we have $(C_{i},C_{i})=-2$ and consequently
$(F^{\prime},F^{\prime})=0$. By [Huy16, Proposition 3.10], the linear system
$\left\lvert F^{\prime}\right\rvert$ has no base points, and the associated
morphism to projective space factors through a genus 1 fibration
$S\to\mathbf{P}^{1}$, with fiber equivalent to $F$.

(2) It is clear that $F^{\prime}\equiv F\pmod{d}$ in $\operatorname{Pic}S$, so
that there exists $D\in\operatorname{Pic}S$ with $(F^{\prime},D)=d$. After
replacing $D$ with $D+nF^{\prime}$ for $n\gg 0$, we may assume that $D$ is
effective. Then $D$ is a multisection of the fibration of degree $d$.

(3) Let ${\mathcal{C}}\subset\left<F,O\right>$ be the set of $C$ that satisfy
$(C,C)=-2$. By Riemann-Roch either $C$ or $-C$ is effective, so let
${\mathcal{C}}^{+}$ be the set of effective elements. We claim that every
element of ${\mathcal{C}}^{+}$ is a sum of classes of curves contained in
fibers.

Indeed, write $C\in{\mathcal{C}}^{+}=\sum a_{i}[C_{i}]$, where the $a_{i}$ are
nonnegative integers and the $C_{i}$ are classes of irreducible curves. Since
$[F]$ moves with empty base locus, it has nonnegative intersection with all
curves, and positive intersection with all curves not contained in any member
of the linear series $|F|$. In particular $O$ is not among the $C_{i}$,
because $[F]\cdot[O]=1$. Therefore $O\cdot C_{i}\geq 0$ for all $i$, and so
$F$ is not one of the $C_{i}$ either. On the other hand it is clear that if
$C_{i}$ is a curve in a fibre that does not meet $O$, then
$[C_{i}]\in{\mathcal{C}}$.

Let $R$ be the root sublattice of $\left<F,O\right>$. From the above we see
that $R$ is freely generated by the classes of curves in fibers that do not
meet $O$. For each reducible fibre we obtain a root lattice as in [SS10,
Corollary 5.17], and distinct fibers give orthogonal sublattices of $R$. ∎

######  Corollary 3.2.3.

Let $S$ be a K3 surface. If the Picard rank $\rho(S)$ is at least 5, then $S$
admits a genus 1 fibration.

###### Proof.

The quadratic form of $\operatorname{Pic}S$ is indefinite and represents zero
over $\mathbf{Q}_{p}$ for all primes $p$. Therefore by Hasse-Minkowski it
represents zero rationally; i.e., there exists $F\in\operatorname{Pic}S$ with
$(F,F)=0$. By Proposition 3.2.2, the surface $S$ admits a genus 1 fibration. ∎

######  Remark 3.2.4.

If the characteristic of $k$ is $2$ or $3$, it is possible that a nef divisor
with self-intersection $0$ does not actually define a genus 1 fibration in the
sense we have defined it; it can happen that the generic fibre has a cusp. See
[Huy16, proof of Proposition 3.10].

In the situation of Proposition 3.2.2, we have a genus 1 fibration on $S$
admitting a multisection $D$ of degree $d$. We may pass to the Jacobian of
this fibration [Huy16, Chapter 11, §4]: this is another K3 surface
$S^{\prime}$ admitting a Jacobian fibration. There is a finite morphism $S\to
S^{\prime}$ of degree $d^{2}$, sending a section $P$ to $dP-v$. The following
lemma identifies the Picard lattice of $S^{\prime}$.

######  Lemma 3.2.5 ([Keu00, Lemma 2.1]).

Let $S$ be a K3 surface admitting a genus 1 fibration of multisection index
$d$ with fiber $F\in\operatorname{Pic}S$, and let $S^{\prime}$ be the Jacobian
of that fibration. Then $\operatorname{Pic}S^{\prime}$ is isomorphic to the
overlattice $(\operatorname{Pic}S)[F/d]$ of $\operatorname{Pic}S$.

