# Geometric quadratic Chabauty

Bas Edixhoven & Guido Lido
<EMAIL_ADDRESS><EMAIL_ADDRESS>Math. Inst., Universiteit
Leiden

###### Abstract

Since Faltings proved Mordell’s conjecture (1983, [16]) we know that the sets
of rational points on curves of genus at least 2 are finite. Determining these
sets, in individual cases, is still an unsolved problem. Chabauty’s method
(1941, [10]) is to intersect, for a prime number $p$, in the $p$-adic Lie
group of $p$-adic points of the jacobian, the closure of the Mordell-Weil
group with the $p$-adic points of the curve. Under the condition that the
Mordell-Weil rank is less than the genus, Chabauty’s method, in combination
with other methods such as the Mordell-Weil sieve, has been applied
successfully to determine all rational points in many cases.

Minhyong Kim’s non-abelian Chabauty programme aims to remove the condition on
the rank. The simplest case, called quadratic Chabauty, was developed by
Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour
de force to the so-called cursed curve (rank and genus both 3).

This article aims to make the quadratic Chabauty method _small_ and
_geometric_ again, by describing it in terms of only ‘simple algebraic
geometry’ (line bundles over the jacobian and models over the integers).

Keywords: rational points, curves, Chabauty, non-abelian, quadratic,
geometric.
2010 Mathematical Subject Classification: 14G05, 11G30.

###### Contents

  1. 1 Introduction
  2. 2 Algebraic geometry
  3. 3 From algebraic geometry to formal geometry
  4. 4 Integral points, closure and finiteness
  5. 5 Parametrisation of integral points, and power series
    1. 5.1 Logarithm and exponential
    2. 5.2 Parametrisation by power series
    3. 5.3 The $p$-adic closure
  6. 6 Explicit description of the Poincaré torsor
    1. 6.1 Norms
    2. 6.2 Norms along finite relative Cartier divisors
    3. 6.3 Explicit description of the Poincaré torsor of a smooth curve
    4. 6.4 Explicit isomorphism for norms along equivalent divisors
    5. 6.5 Symmetry of the Norm for divisors on smooth curves
    6. 6.6 Explicit residue disks and partial group laws
    7. 6.7 Extension of the Poincaré biextension over Néron models
    8. 6.8 Explicit description of the extended Poincaré bundle
    9. 6.9 Integral points of the extended Poincaré torsor
  7. 7 Description of the map from the curve to the torsor
  8. 8 An example with genus 2, rank 2, and 14 points
    1. 8.2 The torsor on the jacobian
    2. 8.3 Some integral points on the biextension
    3. 8.4 Some residue disks of the biextension
    4. 8.5 Geometry mod $p^{2}$ of integral points
    5. 8.6 The rational points with a specific image mod 5.
    6. 8.7 Determination of all rational points
  9. 9 Some further remarks
    1. 9.1 Complex uniformisations of some of the objects involved
    2. 9.2 Finiteness of rational points
    3. 9.3 The relation with $p$-adic heights
  10. References

##  1 Introduction

Faltings proved in 1983, see [16], that for every number field $K$ and every
curve $C$ over $K$ of genus at least $2$, the set of $K$-rational points
$C(K)$ is finite. However, determining $C(K)$, in individual cases, is still
an unsolved problem. For simplicity, we restrict ourselves in this article to
the case $K=\mathbb{Q}$.

Chabauty’s method (1941) for determining $C(\mathbb{Q})$ is to intersect, for
a prime number $p$, in the $p$-adic Lie group of $p$-adic points of the
jacobian, the closure of the Mordell-Weil group with the $p$-adic points of
the curve. There is a fair amount of evidence (mainly hyperelliptic curves of
small genus, see [3]) that Chabauty’s method, in combination with other
methods such as the Mordell-Weil sieve, does determine all rational points
when $r<g$, with $r$ the Mordell-Weil rank and $g$ the genus of $C$.

For a general introduction to Chabauty’s method and Coleman’s effective
version of it, we highly recommend [24], and, for an implementation of it that
is ‘geometric’ in the sense of this article, to [17], in which equations for
the curve embedded in the Jacobian are pulled back via local parametrisations
of the closure of the Mordell-Weil group.

Minhyong Kim’s non-abelian Chabauty programme aims to remove the condition
that $r<g$. The ‘non-abelian’ refers to fundamental groups; the fundamental
group of the jacobian of a curve is the abelianised fundamental group of the
curve. The most striking result in this direction is the so-called quadratic
Chabauty method, applied in [5], a technical tour de force, to the so-called
cursed curve ($r=g=3$). For more details we recommend the introduction of [5].

This article aims to make the quadratic Chabauty method _small_ and
_geometric_ again, by describing it in terms of only ‘simple algebraic
geometry’ (line bundles over the jacobian, models over the integers, and
biextension structures). The main result is Theorem 4.12. It gives a criterion
for a given list of rational points to be complete, in terms of points with
values in $\mathbb{Z}/p^{2}\mathbb{Z}$ only. Section 2 describes the geometric
method in less than 3 pages, Sections 3–5 give the necessary theory, Sections
6–7 give descriptions that are suitable for computer calculations, and Section
8 treats an example with $r=g=2$ and $14$ rational points. As explained in the
remarks following Theorem 4.12, we expect that this approach will make it
possible to treat many more curves. Section 9.1 gives some remarks on the
fundamental groups of the objects we use. They are subgroups of higher
dimensional Heisenberg groups, where the commutator pairing is the
intersection pairing of the first homology group of the curve. Section 9.2
reproves the finiteness of $C(\mathbb{Q})$, for $C$ with $r<g+\rho-1$, with
$\rho$ the rank of the $\mathbb{Z}$-module of symmetric endomorphisms of the
jacobian of $C$. It also shows that a version of Theorem 4.12 that uses higher
$p$-adic precision will always give a finite upper bound for $C(\mathbb{Q})$.
Section 9.3 gives, through an appropriate choice of coordinates that split the
Poincaré biextension, the relation between our geometric approach and the
$p$-adic heights used in the cohomological approach.

Already for the case of classical Chabauty (working with $J$ instead of $T$,
and under the assumption that $r<g$), where everything is linear, the
criterion of Theorem 4.12 can be useful; this has been worked out and
implemented in [30]. We recommend this work as a gentle introduction into the
geometric approach taken in this article. A generalisation from $\mathbb{Q}$
to number fields is given in [13]. For a generalisation of the cohomological
approach, see [2] (quadratic Chabauty) and [14] (non-abelian Chabauty).

Although this article is about geometry, it contains no pictures. Fortunately,
many pictures can be found in [19], and some in [15].

##  2 Algebraic geometry

Let $C$ be a scheme over $\mathbb{Z}$, proper, flat, regular, with
$C_{\mathbb{Q}}$ of dimension one and geometrically connected. Let $n$ be in
$\mathbb{Z}_{\geq 1}$ such that the restriction of $C$ to $\mathbb{Z}[1/n]$ is
smooth. Let $g$ be the genus of $C_{\mathbb{Q}}$. We assume that $g\geq 2$ and
that we have a rational point $b\in C(\mathbb{Q})$; it extends uniquely to a
$b\in C(\mathbb{Z})$. We let $J$ be the Néron model over $\mathbb{Z}$ of the
jacobian $\mathrm{Pic}^{0}_{C_{\mathbb{Q}}/\mathbb{Q}}$. We denote by
$J^{\vee}$ the Néron model over $\mathbb{Z}$ of the dual
$J_{\mathbb{Q}}^{\vee}$ of $J_{\mathbb{Q}}$, and $\lambda\colon J\to J^{\vee}$
the isomorphism extending the canonical principal polarisation of
$J_{\mathbb{Q}}$. We let $P_{\mathbb{Q}}$ be the Poincaré _line bundle_ on
$J_{\mathbb{Q}}\times J_{\mathbb{Q}}^{\vee}$, trivialised on the union of
$\\{0\\}\times J_{\mathbb{Q}}^{\vee}$ and $J_{\mathbb{Q}}\times\\{0\\}$. Then
the Poincaré _torsor_ is the $\mathbb{G}_{\mathrm{m}}$-torsor on
$J_{\mathbb{Q}}\times J_{\mathbb{Q}}^{\vee}$ defined as

$None$ $P^{\times}_{\mathbb{Q}}=\mathbf{Isom}_{J_{\mathbb{Q}}\times
J_{\mathbb{Q}}^{\vee}}(\mathcal{O}_{J_{\mathbb{Q}}\times
J_{\mathbb{Q}}^{\vee}},P_{\mathbb{Q}})\,.$

For every scheme $S$ over $J_{\mathbb{Q}}\times J_{\mathbb{Q}}^{\vee}$,
$P^{\times}_{\mathbb{Q}}(S)$ is the set of isomorphisms from $\mathcal{O}_{S}$
to $(P_{\mathbb{Q}})_{S}$, with a free and transitive action of
$\mathcal{O}_{S}(S)^{\times}$. Locally on $S$ for the Zariski topology,
$(P^{\times}_{\mathbb{Q}})_{S}$ is trivial, and $P^{\times}_{\mathbb{Q}}$ is
represented by a scheme over $J_{\mathbb{Q}}\times J_{\mathbb{Q}}^{\vee}$.

The theorem of the cube gives $P^{\times}_{\mathbb{Q}}$ the structure of a
_biextension_ of $J_{\mathbb{Q}}$ and $J_{\mathbb{Q}}^{\vee}$ by
$\mathbb{G}_{\mathrm{m}}$, a notion for the details of which we recommend
Section I.2.5 of [26], Grothendieck’s Exposés VII and VIII [29], and
references therein. This means the following. For $S$ a $\mathbb{Q}$-scheme,
$x_{1}$ and $x_{2}$ in $J_{\mathbb{Q}}(S)$, and $y$ in
$J^{\vee}_{\mathbb{Q}}(S)$, the theorem of the cube gives a canonical
isomorphism of $\mathcal{O}_{S}$-modules

$None$
$(x_{1},y)^{*}P_{\mathbb{Q}}\otimes_{\mathcal{O}_{S}}(x_{2},y)^{*}P_{\mathbb{Q}}=(x_{1}+x_{2},y)^{*}P_{\mathbb{Q}}\,.$

This induces a morphism of schemes

$None$
$(x_{1},y)^{*}P^{\times}_{\mathbb{Q}}\times_{S}(x_{2},y)^{*}P^{\times}_{\mathbb{Q}}\longrightarrow(x_{1}+x_{2},y)^{*}P^{\times}_{\mathbb{Q}}\,.$

as follows. For any $S$-scheme $T$, and $z_{1}$ in
$((x_{1},y)^{*}P^{\times}_{\mathbb{Q}})(T)$ and $z_{2}$ in
$((x_{2},y)^{*}P^{\times}_{\mathbb{Q}})(T)$, we view $z_{1}$ and $z_{2}$ as
nowhere vanishing sections of the invertible $\mathcal{O}_{T}$-modules
$(x_{1},y)^{*}P_{\mathbb{Q}}$ and $(x_{2},y)^{*}P_{\mathbb{Q}}$. The tensor
product of these two then gives an element of
$((x_{1}+x_{2},y)^{*}P^{\times}_{\mathbb{Q}})(T)$. This gives
$P^{\times}_{\mathbb{Q}}\to J^{\vee}_{\mathbb{Q}}$ the structure of a
commutative group scheme, which is an extension of $J_{\mathbb{Q}}$ by
$\mathbb{G}_{\mathrm{m}}$, over the base $J^{\vee}_{\mathbb{Q}}$. We denote
this group law, and the one on $J_{\mathbb{Q}}\times J^{\vee}_{\mathbb{Q}}$,
as

$None$
${(z_{1},z_{2})}$${z_{1}+_{1}z_{2}}$${((x_{1},y),(x_{2},y))}$${(x_{1},y)+_{1}(x_{2},y)}$${(x_{1}+x_{2},y)\,.}$

In the same way, $P^{\times}_{\mathbb{Q}}\to J_{\mathbb{Q}}$ has a group law
$+_{2}$ that makes it an extension of $J^{\vee}_{\mathbb{Q}}$ by
$\mathbb{G}_{\mathrm{m}}$ over the base $J_{\mathbb{Q}}$. In this way,
$P^{\times}_{\mathbb{Q}}$ is both the universal extension of $J_{\mathbb{Q}}$
by $\mathbb{G}_{\mathrm{m}}$ and the universal extension of
$J^{\vee}_{\mathbb{Q}}$ by $\mathbb{G}_{\mathrm{m}}$. The final ingredient of
the notion of biextension is that the two partial group laws are compatible in
the following sense. For any $\mathbb{Q}$-scheme $S$, for $x_{1}$ and $x_{2}$
in $J_{\mathbb{Q}}(S)$, $y_{1}$ and $y_{2}$ in $J^{\vee}_{\mathbb{Q}}(S)$,
and, for all $i$ and $j$ in $\\{1,2\\}$, $z_{i,j}$ in
$((x_{i},y_{j})^{*}P^{\times}_{\mathbb{Q}})(S)$, we have

$None$
${(z_{1,1}+_{1}z_{2,1})+_{2}(z_{1,2}+_{1}z_{2,2})}$${(z_{1,1}+_{2}z_{1,2})+_{1}(z_{2,1}+_{2}z_{2,2})}$${(x_{1}+x_{2},y_{1})+_{2}(x_{1}+x_{2},y_{2})}$${(x_{1},y_{1}+y_{2})+_{1}(x_{2},y_{1}+y_{2})}$

with the equality in the upper line taking place in
$((x_{1}+x_{2},y_{1}+y_{2})^{*}P^{\times}_{\mathbb{Q}})(S)$.

Now we extend the geometry above over $\mathbb{Z}$. We denote by $J^{0}$ the
fibrewise connected component of $0$ in $J$, which is an open subgroup scheme
of $J$, and by $\Phi$ the quotient $J/J^{0}$, which is an étale (not
necessarily separated) group scheme over $\mathbb{Z}$, with finite fibres,
supported on $\mathbb{Z}/n\mathbb{Z}$. Similarly, we let $J^{\vee 0}$ be the
fibrewise connected component of $J^{\vee}$. Theorem 7.1, in Exposé VIII of
[29] gives that $P^{\times}_{\mathbb{Q}}$ extends uniquely to a
$\mathbb{G}_{\mathrm{m}}$-biextension

$None$ $P^{\times}\longrightarrow J\times J^{\vee 0}$

(Grothendieck’s pairing on component groups is the obstruction to the
existence of such an extension). Note that in this case the existence and the
uniqueness follow directly from the requirement of extending the
rigidification on $J_{\mathbb{Q}}\times\\{0\\}$. For details see Section 6.7.

Our base point $b\in C(\mathbb{Z})$ gives an embedding $j_{b}\colon
C_{\mathbb{Q}}\to J_{\mathbb{Q}}$, which sends, functorially in
$\mathbb{Q}$-schemes $S$, an element $c\in C_{\mathbb{Q}}(S)$ to the class of
the invertible $\mathcal{O}_{C_{S}}$-module $\mathcal{O}_{C_{S}}(c-b)$. Then
$j_{b}$ extends uniquely to a morphism

$None$ $j_{b}\colon C^{\mathrm{sm}}\longrightarrow J$

where $C^{\mathrm{sm}}$ is the open subscheme of $C$ consisting of points at
which $C$ is smooth over $\mathbb{Z}$. Note that
$C_{\mathbb{Q}}(\mathbb{Q})=C(\mathbb{Z})=C^{\mathrm{sm}}(\mathbb{Z})$.

Our next step is to lift $j_{b}$, at least on certain opens of
$C^{\mathrm{sm}}$, to a morphism to a
$\mathbb{G}_{\mathrm{m}}^{\rho-1}$-torsor over $J$, where $\rho$ is the rank
of the free $\mathbb{Z}$-module
$\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})^{+}$, the
$\mathbb{Z}$-module of self-dual morphisms from $J_{\mathbb{Q}}$ to
$J^{\vee}_{\mathbb{Q}}$. This torsor will be the product of pullbacks of
$P^{\times}$ via morphisms

$None$ $(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c}\circ f)\colon J\to J\times
J^{\vee 0}\,,$

with $f\colon J\to J^{\vee}$ a morphism of group schemes, $c\in
J^{\vee}(\mathbb{Z})$, $\mathrm{tr}_{c}$ the translation by $c$, $m$ the least
common multiple of the exponents of all $\Phi({\overline{\mathbb{F}}}_{p})$
with $p$ ranging over all primes, and $m{\cdot}$ the multiplication by $m$ map
on $J^{\vee}$. For such a map $m{\cdot}\circ\mathrm{tr}_{c}\circ f$,
$j_{b}\colon C_{\mathbb{Q}}\to J_{\mathbb{Q}}$ can be lifted to
$(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c}\circ f)^{*}P^{\times}_{\mathbb{Q}}$
if and only if $j_{b}^{*}(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c}\circ
f)^{*}P^{\times}_{\mathbb{Q}}$ is trivial. The degree of this
$\mathbb{G}_{\mathrm{m}}$-torsor on $C_{\mathbb{Q}}$ is minus the trace of
$\lambda^{-1}\circ m{\cdot}\circ(f+f^{\vee})$ acting on
$\mathrm{H}_{1}(J(\mathbb{C}),\mathbb{Z})$. For example, for $f=\lambda$ the
degree is $-4mg$. Note that $j_{b}\colon C_{\mathbb{Q}}\to J_{\mathbb{Q}}$
induces

$None$ $j_{b}^{*}=-\lambda^{-1}\colon J^{\vee}_{\mathbb{Q}}\to
J_{\mathbb{Q}}\,,$

(see [25], Propositions 2.7.9 and 2.7.10). This implies that for $f$ such that
this degree is zero, there is a unique $c$ such that
$j_{b}^{*}(\mathrm{id},\mathrm{tr}_{c}\circ f)^{*}P^{\times}_{\mathbb{Q}}$ is
trivial on $C_{\mathbb{Q}}$, and hence also its $m$th power
$j_{b}^{*}(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c}\circ
f)^{*}P^{\times}_{\mathbb{Q}}$.

The map

$None$
$\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})\longrightarrow\mathrm{Pic}(J_{\mathbb{Q}})\longrightarrow\mathrm{NS}_{J_{\mathbb{Q}}/\mathbb{Q}}(\mathbb{Q})=\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})^{+}$

sending $f$ to the class of $(\mathrm{id},f)^{*}P_{\mathbb{Q}}$ sends $f$ to
$f+f^{\vee}$, hence its kernel is
$\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})^{-}$, the group of
antisymmetric morphisms. But actually, for $f$ antisymmetric, its image in
$\mathrm{Pic}(J_{\mathbb{Q}})$ is already zero (see for example [6] and the
references therein). Hence the image of
$\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})$ in
$\mathrm{Pic}(J_{\mathbb{Q}})$ is free of rank $\rho$, and its subgroup of
classes with degree zero on $C_{\mathbb{Q}}$ is free of rank $\rho{-}1$. Let
$f_{1},\ldots,f_{\rho-1}$ be elements of
$\mathrm{Hom}(J_{\mathbb{Q}},J_{\mathbb{Q}}^{\vee})$ whose images in
$\mathrm{Pic}(J_{\mathbb{Q}})$ form a basis of this subgroup, and let
$c_{1},\ldots,c_{\rho-1}$ be the corresponding elements of
$J^{\vee}(\mathbb{Z})$.

By construction, for each $i$, the morphism $j_{b}\colon C_{\mathbb{Q}}\to
J_{\mathbb{Q}}$ lifts to $(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c_{i}}\circ
f_{i})^{*}P^{\times}_{\mathbb{Q}}$, unique up to $\mathbb{Q}^{\times}$. Now we
spread this out over $\mathbb{Z}$, to open subschemes $U$ of $C^{\mathrm{sm}}$
obtained by removing, for each $q$ dividing $n$, all but one irreducible
components of $C^{\mathrm{sm}}_{\mathbb{F}_{q}}$, with the remaining
irreducible component geometrically irreducible. For such a $U$, the morphism
$\mathrm{Pic}(U)\to\mathrm{Pic}(C_{\mathbb{Q}})$ is an isomorphism, and
$\mathcal{O}_{C}(U)=\mathbb{Z}$, thus, for each $i$, there is a lift

$None$ ${(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c_{i}}\circ
f_{i})^{*}P^{\times}}$${U}$${J}$$\scriptstyle{j_{b}}$$\scriptstyle{\widetilde{j_{b}}}$

unique up to $\mathbb{Z}^{\times}=\\{1,-1\\}$.

At this point we can explain the strategy of our approach to the quadratic
Chabauty method. Let $T$ be the $\mathbb{G}_{\mathrm{m}}^{\rho-1}$-torsor on
$J$ obtained by taking the product of all
$T_{i}:=(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c_{i}}\circ
f_{i})^{*}P^{\times}$:

$None$ ${T}$${P^{\times,\rho-1}}$${U}$${J}$${J\times(J^{\vee
0})^{\rho-1}\,.}$$\scriptstyle{\widetilde{j_{b}}}$$\scriptstyle{j_{b}}$$\scriptstyle{(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c_{i}}\circ
f_{i})_{i}}$

Then each $c\in C_{\mathbb{Q}}(\mathbb{Q})=C^{\mathrm{sm}}(\mathbb{Z})$ lies
in one of the finitely many $U(\mathbb{Z})$’s. For each $U$, we have a lift
$\widetilde{j_{b}}\colon U\to T$, and, for each prime number $p$,
$\widetilde{j_{b}}(U(\mathbb{Z}))$ is contained in the intersection, in
$T(\mathbb{Z}_{p})$, of $\widetilde{j_{b}}(U(\mathbb{Z}_{p}))$ and the closure
$\overline{T(\mathbb{Z})}$ of $T(\mathbb{Z})$ in $T(\mathbb{Z}_{p})$ with the
$p$-adic topology. Of course, one expects this closure to be of dimension at
most $r:=\mathrm{rank}(J(\mathbb{Q}))$, and therefore one expects this method
to be successful if $r<g+\rho-1$, the dimension of $T(\mathbb{Z}_{p})$. The
next two sections make this strategy precise, giving first the necessary
$p$-adic formal and analytic geometry, and then the description of
$\overline{T(\mathbb{Z})}$ as a finite disjoint union of images of
$\mathbb{Z}_{p}^{r}$ under maps constructed from the biextension structure.

##  3 From algebraic geometry to formal geometry

Let $p$ be a prime number. Given $X$ a smooth scheme of relative dimension $d$
over $\mathbb{Z}_{p}$ and $x\in X(\mathbb{F}_{p})$ let us describe the set
$X(\mathbb{Z}_{p})_{x}$ of elements of $X(\mathbb{Z}_{p})$ whose image in
$X(\mathbb{F}_{p})$ is $x$. The smoothness implies that the maximal ideal of
$\mathcal{O}_{X,x}$ is generated by $p$ together with $d$ other elements
$t_{1},\ldots,t_{d}$. In this case we call $p,t_{1},\ldots,t_{d}$ _parameters
at $x$_; if moreover $x_{l}\in X(\mathbb{Z}_{p})_{x}$ is a lift of $x$ such
that $t_{1}(x_{l})=\ldots t_{d}(x_{l})=0$ then we say that the $t_{i}$’s are
_parameters at $x_{l}$_. The $t_{i}$ can be evaluated on all the points in
$X(\mathbb{Z}_{p})_{x}$, inducing a bijection $t:=(t_{1},\ldots,t_{d})\colon
X(\mathbb{Z}_{p})_{x}\to(p\mathbb{Z}_{p})^{d}$. We get a bijection

$None$
$\tilde{t}:=(\tilde{t}_{1},\ldots,\tilde{t}_{d})=\left(\frac{t_{1}}{p},\ldots,\frac{t_{d}}{p}\right)\colon
X(\mathbb{Z}_{p})_{x}\overset{\sim}{\longrightarrow}\mathbb{Z}_{p}^{d}\,.$

This bijection can be interpreted geometrically as follows. Let
$\pi\colon\widetilde{X}_{x}\to X$ denote the blow up of $X$ in $x$. By
shrinking $X$, $X$ is affine and the $t_{i}$ are regular on $X$, $t\colon
X\to\mathbb{A}^{d}_{\mathbb{Z}_{p}}$ is etale, and
$t^{-1}\\{0_{\mathbb{F}_{p}}\\}=\\{x\\}$. Then $\pi\colon\widetilde{X}_{x}\to
X$ is the pull back of the blow up of $\mathbb{A}^{d}_{\mathbb{Z}_{p}}$ at the
origin over $\mathbb{F}_{p}$. The affine open part $\widetilde{X}_{x}^{p}$ of
$\widetilde{X}_{x}$ where $p$ generates the image of the ideal $m_{x}$ of $x$
is the pullback of the corresponding open part of the blow up of
$\mathbb{A}^{d}_{\mathbb{Z}_{p}}$, which is the multiplication by $p$ morphism
$\mathbb{A}^{d}_{\mathbb{Z}_{p}}\to\mathbb{A}^{d}_{\mathbb{Z}_{p}}$ that
corresponds to
$\mathbb{Z}_{p}[t_{1},\ldots,t_{d}]\to\mathbb{Z}_{p}[\tilde{t}_{1},\ldots,\tilde{t}_{d}]$
with $t_{i}\mapsto p\tilde{t}_{i}$. It follows that the $p$-adic completion
$\mathcal{O}(\widetilde{X}_{x}^{p})^{\wedge_{p}}$ of
$\mathcal{O}(\widetilde{X}_{x}^{p})$ is the $p$-adic completion
$\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{d}\rangle$ of
$\mathbb{Z}_{p}[\tilde{t}_{1},\ldots,\tilde{t}_{d}]$. Explicitly, we have

$None$
$\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{d}\rangle=\left\\{\sum_{I\in\mathbb{N}^{d}}a_{I}\tilde{t}^{I}\in\mathbb{Z}_{p}[[\tilde{t}_{1},\ldots,\tilde{t}_{d}]]:\forall
n\geq 0,\,\text{ $\forall^{\text{almost}}I$},\,v_{p}(a_{I})\geq n\right\\}\,.$

With these definitions, we have

$None$ $\displaystyle X(\mathbb{Z}_{p})_{x}$
$\displaystyle=\widetilde{X}_{x}^{p}(\mathbb{Z}_{p})=\mathrm{Hom}(\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{d}\rangle,\mathbb{Z}_{p})=\mathbb{A}^{d}(\mathbb{Z}_{p})\,,$
$\displaystyle(\widetilde{X}_{x}^{p})_{\mathbb{F}_{p}}$
$\displaystyle=\mathrm{Spec}(\mathbb{F}_{p}[\tilde{t}_{1},\ldots,\tilde{t}_{d}]).$

The affine space $(\widetilde{X}_{x}^{p})_{\mathbb{F}_{p}}$ is canonically a
torsor under the tangent space of $X_{\mathbb{F}_{p}}$ at $x$.

This construction is functorial. Let $Y$ be a smooth $\mathbb{Z}_{p}$-scheme,
$f\colon X\to Y$ a morphism over $\mathbb{Z}_{p}$, and $y:=f(x)\in
Y(\mathbb{F}_{p})$. Then the ideal in $\mathcal{O}_{\widetilde{X}_{x}^{p}}$
generated by the image of $m_{f(x)}$ is generated by $p$. That gives us a
morphism $\widetilde{X}_{x}^{p}\to\widetilde{Y}_{f(x)}^{p}$, and then a
morphism from $\mathcal{O}(\widetilde{Y}_{f(x)}^{p})^{\wedge_{p}}$ to
$\mathcal{O}(\widetilde{X}_{x}^{p})^{\wedge_{p}}$. Reduction mod $p$ then
gives a morphism
$(\widetilde{X}_{x}^{p})_{\mathbb{F}_{p}}\to(\widetilde{Y}_{f(x)}^{p})_{\mathbb{F}_{p}}$,
the tangent map of $f$ at $x$, up to a translation.

If this tangent map is injective, and $d_{x}$ and $d_{y}$ denote the
dimensions of $X_{\mathbb{F}_{p}}$ at $x$ and of $Y_{\mathbb{F}_{p}}$ at $y$,
then there are $t_{1},\ldots,t_{d_{y}}$ in $\mathcal{O}_{Y,y}$ such that
$p,t_{1},\ldots,t_{d_{y}}$ are parameters at $y$, and such that
$t_{d_{x}+1},\ldots,t_{d_{y}}$ generate the kernel of
$\mathcal{O}_{Y,y}\to\mathcal{O}_{X,x}$. Then the images in
$\mathcal{O}_{X,x}$ of $p,t_{1},\ldots,t_{d_{x}}$ are parameters at $x$, and
$\mathcal{O}(\widetilde{Y}_{f(x)}^{p})^{\wedge_{p}}\to\mathcal{O}(\widetilde{X}_{x}^{p})^{\wedge_{p}}$
is
$\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{d_{y}}\rangle\to\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{d_{x}}\rangle$,
with kernel generated by $\tilde{t}_{d_{x}+1},\ldots,\tilde{t}_{d_{y}}$.

##  4 Integral points, closure and finiteness

Let us now return to our original problem. The notation $U$, $J$, $T$,
$j_{b}$, $\widetilde{j_{b}}$, $r$, $\rho$ etc., is as at the end of Section 2.
We assume moreover that $p$ does not divide $n$ ($n$ as in the start of
Section 2) and that $p>2$ (for $p=2$ everything that follows can probably be
adapted by working with residue polydiscs modulo $4$).

Let $u$ be in $U(\mathbb{F}_{p})$, and $t:=\widetilde{j_{b}}(u)$. We want a
description of the closure $\overline{T(\mathbb{Z})_{t}}$ of
$T(\mathbb{Z})_{t}$ in $T(\mathbb{Z}_{p})_{t}$. Using the biextension
structure of $P^{\times}$, we will produce, for each element of
$J(\mathbb{Z})_{j_{b}(u)}$, an element of $T(\mathbb{Z})$ over it. Not all of
these points are in $T(\mathbb{Z})_{t}$, but we will then produce a subset of
$T(\mathbb{Z})_{t}$ whose closure is $\overline{T(\mathbb{Z})_{t}}$.

If $T(\mathbb{Z})_{t}$ is empty then $\overline{T(\mathbb{Z})_{t}}$ is empty,
too. So we assume that we have an element $\widetilde{t}\in T(\mathbb{Z})_{t}$
and we define $x_{\widetilde{t}}\in J(\mathbb{Z})$ to be the projection of
$\widetilde{t}$. Let $f=(f_{1},\ldots,f_{\rho-1})\colon J\to J^{\vee,\rho-1}$,
let $c=(c_{1},\ldots,c_{\rho-1})\in J^{\vee,\rho-1}(\mathbb{Z})$. We denote by
$P^{\times,\rho-1}$ the product over $J\times(J^{\vee 0})^{\rho-1}$ of the
$\rho{-}1$ $\mathbb{G}_{\mathrm{m}}$-torsors obtained by pullback of
$P^{\times}$ via the projections to $J\times J^{\vee 0}$; it is a biextension
of $J$ and $(J^{\vee 0})^{\rho-1}$ by $\mathbb{G}_{\mathrm{m}}^{\rho-1}$, and
$T=(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c}\circ f)^{*}P^{\times,\rho-1}$. We
choose a basis $x_{1},\ldots,x_{r}$ of the free $\mathbb{Z}$-module
$J(\mathbb{Z})_{0}$, the kernel of $J(\mathbb{Z})\to J(\mathbb{F}_{p})$. For
each $i,j\in\\{1,\ldots,r\\}$ we choose $P_{i,j}$, $R_{i,\widetilde{t}}$, and
$S_{\widetilde{t},j}$ in $P^{\times,\rho-1}(\mathbb{Z})$ whose images in
$(J\times(J^{\vee 0})^{\rho-1})(\mathbb{Z})$ are $(x_{i},f(mx_{j}))$,
$(x_{i},(m{\cdot}\circ\mathrm{tr}_{c}\circ f)(x_{\widetilde{t}}))$ and
$(x_{\widetilde{t}},f(mx_{j}))$:

$None$
${P_{i,j}}$${R_{i,\widetilde{t}}}$${S_{\widetilde{t},j}}$${P^{\times,\rho-1}}$${(x_{i},f(mx_{j}))}$${(x_{i},(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)(x_{\widetilde{t}}))}$${(x_{\widetilde{t}},f(mx_{j}))}$${J\times(J^{\vee
0})^{\rho-1}\,.}$

For each such choice there are $2^{\rho-1}$ possibilities.

For each $n\in\mathbb{Z}^{r}$ we use the biextension structure on
$P^{\times,\rho-1}\to J\times(J^{\vee 0})^{\rho-1}$ to define the following
points in $P^{\times,\rho-1}(\mathbb{Z})$, with specified images in
$(J\times(J^{\vee 0})^{\rho-1})(\mathbb{Z})$:

$None$ ${\displaystyle
A_{\widetilde{t}}(n)=\sideset{}{{}_{2}}{\sum}_{j=1}^{r}n_{j}\cdot_{2}S_{\widetilde{t},j}\,}$${\displaystyle
B_{\widetilde{t}}(n)=\sideset{}{{}_{1}}{\sum}_{i=1}^{r}n_{i}\cdot_{1}R_{i,\widetilde{t}}\,}$${\displaystyle\left(x_{\widetilde{t}}\,,\sum_{i=1}^{r}n_{i}f(mx_{i})\right)}$${\displaystyle\left(\sum_{i=1}^{r}n_{i}x_{i}\,,(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)(x_{\widetilde{t}})\right)}$ $None$ ${\displaystyle
C(n)=\sideset{}{{}_{1}}{\sum}_{i=1}^{r}n_{i}\cdot_{1}\left(\sideset{}{{}_{2}}{\sum}_{j=1}^{r}n_{j}\cdot_{2}P_{i,j}\right)\,}$${\displaystyle\left(\sum_{i=1}^{r}n_{i}x_{i}\,,\sum_{i=1}^{r}n_{i}f(mx_{i})\right)}$

where $\sum_{1}$ and $\cdot_{1}$ denote iterations of the first partial group
law $+_{1}$ as in ($None$), and analogously for the second group law. We
define, for all $n\in\mathbb{Z}^{r}$,

$None$
$D_{\widetilde{t}}(n):=\left(C(n)+_{2}B_{\widetilde{t}}(n)\right)+_{1}\left(A_{\widetilde{t}}(n)+_{2}\widetilde{t}\right)\,\in\,P^{\times,\rho-1}(\mathbb{Z})\,,$

which is mapped to

$None$
$\left(x_{\widetilde{t}}+\sum_{i=1}^{r}n_{i}x_{i},(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)\left(x_{\widetilde{t}}+\sum_{i=1}^{r}n_{i}x_{i}\right)\right)\in\left(J\times(J^{\vee
0})^{\rho-1}\right)(\mathbb{Z})\,.$

Hence $D_{\widetilde{t}}(n)$ is in $T(\mathbb{Z})$, and its image in
$J(\mathbb{F}_{p})$ is $j_{b}(u)$. We do not know its image in
$T(\mathbb{F}_{p})$.