It is natural to ask about a sort of converse to Lemma 3.2.5. Suppose $S$ is a
K3 surface with Picard lattice $L$, and suppose $L^{\prime}\supset L$ is an
overlattice with $L^{\prime}/L$ cyclic of degree $d$. Does there exist a genus
1 fibration on $S$ whose Jacobian has Picard lattice $L^{\prime}$? This is
true if the rank of $L$ is at least 13, as we will see in Proposition 3.2.8.

######  Lemma 3.2.6.

Let $R$ be a principal ideal domain, let $M$ be a free $R$-module of finite
rank, and let $Q$ be a quadratic form on $M$. Let $N\subset M$ be a submodule
such that the discriminant of $Q|_{N}$ is a unit in $R$. Then $M=N\oplus
N^{\perp}$.

###### Proof.

First note that $N\cap N^{\perp}=0$, as otherwise $Q|_{N}$ would have
discriminant $0$.

Let $F$ be the fraction field of $R$, and let $N^{\vee}\subset M\otimes_{R}F$
be the dual lattice, i.e., the $R$-submodule of $M\otimes_{R}F$ consisting of
those $m$ for which $(m,N)\subset R$. Then $\operatorname{Hom}(N,R)\cong
N^{\vee}$. By hypothesis, we have $N=N^{\vee}$. Let $m\in M$. Then we can find
$n\in N$ such that $(m,x)=(n,x)$ for all $x\in N$. Then $m-n\in N^{\perp}$,
and we conclude $N+N^{\perp}=M$. ∎

In the context of Lemma 3.2.6, we say that $Q$ primitively represents $a\in R$
if there exists a primitive element $x\in M$ with $Q(x)=a$.

######  Lemma 3.2.7.

Let $p$ be prime and let $Q$ be a quadratic form over $\mathbf{Z}_{p}$ of rank
$n\geq 3$ and unit discriminant. If $p=2$, assume further that $Q$ is even.
Then $Q$ represents all elements of $\mathbf{Z}_{p}$ primitively.

###### Proof.

Suppose $p$ is odd. Then $Q$ may be diagonalized [CS99, p. 369, Theorem 2]:
$Q=\sum_{i=1}^{n}a_{i}x_{i}^{2}$. Since $Q$ has unit discriminant, each
$a_{i}\in\mathbf{Z}_{p}^{\times}$. A standard counting argument shows that
$a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}$ primitively represents all
elements of $\mathbf{F}_{p}$. By Hensel’s lemma, $Q$ primitively represents
all of $\mathbf{Z}_{p}$.

If $p=2$, then (loc. cit.) the form $Q$ is isomorphic to the direct sum of a
diagonal form $\sum_{i=1}^{m}a_{i}x_{i}^{2}$ with forms of the shape
$2^{r}Q_{a,b,c}$, where $r\geq 1$ and $Q_{a,b,c}(x,y)=ax^{2}+bxy+cy^{2}$, with
$a,c\in\mathbf{Z}_{2}$ and $b\in\mathbf{Z}_{2}^{\times}$. Note that
$\operatorname{disc}2^{r}Q_{a,b,c}$ has 2-adic valuation $2r$. Since $Q$ has
unit discriminant, all $a_{i}\in\mathbf{Z}_{2}^{\times}$, and all the $r$ are
$0$. But then since $Q$ is even, $m=0$.

Thus $Q$ is the direct sum of at least two forms $Q_{a,b,c}$. For each of
these, we consider two cases. Alternatively, if $2|ac$ then the discriminant
$b^{2}-4ac$ is congruent to $1\bmod 8$. It is therefore the square of a unit,
and so $Q$ is a product of two linear factors
$(rx+sy),(r^{\prime}x+s^{\prime}y)$ that generate the space of linear forms in
two variables over $\mathbf{Z}_{2}$. So $Q$ represents every element of
$\mathbf{Z}_{2}$ primitively in this case. If $a,c\in\mathbf{Z}_{2}^{\times}$,
then $Q_{a,b,c}$ defines a nonsingular conic over $\mathbf{Z}_{2}$, by
Hensel’s lemma $Q_{a,b,c}$ primitively represents all of
$\mathbf{Z}_{2}^{\times}$. Since every element of $\mathbf{Z}_{2}$ is either a
unit or the sum of two units, it follows that $Q$ primitively represents all
of $Z_{2}$ if there are two factors of this type. ∎