We claim that for $n$ in $(p{-}1)\mathbb{Z}^{r}$, $D_{\widetilde{t}}(n)$ is in
$T(\mathbb{Z})_{t}$. Let $n^{\prime}$ be in $\mathbb{Z}^{r}$ and let
$n=(p{-}1)n^{\prime}$. Then, in the trivial
$\mathbb{F}_{p}^{\times,\rho-1}$-torsor $P^{\times,\rho-1}(j_{b}(u),0)$, on
which $+_{2}$ is the group law, we have:

$None$
$A_{\widetilde{t}}(n)=(p{-}1){\cdot}_{2}A_{\widetilde{t}}(n^{\prime})=1\quad\text{in
$\mathbb{F}_{p}^{\times,\rho-1}.$}$

Similarly, in $P^{\times,\rho-1}(0,(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)(j_{b}(u)))=\mathbb{F}_{p}^{\times,\rho-1}$, we have
$B_{\widetilde{t}}(n)=1$, and, similarly, in
$P^{\times,\rho-1}(0,0)=\mathbb{F}_{p}^{\times,\rho-1}$, we have $C(n)=1$. So,
with apologies for the mix of additive and multiplicative notations, in
$P^{\times,\rho-1}(\mathbb{F}_{p})$ we have

$None$
$D_{\widetilde{t}}(n)=\left(1+_{2}1\right)+_{1}\left(1+_{2}t\right)=t\,,$

mapping to the following element in $(J\times J^{\vee
0,\rho-1})(\mathbb{F}_{p})$:

$None$ $\displaystyle\left((0,0)+_{2}((0,(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)(j_{b}(u))))\right)+_{1}\left((j_{b}(u),0)+_{2}(j_{b}(u),(m{\cdot}\circ\mathrm{tr}_{c}\circ
f)(j_{b}(u)))\right)$
$\displaystyle=(j_{b}(u),(m{\cdot}\circ\mathrm{tr}_{c}\circ f)(j_{b}(u)))\,.$

We have proved our claim that $D_{\widetilde{t}}(n)\in T(\mathbb{Z})_{t}$.

So we now have the map

$None$ $\kappa_{\mathbb{Z}}\colon\mathbb{Z}^{r}\to T(\mathbb{Z})_{t},\quad
n\mapsto D_{\widetilde{t}}((p-1)n)\,.$

The following theorem will be proved in Section 5.

###### 4.10 Theorem

Let $x_{1},\ldots,x_{g}$ be in $\mathcal{O}_{J,j_{b}(u)}$ such that together
with $p$ they form a system of parameters of $\mathcal{O}_{J,j_{b}(u)}$, and
let $v_{1},\ldots,v_{\rho-1}$ be in $\mathcal{O}_{T,t}$ such that
$p,x_{1},\ldots,x_{g},v_{1},\ldots,v_{\rho-1}$ are parameters of
$\mathcal{O}_{T,t}$. As in Section 3 these parameters, divided by $p$, give a
bijection

$None$ $T(\mathbb{Z}_{p})_{t}\longrightarrow\mathbb{Z}_{p}^{g+\rho-1}\,.$

The composition of $\kappa_{\mathbb{Z}}$ with the map ($None$) is given by
uniquely determined $\kappa_{1},\ldots,\kappa_{g+\rho-1}$ in
$\mathcal{O}(\mathbb{A}_{\mathbb{Z}_{p}}^{r})^{\wedge_{p}}=\mathbb{Z}_{p}\langle
z_{1},\ldots,z_{r}\rangle$. The images in $\mathbb{F}_{p}[z_{1},\ldots,z_{r}]$
of $\kappa_{1},\ldots,\kappa_{g}$ are of degree at most $1$, and the images of
$\kappa_{g+1},\ldots,\kappa_{g+\rho-1}$ are of degree at most $2$. The map
$\kappa_{\mathbb{Z}}$ extends uniquely to the continuous map

$None$
$\kappa=(\kappa_{1},\ldots,\kappa_{g+\rho-1})\colon\mathbb{A}^{r}(\mathbb{Z}_{p})=\mathbb{Z}_{p}^{r}\longrightarrow
T(\mathbb{Z}_{p})_{t}\,.$

and the image of $\kappa$ is $\overline{T(\mathbb{Z})_{t}}$.

Now the moment has come to confront $U(\mathbb{Z}_{p})_{u}$ with
$\overline{T(\mathbb{Z})_{t}}$. We have $\widetilde{j_{b}}\colon U\to T$,
whose tangent map (mod $p$) at $u$ is injective (here we use that
$C_{\mathbb{F}_{p}}$ is smooth over $\mathbb{F}_{p}$). Then, as at the end of
Section 3,
$\widetilde{j_{b}}\colon\widetilde{U}_{u}^{p}\to\widetilde{T}_{t}^{p}$ is,
after reduction mod $p$, an affine linear embedding of codimension
$g{+}\rho{-}2$,
$\widetilde{j_{b}}^{*}\colon\mathcal{O}(\widetilde{T}_{t}^{p})^{\wedge_{p}}\to\mathcal{O}(\widetilde{U}_{u}^{p})^{\wedge_{p}}$
is surjective and its kernel is generated by elements
$f_{1},\ldots,f_{g+\rho-2}$ (we apologise for using the same letter as for the
components of $f\colon J\to J^{\vee,\rho-1}$), whose images in
$\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{T}_{t}^{p})$ are of degree at
most $1$, and such that $f_{1},\ldots,f_{g-1}$ are in
$\mathcal{O}(\widetilde{J}_{j_{b}(u)}^{p})^{\wedge_{p}}$. The pullbacks
$\kappa^{*}f_{i}$ are in $\mathbb{Z}_{p}\langle z_{1},\ldots,z_{r}\rangle$;
let $I$ be the ideal in $\mathbb{Z}_{p}\langle z_{1},\ldots,z_{r}\rangle$
generated by them, and let

$None$ $A:=\mathbb{Z}_{p}\langle z_{1},\ldots,z_{r}\rangle/I\,.$

Then the elements of $\mathbb{Z}_{p}^{r}$ whose image is in
$U(\mathbb{Z}_{p})_{u}$ are zeros of $I$, hence morphisms of rings from $A$ to
$\mathbb{Z}_{p}$, and hence from the reduced quotient $A_{\mathrm{red}}$ to
$\mathbb{Z}_{p}$.

###### 4.12 Theorem

For $i\in\\{1,\ldots,g{+}\rho{-}2\\}$, let $\kappa^{*}\overline{f_{i}}$ be the
image of $\kappa^{*}f_{i}$ in $\mathbb{F}_{p}[z_{1},\ldots,z_{r}]$, and let
$\overline{I}$ be the ideal of $\mathbb{F}_{p}[z_{1},\ldots,z_{r}]$ generated
by them. Then $\kappa^{*}\overline{f_{1}},\ldots,\kappa^{*}\overline{f_{g-1}}$
are of degree at most $1$, and
$\kappa^{*}\overline{f_{g}},\ldots,\kappa^{*}\overline{f_{g+\rho-2}}$ are of
degree at most $2$. Assume that
$\overline{A}:=A/pA=\mathbb{F}_{p}[z_{1},\ldots,z_{r}]/\overline{I}$ is
finite. Then $\overline{A}$ is the product of its localisations
$\overline{A}_{m}$ at its finitely many maximal ideals $m$. The sum of the
$\dim_{\mathbb{F}_{p}}\overline{A}_{m}$ over the $m$ such that
$\overline{A}/m=\mathbb{F}_{p}$ is an upper bound for the number of elements
of $\mathbb{Z}_{p}^{r}$ whose image under $\kappa$ is in
$U(\mathbb{Z}_{p})_{u}$, and also an upper bound for the number of elements of
$U(\mathbb{Z})$ with image $u$ in $U(\mathbb{F}_{p})$.

###### Proof.

As every $\overline{f_{i}}$ is of degree at most $1$ in
$x_{1},\ldots,x_{g},v_{1},\ldots,v_{\rho-1}$, every
$\kappa^{*}\overline{f_{i}}$ is an $\mathbb{F}_{p}$-linear combination of
$\kappa_{1},\ldots,\kappa_{g+\rho-1}$, hence of degree at most $2$. For $i<g$,
$\overline{f_{i}}$ is a linear combination of $x_{1},\ldots,x_{g}$, and
therefore $\kappa^{*}\overline{f_{i}}$ is of degree at most $1$.

We claim that $A$ is $p$-adically complete. More generally, let $R$ be a
noetherian ring that is $J$-adically complete for an ideal $J$, and let $I$ be
an ideal in $R$. The map from $R/I$ to its $J$-adic completion
$(R/I)^{\wedge}$ is injective ([1, Thm.10.17]). As $J$-adic completion is
exact on finitely generated $R$-modules ([1, Prop.10.12]), it sends the
surjection $R\to R/I$ to a surjection $R=R^{\wedge}\to(R/I)^{\wedge}$ (see [1,
Prop.10.5] for the equality $R=R^{\wedge}$). It follows that
$R/I\to(R/I)^{\wedge}$ is surjective.

Now we assume that $\overline{A}$ is finite. As $A$ is $p$-adically complete,
$A$ is the limit of the system of its quotients by powers of $p$. These
quotients are finite: for every $m\in\mathbb{Z}_{\geq 1}$, $A/p^{m+1}A$ is, as
abelian group, an extension of $A/pA$ by a quotient of $A/p^{m}A$. As
$\mathbb{Z}_{p}$-module, $A$ is generated by any lift of an
$\mathbb{F}_{p}$-basis of $\overline{A}$. Hence $A$ is finitely generated as
$\mathbb{Z}_{p}$-module.

The set of elements of $\mathbb{Z}_{p}^{r}$ whose image under $\kappa$ is in
$U(\mathbb{Z}_{p})$ is in bijection with the set of $\mathbb{Z}_{p}$-algebra
morphisms $\mathrm{Hom}(A,\mathbb{Z}_{p})$. As $A$ is the product of its
localisations $A_{m}$ at its maximal ideals, $\mathrm{Hom}(A,\mathbb{Z}_{p})$
is the disjoint union of the $\mathrm{Hom}(A_{m},\mathbb{Z}_{p})$. For each
$m$, $\mathrm{Hom}(A_{m},\mathbb{Z}_{p})$ has at most
$\mathrm{rank}_{\mathbb{Z}_{p}}(A_{m})$ elements, and is empty if
$\mathbb{F}_{p}\to A/m$ is not an isomorphism. This establishes the upper
bound for the number of elements of $\mathbb{Z}_{p}^{r}$ whose image under
$\kappa$ is in $U(\mathbb{Z}_{p})$. By Theorem 4.10, the elements of
$U(\mathbb{Z})$ with image $u$ in $U(\mathbb{F}_{p})$ are in
$\overline{T(\mathbb{Z})_{t}}$, and therefore of the form $\kappa(x)$ with
$x\in\mathbb{Z}_{p}^{r}$ such that $\kappa(x)$ is in $U(\mathbb{Z}_{p})_{u}$.
This establishes the upper bound for the number of elements of $U(\mathbb{Z})$
with image $u$ in $U(\mathbb{F}_{p})$. ∎

We include some remarks to explain how Theorem 4.12 can be used, and what we
hope that it can do.

###### 4.13 Remark

The $\kappa^{*}\overline{f_{i}}$ in Theorem 4.12 can be computed from the
reduction $\mathbb{F}_{p}^{r}\to T(\mathbb{Z}/p^{2}\mathbb{Z})$ of
$\kappa_{\mathbb{Z}}$ and (to get the $\overline{f_{i}}$) from
$\widetilde{j_{b}}\colon U(\mathbb{Z}/p^{2}\mathbb{Z})_{u}\to
T(\mathbb{Z}/p^{2}\mathbb{Z})_{t}$. For this, one does not need to treat $T$
and $J$ as schemes, one just computes with $\mathbb{Z}/p^{2}\mathbb{Z}$-valued
points. Now assume that $r\leq g+\rho-2$. If, for some prime $p$, the
criterion in Theorem 4.12 fails (that is, $\overline{A}$ is not finite), then
one can try the next prime. We hope (but also expect) that one quickly finds a
prime $p$ such that $\overline{A}$ is finite for every $U$ and for every $u$
in $U(\mathbb{F}_{p})$ such that $\widetilde{j_{b}}(u)$ is in the image of
$T(\mathbb{Z})\to T(\mathbb{F}_{p})$. By the way, note that our notation in
Theorem 4.12 does not show the dependence on $U$ and $u$ of
$\widetilde{j_{b}}$, $\kappa_{\mathbb{Z}}$, $\kappa$ and the
$\overline{f_{i}}$. Instead of varying $p$, one could also increase the
$p$-adic precision, and then the result of Section 9.2 proves that one gets an
upper bound for the number of elements of $U(\mathbb{Z})$.

###### 4.14 Remark

If $r<g+\rho-2$ then we think that it is likely (when varying $p$), for
dimension reasons, unless something special happens as in [3] or Remark 8.9 of
[4], that, for all $u\in U(\mathbb{F}_{p})$, the upper bound in Theorem 4.12
for the number of elements of $U(\mathbb{Z})$ with image $u$ in
$U(\mathbb{F}_{p})$ is sharp. For a precise conjecture in the context of
Chabauty’s method, see the “Strong Chabauty” Conjecture in [31].

###### 4.15 Remark

Suppose that $r=g+\rho-2$. Then we expect, for dimension reasons, that it is
likely (when varying $p$) that, for some $u\in U(\mathbb{F}_{p})$, the upper
bound in Theorem 4.12 for the number of elements of $U(\mathbb{Z})$ with image
$u$ in $U(\mathbb{F}_{p})$ is not sharp. Then, as in the classical Chabauty
method, one must combine the information gotten from several primes, analogous
to ‘Mordell-Weil sieving’; see [27]. In our situation, this amounts to the
following. Suppose that we are given a subset $B$ of $U(\mathbb{Z})$ that we
want to prove to be equal to $U(\mathbb{Z})$. Let $B^{\prime}$ be the
complement in $U(\mathbb{Z})$ of $B$. For every prime $p>2$ not dividing $n$,
Theorem 4.12 gives, interpreting $\overline{A}$ as in the end of the proof of
Theorem 4.12, a subset $O_{p}$ of $J(\mathbb{Z})$, with $O_{p}$ a union of
cosets for the subgroup $p{\cdot}\ker(J(\mathbb{Z})\to J(\mathbb{F}_{p}))$,
that contains $j_{b}(B^{\prime})$. Then one hopes that, taking a large enough
finite set $S$ of primes, the intersection of the $O_{p}$ for $p$ in $S$ is
empty.

##  5 Parametrisation of integral points, and power series

In this section we give a proof of Theorem 4.10. The main tools here are the
formal logarithm and formal exponential of a commutative smooth group scheme
over a $\mathbb{Q}$-algebra ([20], Theorem 1): they give us identities like
$n{\cdot}g=\exp(n{\cdot}\log g)$ that allow us to extend the multiplication to
elements $n$ of $\mathbb{Z}_{p}$.

The evaluation map from $\mathbb{Z}_{p}\langle z_{1},\ldots,z_{n}\rangle$ to
the set of maps $\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}$ is injective (induction
on $n$, non-zero elements of $\mathbb{Z}_{p}\langle z\rangle$ have only
finitely many zeros in $\mathbb{Z}_{p}$).

We say that a map $f\colon\mathbb{Z}_{p}^{n}\to\mathbb{Z}_{p}^{m}$ is _given
by integral convergent power series_ if its coordinate functions are in
$\mathbb{Z}_{p}\langle
z_{1},\ldots,z_{n}\rangle=\mathcal{O}(\mathbb{A}^{n}_{\mathbb{Z}_{p}})^{\wedge_{p}}$.
This property is stable under composition: composition of polynomials over
$\mathbb{Z}/p^{k}\mathbb{Z}$ gives polynomials.

###  5.1 Logarithm and exponential

Let $p$ be a prime number, and let $G$ be a commutative group scheme, smooth
of relative dimension $d$ over a scheme $S$ smooth over $\mathbb{Z}_{p}$, with
unit section $e$ in $G(S)$. For any $s$ in $S(\mathbb{F}_{p})$,
$G(\mathbb{Z}_{p})_{e(s)}$ is a group fibred over $S(\mathbb{Z}_{p})_{s}$. The
fibres have a natural $\mathbb{Z}_{p}$-module structure:
$G(\mathbb{Z}_{p})_{e(s)}$ is the limit of the
$G(\mathbb{Z}/p^{n}\mathbb{Z})_{e(s)}$ ($n\geq 1$), $S(\mathbb{Z}_{p})_{s}$ is
the limit of the $S(\mathbb{Z}/p^{n}\mathbb{Z})_{s}$, and for each $n\geq 1$,
the fibres of $G(\mathbb{Z}/p^{n}\mathbb{Z})_{e(s)}\to
S(\mathbb{Z}/p^{n}\mathbb{Z})_{s}$ are commutative groups annihilated by
$p^{n-1}$. Let $T_{G/S}$ be the relative (geometric) tangent bundle of $G$
over $S$. Then its pullback $T_{G/S}(e)$ by $e$ is a vector bundle on $S$ of
rank $d$.

###### 5.1.1 Lemma

In this situation, and with $n$ the relative dimension of $S$ over
$\mathbb{Z}_{p}$, the formal logarithm and exponential of $G$ base changed to
$\mathbb{Q}\otimes\mathcal{O}_{S,s}$ converge to maps

$\displaystyle\log\colon$
$\displaystyle\widetilde{G}^{p}_{e(s)}(\mathbb{Z}_{p})=G(\mathbb{Z}_{p})_{e(s)}\to(T_{G/S}(e))(\mathbb{Z}_{p})_{0(s)}$
$\displaystyle\exp\colon$
$\displaystyle\widetilde{T}_{G/S}(e)_{0(s)}^{p}(\mathbb{Z}_{p})=(T_{G/S}(e))(\mathbb{Z}_{p})_{0(s)}\to
G(\mathbb{Z}_{p})_{e(s)}\,,$

that are each other’s inverse and, after a choice of parameters for $G\to S$
at $e(s)$ as in ($None$), are given by $n+d$ elements of
$\mathcal{O}(\widetilde{G}^{p}_{e(s)})^{\wedge_{p}}$ and $n+d$ elements of
$\mathcal{O}(\widetilde{T}_{G/S}(e)_{0(s)}^{p})^{\wedge_{p}}$.

For $a$ in $\mathbb{Z}_{p}$ and $g$ in $G(\mathbb{Z}_{p})_{e(s)}$ we have
$a{\cdot}g=\exp(a{\cdot}\log g)$, and, after a choice of parameters for $G\to
S$ at $e(s)$, this map $\mathbb{Z}_{p}\times G(\mathbb{Z}_{p})_{e(s)}\to
G(\mathbb{Z}_{p})_{e(s)}$ is given by $n+d$ elements of
$\mathcal{O}(\mathbb{A}^{1}_{\mathbb{Z}_{p}}\times_{\mathbb{Z}_{p}}\widetilde{G}^{p}_{e(s)})^{\wedge_{p}}$.
The induced morphism
$\mathbb{A}^{1}_{\mathbb{F}_{p}}\times(\widetilde{G}^{p}_{e(s)})_{\mathbb{F}_{p}}\to(\widetilde{G}^{p}_{e(s)})_{\mathbb{F}_{p}}$,
where $(\widetilde{G}^{p}_{e(s)})_{\mathbb{F}_{p}}$ is viewed as the product
of $T_{S_{\mathbb{F}_{p}}}(s)$ and $T_{G/S}(e(s))$, is a morphism over
$T_{S_{\mathbb{F}_{p}}}(s)$, bilinear in $\mathbb{A}^{1}_{\mathbb{F}_{p}}$ and
$T_{G/S}(e(s))$.

###### Proof.

Let $t_{1},\ldots,t_{n}$ be in $\mathcal{O}_{S,s}$ such that
$p,t_{1},\ldots,t_{n}$ are parameters at $s$. Then we have a bijection

$None$ $\tilde{t}\colon S(\mathbb{Z}_{p})_{s}\to\mathbb{Z}_{p}^{n},\quad
a\mapsto p^{-1}{\cdot}(t_{1}(a),\ldots,t_{n}(a))\,.$

Similarly, let $x_{1},\ldots,x_{d}$ be generators for the ideal $I_{e(s)}$ of
$e$ in $\mathcal{O}_{G,e(s)}$. Then $p$, the $t_{i}$ and the $x_{j}$ together
are parameters for $\mathcal{O}_{G,e(s)}$, and give the bijection

$None$ $(t,x)^{\sim}\colon
G(\mathbb{Z}_{p})_{e(s)}\to\mathbb{Z}_{p}^{n+d}\,,\quad b\mapsto
p^{-1}{\cdot}(t_{1}(b),\ldots,x_{d}(b))\,.$

The $\mathrm{d}x_{i}$ form an $\mathcal{O}_{S,s}$-basis of
$\Omega^{1}_{G/S}(e)_{s}$, and so give translation invariant differentials
$\omega_{i}$ on $G_{\mathcal{O}_{S,s}}$. As $G$ is commutative, for all $i$,
$\mathrm{d}\omega_{i}=0$ ([20], Proposition 1.3). We also have the dual
$\mathcal{O}_{S,s}$-basis $\partial_{i}$ of $T_{G/S}(e)$ and the bijection

$None$
$(t,x)^{\sim}\colon(T_{G/S}(e))(\mathbb{Z}_{p})_{0(s)}\to\mathbb{Z}_{p}^{n+d}\,,\quad(a,\sum_{i}v_{i}\partial_{i})\mapsto
p^{-1}{\cdot}(t_{1}(a),\ldots,t_{n}(a),v_{1},\ldots,v_{d})\,.$

Then $\log$ is given by elements $\log_{i}$ in
$(\mathbb{Q}\otimes\mathcal{O}_{S,s})[[x_{1},\ldots,x_{d}]]$ whose constant
term is $0$, uniquely determined (Proposition 1.1 in [20]) by the equality

$None$
$\mathrm{d}\log_{i}=\omega_{i}\,,\quad\text{in}\,\,\oplus_{j}\mathcal{O}_{S,s}[[x_{1},\ldots,x_{d}]]{\cdot}\mathrm{d}x_{j}\,.$

Hence the formula from calculus,
$\log_{i}(x)-\log_{i}(0)=\int_{0}^{1}(t\mapsto tx)^{*}\omega_{i}$, gives us
that, with

$None$ $\log_{i}=\sum_{J\neq
0}\log_{i,J}x^{J}\quad\text{and}\quad\log_{i,J}\in(\mathbb{Q}\otimes\mathcal{O}_{S,s})\,,$

we have, for all $i$ and $J$, with $|J|$ denoting the total degree of $x^{J}$,

$None$ $|J|{\cdot}\log_{i,J}\in\mathcal{O}_{S,s}\,.$

The claim about convergence and definition of $\log\colon
G(\mathbb{Z}_{p})_{e(s)}\to(T_{G/S}(e))(\mathbb{Z}_{p})_{0(s)}$, is now
equivalent to having an analytic bijection
$\mathbb{Z}_{p}^{n+d}\to\mathbb{Z}_{p}^{n+d}$ given by

$None$
${G(\mathbb{Z}_{p})_{e(s)}}$${(T_{G/S}(e))(\mathbb{Z}_{p})_{0(s)}}$${\mathbb{Z}_{p}^{n+d}}$${\mathbb{Z}_{p}^{n+d}}$${(a,b)}$${\left(a,p^{-1}{\cdot}\left(\sum_{J\neq
0}\log_{i,J}(\tilde{t}^{-1}(a))(pb)^{J}\right)_{i}\right)\,.}$$\scriptstyle{?}$$\scriptstyle{(t,x)^{\sim}}$$\scriptstyle{(t,x)^{\sim}}$$\scriptstyle{?}$$\scriptstyle{?}$

We have, for each $i$,

$None$ $p^{-1}{\cdot}\sum_{J\neq
0}\log_{i,J}(\tilde{t}^{-1}(a))(pb)^{J}=\sum_{J\neq
0}\frac{p^{|J|-1}}{|J|}(|J|\log_{i,J})(\tilde{t}^{-1}(a))b^{J}\,.$

For each $i$, this expression is an element of
$\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{n},\tilde{x}_{1},\ldots,\tilde{x}_{d}\rangle=\mathcal{O}(\widetilde{G}^{p}_{e(s)})^{\wedge_{p}}$,
even when $p=2$, because for each $J$, $|J|\log_{i,J}$ is in
$\mathcal{O}_{S,s}$, which is contained in
$\mathbb{Z}_{p}\langle\tilde{t}_{1},\ldots,\tilde{t}_{n}\rangle$, and the
function $\mathbb{Z}_{\geq 1}\to\mathbb{Q}_{p}$, $r\mapsto p^{r-1}/r$ has
values in $\mathbb{Z}_{p}$ and converges to $0$. The existence and analyticity
of $\log$ is now proved (even for $p=2$). As $p>2$, the image of ($None$) in
$\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{G}^{p}_{e(s)})^{\wedge_{p}}$ is
$\tilde{x}_{i}$, and on the first $n$ coordinates, $\log$ is the identity, so,
by applying Hensel modulo powers of $p$, $\log$ is invertible, and the inverse
is also given by $n+d$ elements of
$\mathcal{O}(\widetilde{T}_{G/S}(e)_{0(s)}^{p})^{\wedge_{p}}$.

The function $\mathbb{Z}_{p}\times G(\mathbb{Z}_{p})_{e(s)}\to
G(\mathbb{Z}_{p})_{e(s)}$, $(a,g)\mapsto\exp(a{\cdot}\log g)$ is a composition
of maps given by integral convergent power series, hence it is also of that
form. ∎

###  5.2 Parametrisation by power series

The notation and assumptions are as in the beginning of Section 4, in
particular, $p>2$ and $T$ is as defined in ($None$). We have a $t$ in
$T(\mathbb{F}_{p})$, with image $j_{b}(u)$ in $J(\mathbb{F}_{p})$, and a
$\tilde{t}$ in $T(\mathbb{Z})$ lifting $t$. For every $Q$ in $T(\mathbb{Z})$
mapping to $j_{b}(u)$ in $J(\mathbb{F}_{p})$ there are unique
$\varepsilon\in\mathbb{Z}^{\times,\rho-1}$ and $n\in\mathbb{Z}^{r}$ such that
$Q=\varepsilon{\cdot}D_{\tilde{t}}(n)$: the image of $Q$ in $J(\mathbb{Z})$ is
in $J(\mathbb{Z})_{j_{b}(u)}$, hence differs from the image $x_{\tilde{t}}$ in
$J(\mathbb{Z})$ of $\tilde{t}$ by an element of $J(\mathbb{Z})_{0}$ (with here
$0\in J(\mathbb{F}_{p})$), $\sum_{i}n_{i}x_{i}$ for a unique
$n\in\mathbb{Z}^{r}$, hence $D_{\tilde{t}}(n)$ and $Q$ are in $T(\mathbb{Z})$
and have the same image in $J(\mathbb{Z})$, and that gives the unique
$\varepsilon$. So we have a bijection

$None$ $\mathbb{Z}^{\times,\rho-1}\times\mathbb{Z}^{r}\longrightarrow
T(\mathbb{Z})_{j_{b}(u)}=\\{Q\in T(\mathbb{Z}):Q\mapsto j_{b}(u)\in
J(\mathbb{F}_{p})\\}\,,\quad(\varepsilon,n)\mapsto\varepsilon{\cdot}D_{\tilde{t}}(n)\,.$

But a problem that we are facing is that the map $\mathbb{Z}^{r}\to
T(\mathbb{F}_{p})_{j_{b}(u)}$ sending $n$ to the image of $D_{\tilde{t}}(n)$
depends on the (unknown) images of the $P_{i,j}$, $R_{i,\tilde{t}}$ and
$S_{\tilde{t},j}$ from ($None$) in $P^{\times,\rho-1}(\mathbb{F}_{p})$, and so
we do not know for which $n$ and $\varepsilon$ the point
$\varepsilon{\cdot}D_{\tilde{t}}(n)$ is in $T(\mathbb{Z})_{t}$. Luckily we
have the $\mathbb{Z}_{p}^{\times,\rho-1}$-action on $T(\mathbb{Z}_{p})$. Using
that
$\mathbb{Z}_{p}^{\times}=\mathbb{F}_{p}^{\times}\times(1+p\mathbb{Z}_{p})$ we
have $\mathbb{F}_{p}^{\times,\rho-1}$ acting on
$T(\mathbb{Z}_{p})_{j_{b}(u)}$, compatibly with the torsor structure on
$T(\mathbb{F}_{p})_{j_{b}(u)}$. So, for every $n$ in $\mathbb{Z}^{r}$ there is
a unique $\xi(n)$ in $\mathbb{F}_{p}^{\times,\rho-1}$ such that
$\xi(n){\cdot}D_{\tilde{t}}(n)$ is in $T(\mathbb{Z}_{p})_{t}$. We define

$None$ $D^{\prime}(n):=\xi(n){\cdot}D_{\tilde{t}}(n)\,.$

Then for all $n$ in $\mathbb{Z}^{r}$,

$None$
$\kappa_{\mathbb{Z}}(n)=D_{\tilde{t}}((p-1){\cdot}n)=D^{\prime}((p-1){\cdot}n)\,,$

because $D_{\tilde{t}}((p-1){\cdot}n)$ maps to $t$ in $T(\mathbb{F}_{p})$.
Moreover for every $Q$ in $T(\mathbb{Z})_{t}$ there is a unique
$n\in\mathbb{Z}^{r}$ and a unique $\varepsilon\in\mathbb{Z}^{\times,\rho-1}$
such that
$Q=\varepsilon{\cdot}D_{\tilde{t}}(n)=\xi(n){\cdot}D_{\tilde{t}}(n)=D^{\prime}(n)$.
Hence

$None$ $T(\mathbb{Z})_{t}\subset D^{\prime}(\mathbb{Z}^{r})\,.$

The following lemma proves the existence and uniqueness of the $\kappa_{i}$ of
Theorem 4.10, and the claims on the degrees of the $\overline{\kappa}_{i}$.

###### 5.2.5 Lemma

After any choice of parameters of $\mathcal{O}_{T,t}$ as in Theorem 4.10,
$D^{\prime}$ is given by elements
$\kappa_{1}^{\prime},\ldots,\kappa_{g+\rho-1}^{\prime}$ of
$\mathcal{O}(\mathbb{A}^{r}_{\mathbb{Z}_{p}})^{\wedge_{p}}$, and then
$\kappa_{\mathbb{Z}}$ is given by $\kappa_{1},\ldots,\kappa_{g+\rho-1}$ with,
for all $i\in\\{1,\ldots,g+\rho-1\\}$ and all $a\in\mathbb{Z}_{p}^{r}$,

$\kappa_{i}(a)=\kappa_{i}^{\prime}((p-1)a)\,.$

For all $i$ in $\\{1,\dots,g+\rho-1\\}$ we let
$\overline{\kappa}_{i}^{\prime}$ be the reduction mod $p$ of
$\kappa_{j}^{\prime}$. Then
$\overline{\kappa}_{1}^{\prime},\ldots,\overline{\kappa}_{g}^{\prime}$ are of
degree at most 1, and the remaining $\overline{\kappa}_{j}^{\prime}$ are of
degree at most 2.

###### Proof.

In order to get a formula for $D^{\prime}(n)$, we introduce variants of the
$P_{i,j}$, $R_{i,\widetilde{t}}$, and $S_{\widetilde{t},j}$ as follows. The
images in $(J\times(J^{\vee 0})^{\rho-1})(\mathbb{F}_{p})$ of these points are
of the form $(0,*)$, $(0,*)$, and $(*,0)$, respectively. Hence the fibers over
them of $P^{\times,\rho-1}$ are rigidified, that is, equal to
$\mathbb{F}_{p}^{\times,\rho-1}$. We define their variants $P_{i,j}^{\prime}$,
$R_{i,\widetilde{t}}^{\prime}$, and $S_{\widetilde{t},j}^{\prime}$ in
$P^{\times,\rho-1}(\mathbb{Z}_{p})$ to be the unique elements in their orbits
under $\mathbb{F}_{p}^{\times,\rho-1}$ whose images in
$P^{\times,\rho-1}(\mathbb{F}_{p})$ are equal to the element $1$ in
$\mathbb{F}_{p}^{\times,\rho-1}$. Replacing, in ($None$) and ($None$), these
$P_{i,j}$, $R_{i,\widetilde{t}}$, and $S_{\widetilde{t},j}$ by
$P_{i,j}^{\prime}$, $R_{i,\widetilde{t}}^{\prime}$, and
$S_{\widetilde{t},j}^{\prime}$ gives variants $A^{\prime}$, $B^{\prime}$ and
$C^{\prime}$, and using these in ($None$) gives a variant
$D_{\tilde{t}}^{\prime}(n)$ of $None$.