######  Proposition 3.2.8.

Let $S$ be a K3 surface of Picard number at least $13$. Assume that the ground
field has characteristic 0, or else that $S$ has finite height. Let $p$ be
prime, and let $D$ be a divisor of $S$ which is not divisible by $p$ in
$L=\operatorname{Pic}S$. Assume that $p|(D,x)$ for all $x\in L$, and also that
$p^{2}|(D,D)$. Equivalently,
$L^{\prime}=L[D/p]\subset\operatorname{Pic}S\otimes\mathbf{Q}$ is an
overlattice of $L$ of index $p$. If $p=2$, assume that $L^{\prime}$ is even.
(Since $L$ is even, this condition is equivalent to $8|(D,D)$.)

Then there exists a divisor class $D^{\prime}\in D+pL$ such that the $p$-part
of $\gcd_{x\in L}(D^{\prime},x)$ is $p$, and a genus 1 fibration
$S\to\mathbf{P}^{1}$ with fiber $D^{\prime}$. In particular, if
$S^{\prime}\to\mathbf{P}^{1}$ is the $p$-Jacobian $J^{p}$ ([Huy16, Remark
11.4.1])) of the fibration, then $\operatorname{Pic}S^{\prime}\cong
L^{\prime}$.

###### Proof.

Using Proposition 3.2.1 we assume that the ground field is $\mathbf{C}$.

Let $M\subset\operatorname{Pic}S\otimes\mathbf{Z}_{p}$ be a
$\mathbf{Z}_{p}$-sublattice of maximal rank among those which have unit
discriminant. Write $Q$ for the quadratic form on
$\operatorname{Pic}S\otimes\mathbf{Z}_{p}$. By Lemma 3.2.6, we have
$\operatorname{Pic}S\otimes\mathbf{Z}_{p}=M\oplus N$, with $N=M^{\perp}$. Then
$Q|_{N}$ is divisible by $p$. Indeed, if $x\in N$ satisfies
$Q(x)\in\mathbf{Z}_{p}^{\times}$, then $M\oplus\left<x\right>$ would be a
larger submodule with unit discriminant. In fact we claim that the
intersection form on $N$ is divisible by $p$, i.e., that $N\subset pN^{\vee}$.
This follows automatically if $p$ is odd. If $p=2$, we also need to exclude
the possibility of $x,x^{\prime}\in N$ with
$(x,x^{\prime})\in\mathbf{Z}_{2}^{\times}$. But since we already know that
$(x,x),(x^{\prime},x^{\prime})\in\mathbf{Z}_{2}^{\times}$, we have
$(x,x)(x^{\prime},x^{\prime})-(x,x^{\prime})^{2}\in\mathbf{Z}_{2}^{\times}$,
and again we have constructed a larger submodule
$M\oplus\left<x,x^{\prime}\right>$ with unit discriminant.

Recall that the discriminant group of a lattice $\Lambda$ (over whatever PID
base) is $D(\Lambda)=\Lambda^{\vee}/\Lambda$. Then
$D(L)\otimes\mathbf{Z}_{p}\cong D(L\otimes\mathbf{Z}_{p})$. On the other hand
$L\otimes\mathbf{Z}_{p}=M\oplus N$ as above, and we have $D(M)=0$, whereas

$D(N)\otimes\mathbf{F}_{p}=N^{\vee}/(N,pN^{\vee})=N^{\vee}/pN^{\vee}=N^{\vee}\otimes\mathbf{F}_{p}$

since $N\subset pN^{\vee}$ as we observed above. We conclude from this that
$\dim_{\mathbf{F}_{p}}D(L)\otimes\mathbf{F}_{p}=\operatorname{rank}N$.