Then, for all $n$ in $\mathbb{Z}^{r}$, $D_{\tilde{t}}^{\prime}(n)$ and
$D^{\prime}(n)$ (as in ($None$)) are equal, because both are in
$P^{\times,\rho-1}(\mathbb{Z}_{p})_{t}$, and in the same
$\mathbb{F}_{p}^{\times,\rho-1}$-orbit. Hence we have, for all $n$ in
$\mathbb{Z}^{r}$:

$None$ $\displaystyle A^{\prime}(n)$
$\displaystyle=\sideset{}{{}_{2}}{\sum}_{j=1}^{r}n_{j}\cdot_{2}S_{\widetilde{t},j}^{\prime}\,,\quad
B^{\prime}(n)=\sideset{}{{}_{1}}{\sum}_{i=1}^{r}n_{i}\cdot_{1}R_{i,\widetilde{t}}^{\prime}\,,$
$\displaystyle C^{\prime}(n)$
$\displaystyle=\sideset{}{{}_{1}}{\sum}_{i=1}^{r}n_{i}\cdot_{1}\left(\sideset{}{{}_{2}}{\sum}_{j=1}^{r}n_{j}\cdot_{2}P^{\prime}_{i,j}\right)\,,$
$\displaystyle D^{\prime}(n)$
$\displaystyle=\left(C^{\prime}(n)+_{2}B^{\prime}(n)\right)+_{1}\left(A^{\prime}(n)+_{2}\widetilde{t}\right)\,.$

This shows how the map $n\mapsto D^{\prime}(n)$ is built up from the two
partial group laws $+_{1}$ and $+_{2}$ on $P^{\times,\rho-1}$, and the
iterations ${\cdot}_{1}$ and ${\cdot}_{2}$. Lemma 5.1.1 gives that the
iterations are given by integral convergent power series. The functoriality in
Section 3 gives that the maps induced by $+_{1}$ and $+_{2}$ on residue
polydisks are given by integral convergent power series. Stability under
composition then gives that $n\mapsto D^{\prime}(n)$ is given by elements
$\kappa_{1}^{\prime},\ldots,\kappa_{g+\rho-1}^{\prime}$ of
$\mathbb{Z}_{p}\langle z_{1},\ldots,z_{r}\rangle$.

We call the $\kappa_{i}^{\prime}$ the coordinate functions of the extension
$D^{\prime}\colon\mathbb{Z}_{p}^{r}\to
T(\mathbb{Z}_{p})_{t}=\mathbb{Z}_{p}^{g+\rho-1}$, and their images
$\overline{\kappa}_{1}^{\prime},\ldots,\overline{\kappa}_{g+\rho-1}^{\prime}$
in $\mathbb{F}_{p}[z_{1},\ldots,z_{r}]$ the mod $p$ coordinate functions,
viewed as a morphism
$\overline{D}^{\prime}_{\mathbb{F}_{p}}\colon\mathbb{A}^{r}_{\mathbb{F}_{p}}\to\mathbb{A}^{g+\rho-1}_{\mathbb{F}_{p}}$.

The mod $p$ coordinate functions of $A^{\prime}\colon\mathbb{Z}_{p}^{r}\to
P^{\times,\rho-1}(\mathbb{Z}_{p})=\mathbb{Z}_{p}^{\rho g+\rho-1}$ (after
choosing the necessary parameters) are all of degree at most 1. The same holds
for $B^{\prime}$. We define

$None$ $C^{\prime}_{2}\colon\mathbb{Z}^{r}\times\mathbb{Z}^{r}\longrightarrow
P^{\times,\rho-1}(\mathbb{Z}_{p}),\quad\displaystyle
C^{\prime}_{2}(n,m)=\sideset{}{{}_{1}}{\sum}_{i=1}^{r}n_{i}\cdot_{1}\left(\sideset{}{{}_{2}}{\sum}_{j=1}^{r}m_{j}\cdot_{2}P^{\prime}_{i,j}\right)\,.$

Then the mod $p$ coordinate functions of $C^{\prime}_{2}$, elements of
$\mathbb{F}_{p}[x_{1},\ldots,x_{r},y_{1},\ldots,y_{r}]$, are linear in the
$x_{i}$, and in the $y_{j}$. Hence of degree at most 2, and the same follows
for the mod $p$ coordinate functions of $C^{\prime}$. However, as the first
$\rho g$ parameters for $P^{\times,\rho-1}$ come from $J\times
J^{\vee\rho-1}$, and the 1st and 2nd partial group laws there act on different
factors, the first $\rho g$ mod $p$ coordinate functions of $C^{\prime}$ are
in fact linear. As $D^{\prime}$ is obtained by summing, using the partial
group laws, the results of $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$, we
conclude that
$\overline{\kappa}_{1}^{\prime},\ldots,\overline{\kappa}_{g}^{\prime}$ are of
degree at most 1, and the remaining $\overline{\kappa}_{j}$ are of degree at
most 2. The same holds then for all $\overline{\kappa}_{j}$. ∎

###  5.3 The $p$-adic closure

We know from ($None$) that
$\kappa_{\mathbb{Z}}(\mathbb{Z}^{r})=D^{\prime}((p-1)\mathbb{Z}^{r})$. From
($None$) we know that $\kappa_{\mathbb{Z}}(\mathbb{Z}^{r})\subset
T(\mathbb{Z})_{t}$. From ($None$) we know that $T(\mathbb{Z})_{t}\subset
D^{\prime}(\mathbb{Z}^{r})$. So together we have:

$None$
$D^{\prime}((p-1)\mathbb{Z}^{r})=\kappa_{\mathbb{Z}}(\mathbb{Z}^{r})\subset
T(\mathbb{Z})_{t}\subset D^{\prime}(\mathbb{Z}^{r})\,.$

We have extended $D^{\prime}$ to a continuous map $\mathbb{Z}_{p}^{r}\to
T(\mathbb{Z}_{p})_{t}$. As $\mathbb{Z}_{p}^{r}$ is compact,
$D^{\prime}(\mathbb{Z}_{p}^{r})$ is closed in $T(\mathbb{Z}_{p})_{t}$. As
$\mathbb{Z}^{r}$ and $(p-1)\mathbb{Z}^{r}$ are dense in $\mathbb{Z}_{p}^{r}$,
the closures of their images under $D^{\prime}$ are both equal to
$D^{\prime}(\mathbb{Z}_{p}^{r})$, and equal to $\kappa(\mathbb{Z}_{p}^{r})$.
This finishes the proof of Theorem 4.10.

##  6 Explicit description of the Poincaré torsor

The aim of this section is to give explicit descriptions of the Poincaré
torsor $P^{\times}$ on $J\times J^{\vee,0}$ and its partial group laws, to be
used for doing computations when applying Theorem 4.12. The main results are
as follows. Proposition 6.3.2 describes the fibre of $P$ over a point of
$J\times J^{\vee,0}$, say with values in $\mathbb{Z}/p^{2}\mathbb{Z}$ with $p$
not dividing $n$ or in $\mathbb{Z}[1/n]$, when the corresponding points of $J$
and $J^{\vee,0}$ are given by a line bundle on $C$ (over
$\mathbb{Z}/p^{2}\mathbb{Z}$ or $\mathbb{Z}[1/n]$, and rigidified at $b$) and
an effective relative Cartier divisor on $C$ (over
$\mathbb{Z}/p^{2}\mathbb{Z}$ or $\mathbb{Z}[1/n]$). It also translates the
partial group laws of $P^{\times}$ in terms of such data. Lemma 6.4.8 shows
how to deal with linear equivalence of divisors. Lemma 6.5.4 makes the
symmetry of $P^{\times}$ explicit. Lemma 6.6.8 gives parametrisations of
residue polydisks of $P^{\times}(\mathbb{Z}/p^{2}\mathbb{Z})$, and Lemma
6.6.13 gives partial group laws on these residue polydisks. Proposition 6.8.7
describes the unique extension over $J\times J^{\vee,0}$ of the Poincaré
torsor on $(J\times J^{\vee,0})_{\mathbb{Z}[1/n]}$, in terms of line bundles
and divisors on $C$. Finally, Proposition 6.9.3 describes the fibres of $P$
over $\mathbb{Z}$-points of $J\times J^{\vee,0}$.

In this article, we have chosen to use line bundles and divisors on curves for
describing the jacobian and the Poincaré torsor. Another option is to use line
bundles on curves and the determinant of coherent cohomology, as in Section 2
of [25]. We note that in Section 2, only the restriction of $P$ to
$J^{0}\times J^{\vee,0}$ is treated, and moreover, under the assumption that
$C$ is nodal (that is, all fibres $C_{\mathbb{F}_{p}}$ are reduced and have
only the mildest possible singularities). Another choice we have made is to
develop the basic theory of norms of $\mathbb{G}_{\mathrm{m}}$-torsors under
finite locally free morphisms in this article (Sections 6.1–6.2) and not to
refer, for example, to EGA or SGA, because we think this is easier for the
reader, and because this way we could adapt the definition directly to our use
of it.

###  6.1 Norms

Let $S$ be a scheme, $f\colon S^{\prime}\to S$ be finite and locally free, say
of rank $n$. Then $\mathcal{O}_{S^{\prime}}=f_{*}\mathcal{O}_{S^{\prime}}$ (we
view $\mathcal{O}_{S^{\prime}}$ as a sheaf on $S$) is an
$\mathcal{O}_{S}$-algebra, locally free as $\mathcal{O}_{S}$-module of rank
$n$, and $\mathcal{O}_{S^{\prime}}^{\times}$ is a subsheaf of groups of
$\mathrm{GL}_{\mathcal{O}_{S}}(\mathcal{O}_{S^{\prime}})$. Then the norm
morphism is the composition

$None$
${\mathcal{O}_{S^{\prime}}^{\times}}$${\mathrm{GL}_{\mathcal{O}_{S}}(\mathcal{O}_{S^{\prime}})}$${\mathcal{O}_{S}^{\times}\,.}$$\scriptstyle{\mathrm{Norm}_{S^{\prime}/S}}$$\scriptstyle{\det}$

Our viewing of $\mathcal{O}_{S^{\prime}}$ as a sheaf on $S$ does not change
the notion of $\mathcal{O}_{S^{\prime}}^{\times}$-torsor, because of the
equivalence with invertible $\mathcal{O}_{S^{\prime}}$-modules: triviality
locally on $S^{\prime}$ implies triviality locally on $S$.

For $T$ an $\mathcal{O}_{S^{\prime}}^{\times}$-torsor, we let
$\mathrm{Norm}_{S^{\prime}/S}(T)$ be the $\mathcal{O}_{S}^{\times}$-torsor

$None$
$\mathrm{Norm}_{S^{\prime}/S}(T):=\mathcal{O}_{S}^{\times}\otimes_{\mathcal{O}_{S^{\prime}}^{\times}}T=\left(\mathcal{O}_{S}^{\times}\times
T\right)/\mathcal{O}_{S^{\prime}}^{\times}\,,$

with, for every open $U$ of $S$, and every
$u\in\mathcal{O}_{S^{\prime}}^{\times}(U)$, $u$ acting as
$(v,t)\mapsto(v{\cdot}\mathrm{Norm}_{S^{\prime}/S}(u),u^{-1}{\cdot}t)$. This
is functorial in $T$: a morphism $\varphi\colon T_{1}\to T_{2}$ induces an
isomorphism $\mathrm{Norm}_{S^{\prime}/S}(\varphi)$. It is also functorial for
cartesian diagrams $(S_{2}^{\prime}\to S_{2})\to(S_{1}^{\prime}\to S_{1})$.

For $U\subset S$ open, $T$ an $\mathcal{O}_{S^{\prime}}^{\times}$-torsor, and
$t\in T(U)$, we have the isomorphism of
$\mathcal{O}_{S^{\prime}}^{\times}|_{U}$-torsors
$\mathcal{O}_{S^{\prime}}^{\times}|_{U}\to T|_{U}$ sending $1$ to $t$.
Functoriality gives $\mathrm{Norm}_{S^{\prime}/S}(t)$ in
$(\mathrm{Norm}_{S^{\prime}/S}(T))(U)$, also denoted $1\otimes t$.

The norm functor ($None$) is multiplicative:

$None$
$\mathrm{Norm}_{S^{\prime}/S}(T_{1}\otimes_{\mathcal{O}_{S^{\prime}}}T_{2})=\mathrm{Norm}_{S^{\prime}/S}(T_{1})\otimes_{\mathcal{O}_{S}}\mathrm{Norm}_{S^{\prime}/S}(T_{2})\,,$

such that, if $U\subset S$ is open and $t_{1}$ and $t_{2}$ are in $T_{1}(U)$
and $T_{2}(U)$, then

$None$ $\mathrm{Norm}_{S^{\prime}/S}(t_{1}\otimes
t_{2})\mapsto\mathrm{Norm}_{S^{\prime}/S}(t_{1})\otimes\mathrm{Norm}_{S^{\prime}/S}(t_{2})\,.$

Let $\mathcal{L}$ be an invertible $\mathcal{O}_{S^{\prime}}$-module; locally
on $S$, it is free of rank $1$ as $\mathcal{O}_{S^{\prime}}$-module. This
gives us the $\mathcal{O}_{S^{\prime}}^{\times}$-torsor (on $S$)
$\mathrm{Isom}_{\mathcal{O}_{S^{\prime}}}(\mathcal{O}_{S^{\prime}},\mathcal{L})$,
which gives back $\mathcal{L}$ as
$\mathcal{L}=\mathcal{O}_{S^{\prime}}\otimes_{\mathcal{O}_{S^{\prime}}^{\times}}\mathrm{Isom}_{\mathcal{O}_{S^{\prime}}}(\mathcal{O}_{S^{\prime}},\mathcal{L})$.
The norm of $\mathcal{L}$ via $f\colon S^{\prime}\to S$ is then defined as

$None$
$\mathrm{Norm}_{S^{\prime}/S}(\mathcal{L}):=\mathcal{O}_{S}\otimes_{\mathcal{O}_{S}^{\times}}\mathrm{Norm}_{S^{\prime}/S}(\mathrm{Isom}_{\mathcal{O}_{S^{\prime}}}(\mathcal{O}_{S^{\prime}},\mathcal{L}))\,.$

This construction is functorial for isomorphisms of invertible
$\mathcal{O}_{S^{\prime}}$-modules.

###  6.2 Norms along finite relative Cartier divisors

This part is inspired by [21], section 1.1. Let $S$ be a scheme, let $f\colon
X\to S$ be an $S$-scheme of finite presentation. A finite effective relative
Cartier divisor on $f\colon X\to S$ is a closed subscheme $D$ of $X$ that is
finite and locally free over $S$, and whose ideal sheaf $I_{D}$ is locally
generated by a non-zero divisor (equivalently, $I_{D}$ is locally free of rank
$1$ as $\mathcal{O}_{X}$-module). For such a $D$ and an invertible
$\mathcal{O}_{X}$-module $\mathcal{L}$, the norm of $\mathcal{L}$ along $D$ is
defined, using ($None$), as

$None$
$\mathrm{Norm}_{D/S}(\mathcal{L}):=\mathrm{Norm}_{D/S}(\mathcal{L}|_{D})\,.$

Then $\mathrm{Norm}_{D/S}(\mathcal{L})$ is functorial for cartesian diagrams
$(X^{\prime}\to S^{\prime},\mathcal{L}^{\prime})\to(X\to S,\mathcal{L})$.

###### 6.2.2 Lemma

Let $f\colon X\to S$ be a morphism of schemes that is of finite presentation.
For $D$ a finite effective relative Cartier divisor on $f$, the norm functor
$\mathrm{Norm}_{D/S}$ in ($None$) is multiplicative in $\mathcal{L}$:

$None$
$\mathrm{Norm}_{D/S}(\mathcal{L}_{1}\otimes\mathcal{L}_{2})=\mathrm{Norm}_{D/S}(\mathcal{L}_{1})\otimes_{\mathcal{O}_{S}}\mathrm{Norm}_{D/S}(\mathcal{L}_{2})\,,$

with, for $U\subset S$ open, $V\subset X$ open, containing $f^{-1}U\cap D$ and
$l_{i}\in\mathcal{L}_{i}(V)$ generating $\mathcal{L}_{i}|_{V}$,

$None$ $\mathrm{Norm}_{D/S}(l_{1}\otimes
l_{2})=\mathrm{Norm}_{D/S}(l_{1})\otimes\mathrm{Norm}_{D/S}(l_{2})\,.$

Let $D_{1}$ and $D_{2}$ be finite effective relative Cartier divisors on $f$.
Then the ideal sheaf $I_{D_{1}}I_{D_{2}}\subset\mathcal{O}_{X}$ is locally
free of rank $1$, the closed subscheme $D_{1}+D_{2}$ defined by it is a finite
effective relative Cartier divisor on $f$. The norm functor in ($None$) is
additive in $D$:

$None$
$\mathrm{Norm}_{(D_{1}+D_{2})/S}(\mathcal{L})=\mathrm{Norm}_{D_{1}/S}(\mathcal{L})\otimes_{\mathcal{O}_{S}}\mathrm{Norm}_{D_{2}/S}(\mathcal{L})\,,$

with, for $U\subset S$ open, $V\subset X$ open, containing
$f^{-1}U\cap(D_{1}+D_{2})$ and $l\in\mathcal{L}(V)$ generating
$\mathcal{L}|_{D_{1}+D_{2}}$,

$None$
$\mathrm{Norm}_{(D_{1}+D_{2})/S}(l)=\mathrm{Norm}_{D_{1}/S}(l)\otimes\mathrm{Norm}_{D_{2}/S}(l)\,.$

###### Proof.

Let $D_{1}$ and $D_{2}$ be as stated. If $V\subset X$ is open, and $f_{i}$
generates $I_{D_{i}}|_{V}$, then $f_{1}f_{2}$ generates
$(I_{D_{1}}I_{D_{2}})|_{V}$, and this element of $\mathcal{O}_{X}(V)$ is not a
zero-divisor because $f_{1}$ and $f_{2}$ are not. To show that $D_{1}+D_{2}$
is finite over $S$, we replace $S$ by an affine open of it, and then reduce to
the noetherian case, using the assumption that $f$ is of finite presentation.
Then, $(D_{1}+D_{2})_{\mathrm{red}}$ is the image of
$D_{1,\mathrm{red}}\coprod D_{2,\mathrm{red}}\to X$, and therefore is proper.
Hence $D_{1}+D_{2}$ is proper over $S$, and quasi-finite over $S$, hence
finite over $S$. The short exact sequence

$None$
${I_{D_{2}}/I_{D_{1}+D_{2}}}$${\mathcal{O}_{D_{1}+D_{2}}}$${\mathcal{O}_{D_{2}}}$${(I_{D_{2}})|_{D_{1}}}$

shows that $\mathcal{O}_{D_{1}+D_{2}}$ is locally free as
$\mathcal{O}_{S}$-module, of rank the sum of the ranks of the
$\mathcal{O}_{D_{i}}$. So $D_{1}+D_{2}$ is a finite effective relative Cartier
divisor on $X\to S$.