Now we use that fact that $L=\operatorname{Pic}S=\operatorname{NS}(S)$ embeds
primitively into the K3 lattice $H=H^{2}(X,\mathbf{Z})$. (This is because for
any complex projective surface $S$, we have $\operatorname{NS}(S)\cong
H^{1,1}(S,\mathbf{C})\cap H^{2}(S,\mathbf{Z})$.) Crucially, $H$ is unimodular.
Let $L^{\perp}$ be the orthogonal complement of $L$ in $H$. The intersection
pairing on $H$ induces a pairing $D(L)\otimes H/(L\oplus
L^{\perp})\to\mathbf{Q}/\mathbf{Z}$.

This pairing puts $D(H)$ and $H/(L\oplus L^{\perp})$ into Pontrjagin duality.
Proof: if $h\in H$ satisfies $(L^{\vee},h)\subset\mathbf{Z}$, write
$h=\ell+\ell^{\perp}$, where $\ell\in L\otimes\mathbf{Q}$, $\ell^{\perp}\in
L^{\perp}\otimes\mathbf{Q}$. Then $(L^{\vee},\ell)\subset\mathbf{Z}$ implies
$\ell\in L^{\vee\vee}=L$ and therefore $\ell^{\perp}\in
H\cap(L^{\perp}\otimes\mathbf{Q})=L^{\perp}$ and $h\in L\oplus L^{\perp}$.
Conversely if $\ell^{\vee}\in L^{\vee}$ satisfies
$(\ell^{\vee},H)\in\mathbf{Z}$, then $\ell^{\vee}\in H^{\vee}\cap
L^{\vee}=H\cap L^{\vee}\in H\cap(L\otimes\mathbf{Q})=L$ because $L$ is
embedded primitively in $H$.

In particular $\dim_{\mathbf{F}_{p}}D(L)\otimes\mathbf{F}_{p}=\dim(H/(L\oplus
L^{\perp})\otimes\mathbf{F}_{p}\leq\operatorname{rank}L^{\perp}=22-\operatorname{rank}L$,
as one can see by extending a $\mathbf{Z}$-basis of $L$ to $H$. Thus
$\operatorname{rank}N\leq 22-\operatorname{rank}L$.

We have assumed $\rho(S)=\operatorname{rank}L\geq 13$, and so

$\operatorname{rank}M=\operatorname{rank}L-\operatorname{rank}N\geq
2\operatorname{rank}L-22\geq 4.$

Our divisor $D$ was assumed to satisfy $p|(D,x)$ for all $x\in L$. If we
decompose $D$ as $D=m+n$ with $m\in M$, $n\in N=M^{\perp}$, then $p|(m,x)$ for
all $x\in M$, which is to say, $m\in pM^{\vee}$. But since $M$ is unimodular,
we find that $m\in pM$. We find that $n\equiv D\pmod{L\otimes
p\mathbf{Z}_{p}}$. We have also assumed that $L[D/p]$ is a lattice, which
implies that $p^{2}|(D,D)$; if $p=2$ we have assumed $8|(D,D)$. The same
statements are true when $D$ is replaced with $n$. By Lemma 3.2.7, we can find
a vector $x\in M$ such that $(x,x)=-(n,n)/p^{2}$. Then $D_{0}=n+px\in
L\otimes\mathbf{Z}_{p}$ satisfies $D_{0}\equiv D\pmod{p}$ and
$(D_{0},D_{0})=0$.

By weak approximation on quadric hypersurfaces, there exists $D^{\prime}\in
L\otimes\mathbf{Q}$ satisfying $(D^{\prime},D^{\prime})=0$ which is
$p$-adically close to $D_{0}$. After clearing denominators, we may assume
$D^{\prime}\in L$. Now we may apply Proposition 3.2.2 to obtain the required
genus 1 fibration.

∎

######  Corollary 3.2.9.

Let $S$ be an ordinary K3 surface of Picard number $\geq 13$, and let $p$ be
an odd prime. Let $d=\operatorname{disc}\operatorname{Pic}S$, and let $p^{v}$
be the largest power of $p$ dividing $d$. Suppose we have an isometry
$\operatorname{Pic}S\otimes\mathbf{Q}_{p}\cong L\otimes\mathbf{Q}_{p}$, where
$L$ is a unimodular $\mathbf{Z}_{p}$-lattice. Then there is a map of finite
degree from $S$ to a K3 surface $S^{\prime}$ for which
$\operatorname{disc}\operatorname{Pic}S^{\prime}=d/p^{v}$.

###### Proof.

The hypothesis on $\operatorname{Pic}S$ implies that $v$ is even, as the
discriminant of a quadratic form is well-defined up to a square. It also
implies that the Hasse invariant of $\operatorname{Pic}S\otimes\mathbf{Q}_{p}$
is trivial, as this is the case for $L\otimes\mathbf{Q}_{p}$. Indeed, we may
diagonalize the quadratic form on $L$ as $\sum_{i=1}^{n}a_{i}x_{i}^{2}$, with
$a_{i}\in\mathbf{Z}_{p}^{\times}$, and then the Hasse invariant is
$\prod_{i<j}(a_{i},a_{j})$ (Hilbert symbol). Each factor in the product is
trivial (note that $p$ is odd).

We proceed by induction on $v$, the case $v=0$ being obvious. First suppose
that $D(\operatorname{Pic}S)$ contains an element of order $p^{2}$,
represented by $D^{\vee}\in(\operatorname{Pic}S)^{\vee}$. Then
$D=p^{2}D^{\vee}\in\operatorname{Pic}S$ satisfies the hypotheses of
Proposition 3.2.8, in which case there is a degree $p^{2}$ map $S\to
S^{\prime}$ with $\operatorname{disc}\operatorname{Pic}S^{\prime}=d/p^{2}$.

Therefore assume that $D(\operatorname{Pic}S)\otimes\mathbf{Z}_{p}$ is
$p$-torsion, in which case its $\mathbf{F}_{p}$-dimension is $v$. The
quadratic form on $\operatorname{Pic}S\otimes\mathbf{Z}_{p}$ is equivalent to
the diagonal form

$pa_{1}x_{1}^{2}+\cdots+pa_{v}x_{v}^{2}+a_{v+1}x_{v+1}^{2}+\cdots+a_{n}x_{n}^{2},$
(3.2.6)

with each $a_{i}\in\mathbf{Z}_{p}^{\times}$. If $v>2$, there exist
$x_{1},x_{2},x_{3}\in\mathbf{Z}_{p}$ such that
$a_{1}x_{1}^{2}+a_{2}x_{2}^{2}+a_{3}x_{3}^{2}\equiv 0\pmod{p}$ since every
nonsingular conic over $\mathbf{F}_{p}$ has a rational point. Let
$D\in\operatorname{Pic}S$ be $p$-adically close to
$(x_{1},x_{2},x_{3},0,\dots)\in\operatorname{Pic}S\otimes\mathbf{Z}_{p}$. Then
$D$ satisfies the hypotheses of Proposition 3.2.8, and again we can remove a
factor of $p^{2}$ from $\operatorname{disc}\operatorname{Pic}S$.

We are now reduced to the case $v=2$. Standard formulas for the Hilbert symbol
reveal that $(pa_{1},a_{i})=(pa_{2},a_{i})$ for all $i\geq 3$, so that the
Hasse invariant of (3.2.6) is $(pa_{1},pa_{2})=(-a_{1}a_{2}/p)$ (Legendre
symbol). But by the observation in the first paragraph, the Hasse invariant is
trivial; i.e., $-a_{1}a_{2}$ is a square modulo $p$. Thus
$a_{1}x_{1}^{2}+a_{2}x_{2}^{2}$ represents 0 modulo $p$, and we proceed as in
the previous paragraph. ∎

###  3.3 Kummer surfaces $\operatorname{Km}(E_{1}\times E_{2})$ of Picard rank
18

Here we recall the basic constructions and properties of the Kummer surface
$\operatorname{Km}(A)$ attached to an abelian surface $A$. For the moment let
us suppose we are in characteristic not 2. Denoting by $\iota$ the involution
$x\mapsto-x$ on $A$, the quotient $A/\iota$ has rational double point
singularities at each of the 16 fixed points of $\iota$ (namely, the 2-torsion
points of $A$). Let $\operatorname{Km}(A)$ be the minimal resolution of
$A/\iota$. Then $\operatorname{Km}(A)$ is a K3 surface.