We prove ($None$), by proving the required statement about sheaves of groups.
The diagram

$None$
${\mathcal{O}_{D_{1}+D_{2}}^{\times}}$${\mathcal{O}_{D_{1}}^{\times}\times\mathcal{O}_{D_{2}}^{\times}}$${\mathcal{O}_{S}^{\times}\times\mathcal{O}_{S}^{\times}}$${\mathcal{O}_{S}^{\times}}$${u}$${\mathrm{Norm}_{D_{1}/S}(u)\mathrm{Norm}_{D_{2}/S}(u)\,.}$$\scriptstyle{\mathrm{Norm}_{(D_{1}+D_{2})/S}}$$\scriptstyle{\mathrm{Norm}_{D_{1}/S}\times\mathrm{Norm}_{D_{2}/S}}$$\scriptstyle{\cdot}$

commutes, because multiplication by $u$ on $\mathcal{O}_{D_{1}+D_{2}}$
preserves the short exact sequence ($None$), multiplying on the sub and
quotient by its images in $\mathcal{O}_{D_{1}}^{\times}$ and in
$\mathcal{O}_{D_{2}}^{\times}$; note that the sub is an invertible
$\mathcal{O}_{D_{1}}$-module. ∎

###  6.3 Explicit description of the Poincaré torsor of a smooth curve

Let $g$ be in $\mathbb{Z}_{\geq 1}$, let $S$ be a scheme, and $\pi\colon C\to
S$ be a proper smooth curve, with geometrically connected fibres of genus $g$,
with a section $b\in C(S)$. Let $J\to S$ be its jacobian. On $C\times_{S}J$ we
have $\mathcal{L}^{\mathrm{univ}}$, the universal invertible
$\mathcal{O}$-module of degree zero on $C$, rigidified at $b$.

Let $d\geq 0$, and $C^{(d)}$ the $d$th symmetric power of $C\to S$ (we note
that the quotient $C^{d}\to C^{(d)}$ is finite, locally free of rank $d!$, and
commutes with base change on $S$). Then on $C\times_{S}C^{(d)}$ we have $D$,
the universal effective relative Cartier divisor on $C$ of degree $d$. Hence,
on $C\times_{S}J\times_{S}C^{(d)}$ we have their pullbacks $D_{J}$ and
$\mathcal{L}^{\mathrm{univ}}_{C^{(d)}}$, giving us

$None$
$\mathcal{N}_{d}:=\mathrm{Norm}_{D_{J}/(J\times_{S}C^{(d)})}(\mathcal{L}^{\mathrm{univ}}_{C^{(d)}})\,.$

This invertible $\mathcal{O}$-module $\mathcal{N}_{d}$ on
$J\times_{S}C^{(d)}$, rigidified at the zero-section of $J$, gives us a
morphism of $S$-schemes $C^{(d)}$ to $\mathrm{Pic}_{J/S}$. The point $db$ (the
divisor $d$ times the base point $b$) in $C^{(d)}(S)$ is mapped to $0$,
precisely because $\mathcal{L}^{\mathrm{univ}}$ is rigidified at $b$, and
$None$. Hence there is a unique morphism $\Box\colon C^{(d)}\to
J^{\vee}=\mathrm{Pic}^{0}_{J/S}$ such that the pullback of the Poincaré bundle
$P$ on $J\times J^{\vee}$ by $(\mathrm{id},\Box)\colon J\times C^{(d)}\to
J\times J^{\vee}$, with its rigidifications, is the same as $\mathcal{N}_{d}$.
The following proposition tells us what the morphism $\Box$ is, and the next
section tells us what the induced isomorphism is between the fibres of
$\mathcal{N}_{d}$ at points of $J\times C^{(d)}$ with the same image in
$J\times_{S}J$.

###### 6.3.2 Proposition

The pullback of $P$ by $(j_{b},j_{b}^{*,-1})\colon C\times_{S}J\to
J\times_{S}J^{\vee}$ together with its rigidifications at $b$ and $0$, is
equal to $\mathcal{L}^{\mathrm{univ}}$.

Let $d$ be in $\mathbb{Z}_{\geq 0}$. The morphism $\Box\colon C^{(d)}\to
J^{\vee}=\mathrm{Pic}^{0}_{J/S}$ is the composition of first $\Sigma\colon
C^{(d)}\to J$, sending, for every $S$-scheme $T$, each point $D\in C^{(d)}(T)$
to the class of $\mathcal{O}_{C_{T}}(D-db)$ twisted by the pullback from $T$
that makes it rigidified at $b$, followed by $j_{b}^{*,-1}\colon J\to
J^{\vee}$. Summarised in a diagram, with $\mathcal{M}:=(\mathrm{id}\times
j_{b}^{*,-1})^{*}P$:

$None$
${\mathcal{L}^{\mathrm{univ}}}$${P}$${\mathcal{M}}$${\mathcal{N}_{d}}$${C\times_{S}J}$${J\times_{S}J^{\vee}}$${J\times_{S}J}$${J\times_{S}C^{(d)}\,.}$$\scriptstyle{\widetilde{\mathrm{id}\times\Sigma}}$$\scriptstyle{j_{b}\times
j_{b}^{*,-1}}$$\scriptstyle{\mathrm{id}\times
j_{b}^{*,-1}}$$\scriptstyle{\mathrm{id}\times\Sigma}$

Then $\mathcal{M}$, with its rigidifications at $\\{0\\}\times_{S}J$ and
$J\times_{S}\\{0\\}$, is symmetric. For $T\to S$, $x$ in $J(T)$ given by an
invertible $\mathcal{O}$-module $\mathcal{L}$ on $C_{T}$ rigidified at $b$,
and $y=\Sigma(D)$ in $J(T)$ given by an effective relative divisor $D$ of
degree $d$ on $C_{T}$ we have

$None$
$P\left(x,j_{b}^{*,-1}(y)\right)=\mathcal{M}(x,y)=\mathrm{Norm}_{D/T}(\mathcal{L})\,.$

For $c_{1}$ and $c_{2}$ in $C(S)$, we have

$None$
$\mathcal{M}\left(j_{b}(c_{1}),j_{b}(c_{2})\right)=c_{2}^{*}\left(\mathcal{O}_{C}(c_{1}-b)\right)\otimes
b^{*}\left(\mathcal{O}_{C}(b-c_{1})\right)\,,$

and, as invertible $\mathcal{O}$-modules on $C\times_{S}C$, with $\Delta$ the
diagonal and $\mathrm{pr}_{\emptyset}\colon C\times_{S}C\to S$ the structure
morphism, we have

$None$ $(j_{b}\times
j_{b})^{*}\mathcal{M}=\mathcal{O}(\Delta)\otimes\mathrm{pr}_{1}^{*}\mathcal{O}(-b)\otimes\mathrm{pr}_{2}^{*}\mathcal{O}(-b)\otimes\mathrm{pr}_{\emptyset}^{*}b^{*}T_{C/S}\,.$

For $d>2g-2$, $\widetilde{\mathrm{id}\times\Sigma}$ gives $\mathcal{N}_{d}$ a
descent datum along $\mathrm{id}\times\Sigma$ that gives $\mathcal{M}$ on
$J\times_{S}J$. For $T$ an $S$-scheme, $x\in J(S)$ given by $\mathcal{L}$ on
$C_{T}$, rigidified at $b$, $D_{1}$ and $D_{2}$ in $C^{(d_{1})}(S)$ and
$C^{(d_{2})}(S)$, the isomorphism

$None$
$\mathcal{M}(x,\Sigma(D_{1}+D_{2}))=\mathcal{M}(x,\Sigma(D_{1}))\otimes\mathcal{M}(x,\Sigma(D_{2}))$

corresponds, via $\widetilde{\mathrm{id}\times\Sigma}$, to

$None$ $\displaystyle\mathcal{N}_{d_{1}+d_{2}}(x,D_{1}+D_{2})$
$\displaystyle=\mathrm{Norm}_{(D_{1}+D_{2})/T}(\mathcal{L})=\mathrm{Norm}_{D_{1}/T}(\mathcal{L})\otimes\mathrm{Norm}_{D_{2}/T}(\mathcal{L})$
$\displaystyle=\mathcal{N}_{d_{1}}(x,D_{1})\otimes\mathcal{N}_{d_{2}}(x,D_{2})\,,$

using Lemma 6.2.2.

For $T$ an $S$-scheme and $x_{1}$ and $x_{2}$ in $J(T)$ given by
$\mathcal{O}$-modules $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ on $C_{T}$,
rigidified at $b$, and $D$ in $C^{(d)}(T)$, the isomorphism

$None$
$\mathcal{M}(x_{1}+x_{2},\Sigma(D))=\mathcal{M}(x_{1},\Sigma(D))\otimes\mathcal{M}(x_{2},\Sigma(D))$

corresponds, via $\widetilde{\mathrm{id}\times\Sigma}$, to

$None$ $\displaystyle\mathcal{N}_{d}(x_{1}+x_{2},D)$
$\displaystyle=\mathrm{Norm}_{D/T}(\mathcal{L}_{1}\otimes\mathcal{L}_{2})=\mathrm{Norm}_{D/T}(\mathcal{L}_{1})\otimes\mathrm{Norm}_{D/T}(\mathcal{L}_{2})$
$\displaystyle=\mathcal{N}_{d}(x_{1},D)\otimes\mathcal{N}_{d}(x_{2},D)\,,$

using Lemma 6.2.2.

###### Proof.

Let $T$ be an $S$-scheme, and $x$ be in $J(T)$. Then $x$ corresponds to the
invertible $\mathcal{O}$-module $(\mathrm{id}\times
x)^{*}\mathcal{L}^{\mathrm{univ}}$ on $C_{T}$, rigidified at $b$. Let
$z:=j_{b}^{*,-1}(x)$ in $J^{\vee}(T)$. Then $j_{b}^{*}(z)=x$, meaning that the
pullback of $(\mathrm{id}\times z)^{*}P$ on $J_{T}$ rigidified at $0$ by
$j_{b}$ equals $(\mathrm{id}\times x)^{*}\mathcal{L}^{\mathrm{univ}}$ on
$C_{T}$ rigidified at $b$. Taking $T:=J$ and $x$ the tautological point gives
the first claim of the proposition.

The symmetry of $\mathcal{M}$ with its rigidifications follows from [25],
(2.7.1) and Lemma 2.7.5, and (2.7.7), using $None$.

Now we prove ($None$). So let $T$ and $x$ be as above, and $y=\Sigma(D)$ in
$J(T)$ given by a relative divisor $D$ of degree $d$ on $C_{T}$. As $C^{d}\to
C^{(d)}$ is finite and locally free of rank $d!$, we may and do suppose that
$D$ is a sum of sections, say $D=\sum_{i=1}^{d}(c_{i})$, with $c_{i}\in C(T)$.
Then we have, functorially:

$None$ $\displaystyle P(x,j_{b}^{*,-1}(y))$
$\displaystyle=P(y,j_{b}^{*,-1}(x))=P(\Sigma(D),j_{b}^{*,-1}(x))$
$\displaystyle=P\left(\sum_{i}j_{b}(c_{i}),j_{b}^{*,-1}(x)\right)=\bigotimes_{i}P(j_{b}(c_{i}),j_{b}^{*,-1}(x))$
$\displaystyle=\bigotimes_{i}\mathcal{L}^{\mathrm{univ}}(c_{i},x)=\bigotimes_{i}\mathcal{L}(c_{i})=\mathrm{Norm}_{D/T}(\mathcal{L})\,.$

Identities ($None$) and ($None$) follow directly from ($None$).

Now we prove the claimed compatibility between ($None$) and ($None$). We do
this by considering the case where $\mathcal{L}$ is universal, that is, base
changing to $J_{T}$ and $x$ the universal point. Then, on $J_{T}$, we have 2
isomorphisms from $\mathrm{Norm}_{(D_{1}+D_{2})/J_{T}}(\mathcal{L})$ to
$\mathrm{Norm}_{D_{1}/J_{T}}(\mathcal{L})\otimes\mathrm{Norm}_{D_{2}/J_{T}}(\mathcal{L})$.
These differ by an element of
$\mathcal{O}(J_{T})^{\times}=\mathcal{O}(T)^{\times}$. Hence it suffices to
check that this element equals $1$ at $0\in J(T)$. This amounts to checking
that the 2 isomorphisms are equal for $\mathcal{L}=\mathcal{O}_{C_{T}}$ with
the standard rigidification at $b$. Then, both isomorphisms are the
multiplication map
$\mathcal{O}_{T}\otimes_{\mathcal{O}_{T}}\mathcal{O}_{T}\to\mathcal{O}_{T}$.

The compatibility between ($None$) and ($None$) is proved analogously. ∎

###### 6.3.12 Remark

From Proposition 6.3.2 one easily deduces, in that situation, for $T$ an
$S$-scheme, $x$ in $J(T)$ given by an invertible $\mathcal{O}$-module
$\mathcal{L}$ on $C_{T}$, and $D_{1}$ and $D_{2}$ effective relative Cartier
divisors on $C_{T}$, of the same degree, a canonical isomorphism

$None$
$\mathcal{M}(x,\Sigma(D_{1})-\Sigma(D_{2}))=\mathrm{Norm}_{D_{1}/T}(\mathcal{L})\otimes\mathrm{Norm}_{D_{2}/T}(\mathcal{L})^{-1}\,,$

satisfying the analogous compatibilities as in Proposition 6.3.2. No
rigidification of $\mathcal{L}$ at $b$ is needed. In fact, for
$\mathcal{L}_{0}$ an invertible $\mathcal{O}_{T}$-module, we have
$\mathrm{Norm}_{D_{1}/T}(\pi^{*}\mathcal{L}_{0})=\mathcal{L}_{0}^{\otimes d}$,
where $\pi\colon C_{T}\to T$ is the structure morphism and $d$ is the degree
of $D_{1}$. Hence the right hand side of ($None$) is independent of the choice
of $\mathcal{L}$, given $x$.

###  6.4 Explicit isomorphism for norms along equivalent divisors

Let $g$ be in $\mathbb{Z}_{\geq 1}$, let $S$ be a scheme, and $p\colon C\to S$
be a proper smooth curve, with geometrically connected fibres of genus $g$,
with a section $b\in C(S)$. Let $D_{1},D_{2}$ be effective relative Cartier
divisors of degree $d$ on $C$, that we also view as elements of $C^{(d)}(S)$.
Recall from Proposition 6.3.2 the morphism $\Sigma\colon C^{(d)}\to J$. Then
$\Sigma(D_{1})=\Sigma(D_{2})$ if and only if $D_{1},D_{2}$ are linearly
equivalent in the following sense: locally on $S$, there exists an $f$ in
$\mathcal{O}_{C}(U)^{\times}$, with $U:=C\setminus(D_{1}\cup D_{2})$, such
that $f{\cdot}\colon\mathcal{O}_{U}\to\mathcal{O}_{U}$ extends to an
isomorphism $f{\cdot}\colon\mathcal{O}_{C}(D_{1})\to\mathcal{O}_{C}(D_{2})$.
In this case, we define $\mathrm{div}(f)=D_{2}-D_{1}$. Proposition 6.3.2 gives
us, for each invertible $\mathcal{O}$-module $\mathcal{L}$ of degree $0$ on
$C$ rigidified at $b$ (viewed as an element of $J(S)$) specific isomorphisms

$None$ $\displaystyle\mathrm{Norm}_{D_{1}/S}(\mathcal{L})$
$\displaystyle=\mathcal{N}_{d}(\mathcal{L},D_{1})=\mathcal{M}(\mathcal{L},\Sigma(D_{1}))=\mathcal{M}(\mathcal{L},\Sigma(D_{2}))=\mathcal{N}_{d}(\mathcal{L},D_{2})$
$\displaystyle=\mathrm{Norm}_{D_{2}/S}(\mathcal{L})\,.$

Now we describe explicitly this isomorphism
$\mathrm{Norm}_{D_{1}/S}(\mathcal{L})\to\mathrm{Norm}_{D_{2}/S}(\mathcal{L})$.
To do so we first describe _an_ isomorphism

$None$
$\varphi_{\mathcal{L},D_{1},D_{2}}\colon\mathrm{Norm}_{D_{1}/S}(\mathcal{L})\longrightarrow\mathrm{Norm}_{D_{2}/S}(\mathcal{L})$

that is functorial for Cartesian diagrams $(C^{\prime}\to
S^{\prime},\mathcal{L}^{\prime},D_{1}^{\prime},D_{2}^{\prime})\to(C\to
S,\mathcal{L},D_{1},D_{2})$ and then we prove that _this_ isomorphism is the
one in ($None$).

We construct $\varphi_{\mathcal{L},D_{1},D_{2}}$ locally on $S$ and the
functoriality of the construction takes care of making it global. So, suppose
that $f$ is as above: $f\in\mathcal{O}_{C}(U)^{\times}$, and
$f{\cdot}\colon\mathcal{O}_{U}\to\mathcal{O}_{U}$ extends to an isomorphism
$f{\cdot}\colon\mathcal{O}_{C}(D_{1})\to\mathcal{O}_{C}(D_{2})$. Let
$n\in\mathbb{Z}$ with $n>2g-2+2d$. Then $p_{*}(\mathcal{L}(nb))\to
p_{*}\mathcal{L}(nb)|_{D_{1}+D_{2}}$ and $p_{*}(\mathcal{O}_{C}(nb))\to
p_{*}\mathcal{O}_{C}(nb)|_{D_{1}+D_{2}}$ are surjective, and (still localising
on $S$) $p_{*}(\mathcal{L}(nb))$ and $p_{*}(\mathcal{O}_{C}(nb))$ are free
$\mathcal{O}_{S}$-modules and $\mathcal{L}(nb)|_{D_{1}+D_{2}}$ and
$\mathcal{O}_{C}(nb)|_{D_{1}+D_{2}}$ are free
$\mathcal{O}_{D_{1}+D_{2}}$-modules of rank 1. Then we have $l_{0}$ in
$(\mathcal{L}(nb))(C)$ and $l_{1}$ in $(\mathcal{O}_{C}(nb))(C)$ restricting
to generators on $D_{1}+D_{2}$. Let $D^{-}:=\mathrm{div}(l_{1})$ and
$D^{+}:=\mathrm{div}(l_{0})$, and let $V:=C\setminus(D^{+}+D^{-})$. Note that
$V$ contains $D_{1}+D_{2}$ and that $U$ contains $D^{+}+D^{-}$. Then, on $V$,
$l:=l_{0}/l_{1}$ is in $\mathcal{L}(V)$, generates
$\mathcal{L}|_{D_{1}+D_{2}}$, and multiplication by $l$ is an isomorphism
${\cdot}l\colon\mathcal{O}_{C}(D^{+}-D^{-})\to\mathcal{L}$, that is,
$\mathrm{div}(l)=D^{+}-D^{-}$. Let

$None$
$f(\mathrm{div}(l))=f(D^{+}-D^{-}):=\mathrm{Norm}_{D^{+}/S}(f|_{D^{+}})\cdot\mathrm{Norm}_{D^{-}/S}(f|_{D^{-}})^{-1}\in\mathcal{O}_{S}(S)^{\times}\,,$

and let $\varphi_{\mathcal{L},l,f}$ be the isomorphism, given in terms of
generators

$None$
$\displaystyle\varphi_{\mathcal{L},l,f}\colon\mathrm{Norm}_{D_{1}/S}(\mathcal{L})$
$\displaystyle\longrightarrow\mathrm{Norm}_{D_{2}/S}(\mathcal{L})$
$\displaystyle\mathrm{Norm}_{D_{1}/S}(l)$ $\displaystyle\longmapsto
f(\mathrm{div}(l))^{-1}\cdot\mathrm{Norm}_{D_{2}/S}(l)\,.$

Now suppose that we made other choices $n^{\prime}$, $l_{0}^{\prime}$,
$l_{1}^{\prime}$. Then we get ${D^{-}}^{\prime}$, ${D^{+}}^{\prime}$,
$V^{\prime}$, $l^{\prime}$ and $\varphi_{\mathcal{L},l^{\prime},f}$. Then
there is a unique function $g\in\mathcal{O}_{C}(V\cap V^{\prime})^{\times}$
such that $l^{\prime}=gl$ in $\mathcal{L}(V\cap V^{\prime})$. Then

$None$
$\displaystyle\varphi_{\mathcal{L},l^{\prime},f}(\mathrm{Norm}_{D_{1}/S}(l))$
$\displaystyle=\varphi_{\mathcal{L},l^{\prime},f}(\mathrm{Norm}_{D_{1}/S}(g^{-1}l^{\prime}))$
$\displaystyle=\varphi_{\mathcal{L},l^{\prime},f}(g^{-1}(D_{1})\mathrm{Norm}_{D_{1}/S}(l^{\prime}))$
$\displaystyle=g^{-1}(D_{1}){\cdot}\varphi_{\mathcal{L},l^{\prime},f}(\mathrm{Norm}_{D_{1}/S}(l^{\prime}))$
$\displaystyle=g^{-1}(D_{1}){\cdot}f(\mathrm{div}(l^{\prime}))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(l^{\prime})$
$\displaystyle=g^{-1}(D_{1}){\cdot}f(\mathrm{div}(gl))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(gl)$
$\displaystyle=g^{-1}(D_{1}){\cdot}f(\mathrm{div}(g)+\mathrm{div}(l))^{-1}{\cdot}g(D_{2}){\cdot}\mathrm{Norm}_{D_{2}/S}(l)$
$\displaystyle=g^{-1}(D_{1}){\cdot}f(\mathrm{div}(g))^{-1}{\cdot}g(D_{2}){\cdot}f(\mathrm{div}(l))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(l)$
$\displaystyle=g(\mathrm{div}(f)){\cdot}f(\mathrm{div}(g))^{-1}{\cdot}\varphi_{\mathcal{L},l,f}(\mathrm{Norm}_{D_{1}/S}(l))$
$\displaystyle=\varphi_{\mathcal{L},l,f}(\mathrm{Norm}_{D_{1}/S}(l))\,,$

where, in the last step, we used Weil reciprocity, in a generality for which
we do not know a reference. The truth in this generality is clear from the
classical case by reduction to the universal case, in which the base scheme is
integral: take a suitable level structure on $J$, then consider the universal
curve with this level structure, and the universal 4-tuple of effective
divisors with the necessary conditions. We conclude that
$\varphi_{\mathcal{L},l,f}=\varphi_{\mathcal{L},l^{\prime},f}$.

Now suppose that $f^{\prime}$ is in $\mathcal{O}_{C}(U)^{\times}$ with
$\mathrm{div}(f^{\prime})=\mathrm{div}(f)$. Then there is a unique
$u\in\mathcal{O}_{S}(S)^{\times}$ such that $f^{\prime}=u{\cdot}f$, and since
$\mathcal{L}$ has degree $0$ on $C$

$None$
$\displaystyle\varphi_{\mathcal{L},l,f^{\prime}}\left(\mathrm{Norm}_{D_{1}/S}(l)\right)$
$\displaystyle=(u{\cdot}f)(\mathrm{div}(l))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(l)$
$\displaystyle=u^{-\deg(\mathrm{div}(l))}f(\mathrm{div}(l))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(l)$
$\displaystyle=f(\mathrm{div}(l))^{-1}{\cdot}\mathrm{Norm}_{D_{2}/S}(l)=\varphi_{\mathcal{L},l,f}\left(\mathrm{Norm}_{D_{1}/S}(l)\right)\,.$

Hence $\varphi_{\mathcal{L},l,f^{\prime}}=\varphi_{\mathcal{L},l,f}$. We
define

$None$
$\varphi_{D_{1},D_{2},\mathcal{L}}\colon\mathrm{Norm}_{D_{1}/S}(\mathcal{L})\longrightarrow\mathrm{Norm}_{D_{2}/S}(\mathcal{L})$

as the isomorphism $\varphi_{\mathcal{L},l,f}$ in ($None$) for any local
choice of $f$ and $l$.

###### 6.4.8 Lemma

With the assumptions as in the beginning of Section 6.4, the isomorphism
$\varphi_{\mathcal{L},D_{1},D_{2}}$ in ($None$) is equal to the isomorphism in
($None$).

###### Proof.

We do this, as in the proof of Proposition 6.3.2, by considering the case of
the universal $\mathcal{L}$, that is, we base change via $J\to S$, and then
restricting to $0\in J(S)$. This amounts to checking that the 2 isomorphisms
are equal for $\mathcal{L}=\mathcal{O}_{C}$ with the standard rigidification
at $b$. In this case,
$\mathrm{Norm}_{D_{i}/S}(\mathcal{O}_{C})=\mathcal{O}_{S}$, with
$\mathrm{Norm}_{D_{i}/S}(1)=1$. Hence
$\varphi_{D_{1},D_{2},\mathcal{O}_{C}}=\varphi_{\mathcal{O}_{C},1,f}$ is the
identity on $\mathcal{O}_{S}$ (use ($None$)). The other isomorphism is the
identity on $\mathcal{O}_{S}$ because of the rigidifications of $\mathcal{M}$
and $\mathcal{N}_{d}$ on $0\times J$ and $0\times C^{(d)}$. ∎

###  6.5 Symmetry of the Norm for divisors on smooth curves

Let $C\to S$ be a proper and smooth curve with geometrically connected fibres.
For $D_{1}$, $D_{2}$ effective relative Cartier divisors on $C$ we define an
isomorphism

$None$
$\varphi_{D_{1},D_{2}}\colon\mathrm{Norm}_{D_{1}/S}(\mathcal{O}_{C}(D_{2}))\longrightarrow\mathrm{Norm}_{D_{2}/S}(\mathcal{O}_{C}(D_{1}))$

that is functorial for cartesian diagrams
$(C^{\prime}/S^{\prime},D^{\prime}_{1},D^{\prime}_{2})\to(C/S,D_{1},D_{2})$.

If suffices to define this isomorphism in the universal case, that is, over
the scheme that parametrises all $D_{1}$ and $D_{2}$. Let $d_{1}$ and $d_{2}$
be in $\mathbb{Z}_{\geq 0}$, and let $U:=C^{(d_{1})}\times_{S}C^{(d_{2})}$,
and let $D_{1}$ and $D_{2}$ be the universal divisors on $C_{U}$. Then we have
the invertible $\mathcal{O}_{U}$-modules
$\mathrm{Norm}_{D_{1}/U}(\mathcal{O}_{C}(D_{2}))$ and
$\mathrm{Norm}_{D_{2}/U}(\mathcal{O}_{C}(D_{1}))$. The image of $D_{1}\cap
D_{2}$ in $U$ is closed, let $U^{0}$ be its complement. Then, over $U^{0}$,
$D_{1}$ and $D_{2}$ are disjoint, and the restrictions of
$\mathrm{Norm}_{D_{1}/U}(\mathcal{O}_{C}(D_{2}))$ and
$\mathrm{Norm}_{D_{2}/U}(\mathcal{O}_{C}(D_{1}))$ are generated by
$\mathrm{Norm}_{D_{1}/U}(1)$ and $\mathrm{Norm}_{D_{2}/U}(1)$, and there is a
unique isomorphism $(\varphi_{D_{1},D_{2}})_{U^{0}}$ that sends
$\mathrm{Norm}_{D_{1}/U}(1)$ to $\mathrm{Norm}_{D_{2}/U}(1)$.

We claim that this isomorphism extends to an isomorphism over $U$. To see it,
we base change by $U^{\prime}\to U$, where
$U^{\prime}=C^{d_{1}}\times_{S}C^{d_{2}}$, then $U^{\prime}\to U$ is finite,
locally free of rank $d_{1}!{\cdot}d_{2}!$. Then
$D_{1}=P_{1}+\cdots+P_{d_{1}}$ and $D_{2}=Q_{1}+\cdots+Q_{d_{2}}$ with the
$P_{i}$ and $Q_{j}$ in $C(U^{\prime})$. The complement of the inverse image
$U^{\prime 0}$ in $U^{\prime}$ of $U^{0}$ is the union of the pullbacks
$D_{i,j}$ under $\mathrm{pr}_{i,j}\colon U^{\prime}\to C\times_{S}C$ of the
diagonal, that is, the locus where $P_{i}=Q_{j}$. Each $D_{i,j}$ is an
effective relative Cartier divisor on $U^{\prime}$, isomorphic as $S$-scheme
to $C^{d_{1}+d_{2}-1}$, hence smooth over $S$. Now

$None$
$\mathrm{Norm}_{D_{1}/U^{\prime}}(\mathcal{O}(D_{2}))=\bigotimes_{i,j}P_{i}^{*}\mathcal{O}(Q_{j})\,,\quad\mathrm{Norm}_{D_{2}/U^{\prime}}(\mathcal{O}(D_{1}))=\bigotimes_{i,j}Q_{j}^{*}\mathcal{O}(P_{i})\,,$

and, on $U^{\prime 0}$,

$None$
$\mathrm{Norm}_{D_{1}/U^{\prime}}(1)=\bigotimes_{i,j}1\,,\quad\mathrm{Norm}_{D_{2}/U^{\prime}}(1)=\bigotimes_{i,j}1\,,\quad\text{in
$\mathcal{O}(U^{\prime 0})$.}$

The divisor on $U^{\prime}$ of the tensor-factor $1$ at $(i,j)$, both in
$\mathrm{Norm}_{D_{1}/U^{\prime}}(1)$ and in
$\mathrm{Norm}_{D_{2}/U^{\prime}}(1)$, is $D_{i,j}$. Therefore, the
isomorphism $(\varphi_{D_{1},D_{2}})_{U^{0}}$ extends, uniquely, to an
isomorphism $\varphi_{D_{1},D_{2}}$ over $U^{\prime}$, which descends uniquely
to $U$.

Our description of $\varphi_{D_{1},D_{2}}$ allows us to compute it in the
trivial case where $D_{1}$ and $D_{2}$ are disjoint. One should be a bit
careful in other cases. For example, when $d_{1}=d_{2}=1$ and $P=Q$, we have
$P^{*}\mathcal{O}_{C}(Q)=P^{*}\mathcal{O}_{C}(P)$ is the tangent space of
$C\to S$ at $P$, and hence also at $Q$, but $\varphi_{P,Q}$ is multiplication
by $-1$ on that tangent space. The reason for that is that the switch
automorphism on $C\times_{S}C$ induces $-1$ on the normal bundle of the
diagonal.

###### 6.5.4 Lemma

Let $b$ be an $S$-point on $C$. Because of the symmetry in Proposition 6.3.2,
using ($None$), for $D_{1}$, $D_{2}$ relative effective divisors on $C$ of
degree $d_{1},d_{2}$ over $S$ we have the following diagram of isomorphisms
defining $\psi_{D_{1},D_{2}}$

${\mathcal{M}(\Sigma(D_{2}),\Sigma(D_{1}))}$${\mathrm{Norm}_{D_{1}/S}(\mathcal{O}_{C}(D_{2}-d_{2}b))\otimes
b^{*}\mathcal{O}_{C}(D_{2}-d_{2}b)^{-d_{1}}}$${\mathcal{M}(\Sigma(D_{1}),\Sigma(D_{2}))}$${\mathrm{Norm}_{D_{2}/S}(\mathcal{O}_{C}(D_{1}-d_{1}b))\otimes
b^{*}\mathcal{O}_{C}(D_{1}-d_{1}b)^{-d_{2}}\,.}$$\scriptstyle{\psi_{D_{1},D_{2}}}$

Then

$None$
$\psi_{D_{1},D_{2}}=\varphi_{D_{1},D_{2}}\otimes\varphi_{D_{1},d_{2}b}^{-1}\otimes\varphi_{d_{1}b,D_{2}}^{-1}\otimes\varphi_{d_{1}b,d_{2}b}\,.$

Moreover the isomorphisms $\varphi_{D_{1},D_{2}}$, and consequently
$\psi_{D_{1},D_{2}}$, are compatible with addition of divisors, that is, under
($None$) and ($None$), for every triple $D_{1},D_{2},D_{3}$ of relative
Cartier divisors on $C$ we have

$None$
$\varphi_{D_{1}+D_{2},D_{3}}=\varphi_{D_{1},D_{3}}\otimes\varphi_{D_{2},D_{3}}\,,\quad\varphi_{D_{1},D_{2}+D_{3}}=\varphi_{D_{1},D_{2}}\otimes\varphi_{D_{1},D_{3}}\,.$

###### Proof.

It is enough to prove it in the universal case, that is, when $D_{1}$ and
$D_{2}$ are the universal divisors on $C_{U}$, and there we know that there
exists a $u$ in $\mathcal{O}_{U}(U)^{\times}=\mathcal{O}_{S}(S)^{\times}$ such
that

$None$
$u\cdot\psi_{D_{1},D_{2}}=\varphi_{D_{1},D_{2}}{\otimes}\varphi_{D_{1},d_{2}b}^{-1}{\otimes}\varphi_{d_{1}b,D_{2}}^{-1}{\otimes}\varphi_{d_{1}b,d_{2}b}\,.$

Since the symmetry in Proposition 6.3.2 is compatible with the rigidification
at $(0,0)\in(J\times J)(S)$ then $\psi_{d_{1}b,d_{2}b}$ is the identity on
$\mathcal{O}_{U}$, as well as the right hand side of ($None$) when
$D_{i}=d_{i}b$. Hence $u=u(d_{1}b,d_{2}b)=1$, proving ($None$).