The Picard lattice
$\operatorname{Pic}\operatorname{Km}(A)=\operatorname{NS}\operatorname{Km}(A)$
contains both $\operatorname{NS}(A)$ and the classes of the 16 exceptional
divisors. In fact these generate $\operatorname{Pic}\operatorname{Km}(A)$ and
we have:

$\rho(\operatorname{Km}(A))=16+\rho(A)$ (3.3.1)

We are mostly interested in the case that $A=E_{1}\times E_{2}$ is a product
of non-isogenous elliptic curves $E_{1},E_{2}$. In this case, (3.3.1) gives
$\rho(\operatorname{Km}(A))=18$. There is an elliptic fibration
$\operatorname{Km}(E_{1}\times E_{2})\to\mathbf{P}^{1}$ given by projecting
onto $E_{2}/\iota\cong\mathbf{P}^{1}$; the geometric fibers are all isomorphic
to $E_{1}$. In fact this gives an alternate construction of
$\operatorname{Km}(E_{1}\times E_{2})$: it is the quadratic twist of the
constant elliptic fibration $E_{1}\times\mathbf{P}^{1}\to\mathbf{P}^{1}$ by
the double cover $E_{2}\to\mathbf{P}^{1}$. (Of course, we could have reversed
the roles of $E_{1}$ and $E_{2}$ in this construction.)

The bad fibers of $\operatorname{Km}(E_{1}\times E_{2})\to\mathbf{P}^{1}$
occur at the four branch points of $E_{2}\to\mathbf{P}^{1}$, and they are all
of type $I_{0}^{\ast}=\tilde{D}_{4}$. Therefore the trivial lattice of the
fibration is $U\oplus D_{4}^{\oplus 4}$. We note here that
$\operatorname{disc}D_{4}=4$. On the other hand the Mordell-Weil group of this
fibration has order 4 (coming from the 2-torsion in $E_{1}$). We conclude from
(3.2.1) that $\operatorname{Pic}\operatorname{Km}(E_{1}\times E_{2})$ has
discriminant $-16$. The lattice
$\operatorname{Pic}\operatorname{Km}(E_{1}\times E_{2})$ does not depend on
the elliptic curves $E_{1}$ or $E_{2}$, so long as they are nonisogenous. In
the complex setting, the transcendental lattice of
$\operatorname{Km}(E_{1}\times E_{2})$ is $T_{E_{1}\times
E_{2}}(2)\cong(H^{1}(E_{1},\mathbf{Z})\otimes H^{1}(E_{2},\mathbf{Z}))(2)\cong
U^{\oplus 2}(2)$.

We are interested in the question of whether a given K3 surface $S$ of rank 18
is isogenous to a Kummer surface of the form $\operatorname{Km}(E_{1}\times
E_{2})$, and if so, how to find the elliptic curves $E_{1}$ and $E_{2}$. The
following result relies on a deep theorem of Mukai.

######  Proposition 3.3.1.

Let $S$ be a complex K3 surface such that
$\operatorname{Pic}S\otimes\mathbf{Q}$ is isometric to
$\operatorname{Pic}\operatorname{Km}(E_{1}\times E_{2})\otimes\mathbf{Q}$ for
some nonisogenous elliptic curves $E_{1},E_{2}$. Then there exists an isogeny
between $S$ and a Kummer surface of that form. (An isogeny between K3 surfaces
$S$ and $S^{\prime}$ is an algebraic cycle on $S\times S^{\prime}$ inducing an
isometry
$H^{2}(S,\mathbf{Q})\stackrel{{\scriptstyle\sim}}{{\smash{\longrightarrow}\rule{0.0pt}{1.72218pt}}}H^{2}(S^{\prime},\mathbf{Q})$.)

###### Proof.

(Sketch.) Let $T_{S}$ be the transcendental lattice of $S$. Then $T_{S}$ has
signature $(2,-2)$, and is the complement of
$\operatorname{Pic}\operatorname{Km}(E_{1}\times E_{2})$ in
$H^{2}(S,\mathbf{Z})$. A calculation involving Hasse-Minkowski invariants
shows that $T_{S}\otimes\mathbf{Q}$ is isometric to $U^{\oplus
2}\otimes\mathbf{Q}$. Via this isometry, the Hodge structure on $T_{S}$ now
determines a Hodge structure on $U^{\oplus 2}$, which is to say, a morphism of
real groups $h\colon\mathbf{S}\to O(2,2)$, where $\mathbf{S}$ is the Deligne
torus. Now observe that there is an isomorphism $(g_{1},g_{2})\mapsto
g_{1}\otimes g_{2}$ from
$(\operatorname{Sp}(2)\times\operatorname{Sp}(2))/\left\\{\pm 1\right\\}$ onto
the neutral component of $O(2,2)$. Therefore $h$ factors through a morphism
$\mathbf{S}\to(\operatorname{Sp}(2)\times\operatorname{Sp}(2))/\left\\{\pm
1\right\\}$. For $i=1,2$, let $h_{i}$ be the projection of this morphism onto
the $i$th copy of $\operatorname{Sp}(2)/\left\\{\pm
1\right\\}\cong\operatorname{PGL}_{2}$; then $h_{i}$ determines an elliptic
curve $E_{i}$. Thus we have an isometry of rational Hodge structures:

$T_{S}\otimes\mathbf{Q}\cong H^{1}(E_{1},\mathbf{Q})\otimes
H^{1}(E_{2},\mathbf{Q})\cong T_{\operatorname{Km}(E_{1}\times
E_{2})}\otimes\mathbf{Q}$

We now invoke a theorem of Mukai [Muk87, Corollary 1.10]: an isometry between
rational Hodge structures of two K3 surfaces of rank $\geq 11$ is always
induced from an isogeny. ∎

In this section we prove an effective version of Proposition 3.3.1, whereby
the elliptic curves $E_{1}$ and $E_{2}$ can in theory be computed from $S$,
and where the isogeny is simply a finite rational map (which can also be
computed). The idea is to leverage operations on elliptic fibrations. We build
up the result in stages.

######  Proposition 3.3.2.

Let $S$ be a K3 surface whose Picard lattice is isometric to
$\operatorname{Km}(E_{1}\times E_{2})$ for some nonisogenous elliptic curves
$E_{1},E_{2}$. Then $S$ is isomorphic to a Kummer surface of that form.

###### Proof.

Proposition 3.2.2 produces an elliptic fibration $S\to\mathbf{P}^{1}$ with
$I_{0}^{\ast}$ fibers at four points of $\mathbf{P}^{1}$. Let
$E_{2}\to\mathbf{P}^{1}$ be the elliptic curve branched at exactly these four
points. Then the quadratic twist of $S\to\mathbf{P}^{1}$ by
$E_{2}\to\mathbf{P}^{1}$ has $I_{0}$ reduction at these points; i.e., it has
good reduction everywhere and therefore must be a constant elliptic curve
$E_{1}\times\mathbf{P}^{1}$. Thus $S$ is a quadratic twist of
$E_{1}\times\mathbf{P}^{1}$ by $E_{2}\to\mathbf{P}^{1}$, and this is exactly
$\operatorname{Km}(E_{1}\times E_{2})$. ∎

######  Remark 3.3.3.

Recall that a Shioda-Inose structure ([Mor84, Definition 6.1]) on a complex K3
surface $S$ is a rational map $S\to S^{\prime}$ of degree $2$ such that
$S^{\prime}$ is a Kummer surface and $T_{S^{\prime}}\cong 2T_{S}$, meaning
that that the Gram matrix of $T_{S^{\prime}}$ is twice that of $T_{S}$.
Morrison proves [Mor84, Theorem 6.3] that $S$ has a Shioda-Inose structure if
and only if $\operatorname{Pic}S\otimes\mathbf{Q}$ is isometric to
$\operatorname{Pic}K\otimes\mathbf{Q}$ for some Kummer surface