Now we prove ($None$). As for ($None$), it is enough to prove it in the
universal case and then we can reduce to the case where $D_{1}=d_{1}b$,
$D_{2}=d_{2}b$ and $D_{3}=d_{3}b$ for $d_{i}$ positive integers where we have

$None$ $\displaystyle\varphi_{d_{1}b+d_{2}b,d_{3}b}$
$\displaystyle=\varphi_{d_{1}b,d_{3}b}\otimes\varphi_{d_{2}b,d_{3}b}=(-1)^{(d_{1}+d_{2})d_{3}}\,,$
$\displaystyle\varphi_{d_{1}b,d_{2}b+d_{3}b}$
$\displaystyle=\varphi_{d_{1}b,d_{2}b}\otimes\varphi_{d_{1}b,d_{3}b}=(-1)^{d_{1}(d_{2}+d_{3})}\,.$

∎

###  6.6 Explicit residue disks and partial group laws

Let $C$ be a smooth, proper, geometrically connected curve over
${\mathbb{Z}/p^{2}}$, with a $b\in C({\mathbb{Z}/p^{2}})$, let $g$ be the
genus, and let $\mathcal{M}$ be as in Proposition 6.3.2. Let $D=D^{+}-D^{-}$
and $E=E^{+}-E^{-}$ be relative Cartier divisors of degree $0$ on $C$. For
each $\alpha$ in $\mathcal{M}^{\times}(\mathbb{F}_{p})$ whose image in
$(J\times J)(\mathbb{F}_{p})$ is given by $(D,E)$ we parametrise
$\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{\alpha}$, under the assumption that
there exists a non-special split reduced divisor of degree $g$ on
$C_{\mathbb{F}_{p}}$.

Let $b_{1},\ldots,b_{g}$ in $C({\mathbb{Z}/p^{2}})$ have distinct images
$\overline{b}_{i}$ in $C(\mathbb{F}_{p})$ such that
$h^{0}(C_{\mathbb{F}_{p}},\overline{b}_{1}+\cdots+\overline{b}_{g})=1$, and
let $b_{g+1},\ldots,b_{2g}$ in $C({\mathbb{Z}/p^{2}})$ be such that the
$\overline{b}_{g+i}$ are distinct and
$h^{0}(C_{\mathbb{F}_{p}},\overline{b}_{g+1}+\cdots+\overline{b}_{2g})=1$.
Then the maps

$None$ $\displaystyle f_{1}\colon C^{g}$ $\displaystyle\longrightarrow
J\,,\quad(c_{1},\ldots,c_{g})\longmapsto\left[\mathcal{O}_{C}(c_{1}+\cdots+c_{g}-(b_{1}+\cdots+b_{g})+D)\right]$
$\displaystyle f_{2}\colon C^{g}$ $\displaystyle\longrightarrow
J\,,\quad(c_{1},\ldots,c_{g})\longmapsto\left[\mathcal{O}_{C}(c_{1}+\cdots+c_{g}-(b_{g+1}+\cdots+b_{2g})+E)\right]\,,$

are étale respectively in the points
$(\overline{b_{1}},\ldots,\overline{b_{g}})\in C^{g}(\mathbb{F}_{p})$ and
$(\overline{b_{g+1}},\ldots,\overline{b_{2g}})\in C^{g}(\mathbb{F}_{p})$ and
consequently give bijections
$C^{g}({\mathbb{Z}/p^{2}})_{(\overline{b_{1}},\ldots,\overline{b_{g}})}\to
J({\mathbb{Z}/p^{2}})_{\overline{D}}$ and
$C^{g}({\mathbb{Z}/p^{2}})_{(\overline{b_{g+1}},\ldots,\overline{b_{2g}})}\to
J({\mathbb{Z}/p^{2}})_{\overline{E}}$. For each point $c\in C(\mathbb{F}_{p})$
we choose

$None$ $x_{D,c}\in\mathcal{O}_{C}(-D)_{c}\text{ a generator}\,,\quad\text{and
$x_{c}\in\mathcal{O}_{C,c}$,}$

such that $p$ and $x_{c}$ generate the maximal ideal of $\mathcal{O}_{C,c}$.

For each $i=1,\ldots,2g$ we choose $x_{b_{i}}$ so that $x_{b_{i}}(b_{i})=0$.
For each $({\mathbb{Z}/p^{2}})$-point $c\in C({\mathbb{Z}/p^{2}})$ with image
$\overline{c}$ in $C(\mathbb{F}_{p})$ and for each $\lambda\in\mathbb{F}_{p}$
let $c_{\lambda}$ be the unique point in
$C({\mathbb{Z}/p^{2}})_{\overline{c}}$ with
$x_{\overline{c}}(c_{\lambda})=\lambda p$. Then the map $\lambda\mapsto
c_{\lambda}$ is a bijection $\mathbb{F}_{p}\to
C({\mathbb{Z}/p^{2}})_{\overline{c}}$ hence the maps $f_{1},f_{2}$ induce
bijections

$None$ $\displaystyle\mathbb{F}_{p}^{g}$ $\displaystyle\longrightarrow
J({\mathbb{Z}/p^{2}})_{\overline{D}}\,,\quad\lambda\longmapsto
D_{\lambda}:=D+(b_{1,\lambda_{1}}-b_{1})+\cdots+(b_{g,\lambda_{g}}-b_{g})$
$\displaystyle\mathbb{F}_{p}^{g}$ $\displaystyle\longrightarrow
J({\mathbb{Z}/p^{2}})_{\overline{E}}\,,\quad\mu\longmapsto
E_{\mu}:=E+(b_{g+1,\mu_{1}}-b_{g+1})+\cdots+(b_{2g,\mu_{g}}-b_{2g})\,.$

Hence $\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{\overline{D},\overline{E}}$
is the union of $\mathcal{M}^{\times}(D_{\lambda},E_{\mu})$ as $\lambda$ and
$\mu$ vary in $\mathbb{F}_{p}^{g}$ and by Proposition 6.3.2 and Remark 6.3.12
we have

$None$ $\displaystyle\mathcal{M}(D_{\lambda},E_{\mu})=$
$\displaystyle\mathrm{Norm}_{E^{+}/({\mathbb{Z}/p^{2}})}(\mathcal{O}_{C}(D_{\lambda}))\otimes\mathrm{Norm}_{E^{-}/({\mathbb{Z}/p^{2}})}(\mathcal{O}_{C}(D_{\lambda}))^{-1}\otimes$
$\displaystyle\otimes\bigotimes_{i=1}^{g}\left(b_{g+i,\mu_{i}}^{*}\mathcal{O}_{C}(D_{\lambda})\otimes
b_{g+i}^{*}\mathcal{O}_{C}(D_{\lambda})^{-1}\right)\,.$

For each $i\in\\{1,\ldots,g\\}$, $c\in C({\mathbb{Z}/p^{2}})$ and
$\lambda\in\mathbb{F}_{p}$ we define $x_{i}(c,\lambda):=1$ if
$\overline{c}\neq\overline{b}_{i}$ and $x_{i}(c,\lambda):=x_{b_{i}}-\lambda p$
if $\overline{c}=\overline{b}_{i}$, so that $c^{*}x_{i}(c,\lambda)^{-1}$
generates $c^{*}\mathcal{O}(b_{i,\lambda})$. Then, for each $c\in
C({\mathbb{Z}/p^{2}})$ and each $\lambda\in\mathbb{F}_{p}^{g}$,

$None$
$c^{*}\left(x_{D,c}^{-1}\cdot\prod_{i=1}^{g}\frac{x_{i}(c,0)}{x_{i}(c,\lambda_{i})}\right)\quad\text{generates
}c^{*}\mathcal{O}_{C}(D_{\lambda})\,.$

We write $E^{\pm}=E^{0,\pm}+\cdots+E^{g,\pm}$ so that $E^{0,\pm}$ is disjoint
from $\\{\overline{b}_{1},\ldots,\overline{b}_{g}\\}$, and $E^{i,\pm}$,
restricted to $C_{\mathbb{F}_{p}}$, is supported on $\overline{b}_{i}$. Let
$x_{D,E}$ be a generator of $\mathcal{O}_{C}(-D)$ in a neighborhood of
$E^{+}\cup E^{-}$. Then, for each $\lambda$ in $\mathbb{F}_{p}^{g}$,

$None$
$\mathrm{Norm}_{E^{0,\pm}/({\mathbb{Z}/p^{2}})}(x_{D,E}^{-1})\otimes\bigotimes_{i=1}^{g}\mathrm{Norm}_{E^{i,\pm}/({\mathbb{Z}/p^{2}})}\left(x_{D,E}^{-1}\cdot\frac{x_{b_{i}}}{x_{b_{i}}-\lambda_{i}p}\right)$

generates
$\mathrm{Norm}_{E^{\pm}/({\mathbb{Z}/p^{2}})}(\mathcal{O}_{C}(D_{\lambda}))$.
By ($None$), ($None$) and ($None$) we see that, for $\lambda$ and $\mu$ in
$\mathbb{F}_{p}^{g}$,

$None$ $\displaystyle s_{D,E}(\lambda,$
$\displaystyle\mu):=\mathrm{Norm}_{E^{0,+}/({\mathbb{Z}/p^{2}})}(x_{D,E}^{-1})\otimes\bigotimes_{i=1}^{g}\mathrm{Norm}_{E^{i,+}/({\mathbb{Z}/p^{2}})}\left(x_{D,E}^{-1}\cdot\frac{x_{b_{i}}}{x_{b_{i}}-\lambda_{i}p}\right)\otimes$
$\displaystyle\otimes\mathrm{Norm}_{E^{0,-}/({\mathbb{Z}/p^{2}})}(x_{D,E}^{-1})^{-1}\otimes\bigotimes_{i=1}^{g}\mathrm{Norm}_{E^{i,-}/({\mathbb{Z}/p^{2}})}\left(x_{D,E}^{-1}\cdot\frac{x_{b_{i}}}{x_{b_{i}}-\lambda_{i}p}\right)^{-1}\otimes$
$\displaystyle\otimes\bigotimes_{i=1}^{g}\left(b_{g+i,\mu_{i}}^{*}\left(x_{D,b_{g+i}}^{-1}\cdot\prod_{j=1}^{g}\frac{x_{j}(b_{g+i,\mu_{i}},0)}{x_{j}(b_{g+i,\mu_{i}},\lambda_{j})}\right)\otimes
b_{g+i}^{*}\left(x_{D,b_{g+i}}^{-1}\cdot\prod_{j=1}^{g}\frac{x_{j}(b_{g+i},0)}{x_{j}(b_{g+i},\lambda_{j})}\right)^{-1}\right)$

generates the free rank one ${\mathbb{Z}/p^{2}}$-module
$\mathcal{M}(D_{\lambda},E_{\mu})$. The fibre
$\mathcal{M}^{\times}(\overline{D},\overline{E})$ over
$(\overline{D},\overline{E})$ in $(J\times J)(\mathbb{F}_{p})$ is an
$\mathbb{F}_{p}^{\times}$-torsor, containing $\overline{s_{D,E}(0,0)}$, hence
in bijection with $\mathbb{F}_{p}^{\times}$ by sending $\xi$ in
$\mathbb{F}_{p}^{\times}$ to $\xi{\cdot}\overline{s_{D,E}(0,0)}$. Using that
$({\mathbb{Z}/p^{2}})^{\times}=\mathbb{F}_{p}^{\times}\times(1+p\mathbb{F}_{p})$,
we conclude the following lemma.

###### 6.6.8 Lemma

With the assumptions and definitions from the start of Section 6.6, we have,
for each $\xi\in\mathbb{F}_{p}^{\times}$, a parametrisation of the mod $p^{2}$
residue polydisk of $\mathcal{M}^{\times}$ at
$\xi{\cdot}\overline{s_{D,E}(0,0)}$ by the bijection

$\displaystyle\mathbb{F}_{p}^{g}\times\mathbb{F}_{p}^{g}\times\mathbb{F}_{p}\longrightarrow\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{\xi{\cdot}\overline{s_{D,E}(0,0)}}\,,\quad(\lambda,\mu,\tau)\longmapsto(1+p\tau){\cdot}\xi{\cdot}s_{D,E}(\lambda,\mu)\,.$

Using this parametrization it easy to describe the two partial group laws on
$\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})$ when one of the two points we are
summing lies over $(\overline{D},\overline{E})$ and the other lies over
$(\overline{D},0)$ or $(0,\overline{E})$. To compute the group law in
$J({\mathbb{Z}/p^{2}})$ we notice that for each $c\in C({\mathbb{Z}/p^{2}})$
such that $x_{c}(c)=0$ and for each $\lambda,\mu\in\mathbb{F}_{p}$ we have

$None$ $\frac{x_{c}^{2}}{(x_{c}{-}\lambda p)(x_{c}{-}\mu
p)}=\frac{x_{c}^{2}}{x_{c}^{2}-\lambda px_{c}-\mu
px_{c}}=\frac{x_{c}}{x_{c}-(\lambda{+}\mu)p}$

and since these rational functions generate
$\mathcal{O}_{C}(c_{\lambda}-c+c_{\mu}-c)$ and
$\mathcal{O}_{C}(c_{\lambda+\mu}-c)$ in a neighborhood of $c$, we have the
_equality_ of relative Cartier divisors on $C$

$None$ $(c_{\lambda}-c)+(c_{\mu}-c)=c_{\lambda+\mu}-c\,.$

Hence, under the definition for $\lambda\in\mathbb{F}_{p}^{g}$ of

$None$
$D_{\lambda}^{0}:=(b_{1,\lambda_{1}}-b_{1})+\cdots+(b_{g,\lambda_{g}}-b_{g})\,,\quad
E_{\lambda}^{0}:=(b_{g+1,\lambda_{1}}-b_{g+1})+\cdots+(b_{2g,\lambda_{g}}-b_{2g})\,,$

we have, for all $\lambda,\mu\in\mathbb{F}_{p}^{g}$, that
$D_{\lambda}+D_{\mu}^{0}=D_{\lambda+\mu}$ and
$E_{\lambda}+E_{\mu}^{0}=E_{\lambda+\mu}$. Definition $None$, applied with
$(D,0)$ and $(0,E)$, with $x_{0,E}=1$ and, for every $c\in C(\mathbb{F}_{p})$,
with $x_{0,c}=1$, gives, for all $\lambda,\mu$ in $\mathbb{F}_{p}^{g}$, the
elements

$None$
$s_{D,0}(\lambda,\mu)\in\mathcal{M}^{\times}(D_{\lambda},E_{\mu}^{0})\,,\quad
s_{0,E}(\lambda,\mu)\in\mathcal{M}^{\times}(D_{\lambda}^{0},E_{\mu})\,.$

With these definitions, we have the following lemma for the partial group laws
of $\mathcal{M}$.

###### 6.6.13 Lemma

With the assumptions and definitions from the start of Section 6.6, we have,
for all $\lambda,\lambda_{1},\lambda_{2},\mu,\mu_{1},\mu_{2}$ in
$\mathbb{F}_{p}^{g}$, that

$\displaystyle s_{D,0}(\lambda,\mu_{1})+_{2}s_{D,E}(\lambda,\mu_{2})$
$\displaystyle=s_{D,0}(\lambda,\mu_{1})\otimes
s_{D,E}(\lambda,\mu_{2})=s_{D,E}(\lambda,\mu_{1}+\mu_{2})$ $\displaystyle
s_{0,E}(\lambda_{1},\mu)+_{1}s_{D,E}(\lambda_{2},\mu)$
$\displaystyle=s_{D,0}(\lambda_{1},\mu)\otimes
s_{D,E}(\lambda_{2},\mu)=s_{D,E}(\lambda_{1}+\lambda_{2},\mu)\,,$

and, consequently, for all $\tau_{1},\tau_{2}\in\mathbb{F}_{p}$ and
$\xi_{1},\xi_{2}\in\mathbb{F}_{p}^{\times}$, that

$None$
$\displaystyle\xi_{1}(1{+}\tau_{1}p){\cdot}s_{D,0}(\lambda,\mu_{1})+_{2}\xi_{2}(1{+}\tau_{2}p){\cdot}s_{D,E}(\lambda,\mu_{2})$
$\displaystyle=\xi_{1}(1{+}\tau_{1}p)\xi_{2}(1{+}\tau_{2}p){\cdot}s_{D,E}(\lambda,\mu_{1}{+}\mu_{2})$
$\displaystyle=\xi_{1}\xi_{2}(1{+}(\tau_{1}{+}\tau_{2})p){\cdot}s_{D,E}(\lambda,\mu_{1}{+}\mu_{2})\,,$
$\displaystyle\xi_{1}(1{+}\tau_{1}p){\cdot}s_{0,E}(\lambda_{1},\mu)+_{1}\xi_{2}(1{+}\tau_{2}p){\cdot}s_{D,E}(\lambda_{2},\mu)$
$\displaystyle=\xi_{1}\xi_{2}(1{+}(\tau_{1}{+}\tau_{2})p){\cdot}s_{D,E}(\lambda_{1}{+}\lambda_{2},\mu)\,.$

###### Proof.

This follows from ($None$) and ($None$), together with the equivalence of
($None$) and ($None$) and the equivalence of ($None$) and ($None$) in
Proposition 6.3.2. ∎

We end this section with one more lemma.

###### 6.6.15 Lemma

The parametrization in Lemma 6.6.8 is the inverse of a bijection given by
parameters on $\mathcal{M}^{\times}$ analogously to ($None$).

###### Proof.

Let $\mathcal{Q}$ be the pullback of $\mathcal{M}$ by $f_{1}{\times}f_{2}$
with $f_{1}$ and $f_{2}$ as in ($None$). Then the lift
$\widetilde{f_{1}{\times}f_{2}}\colon\mathcal{Q}^{\times}\to\mathcal{M}^{\times}$
is étale at any point $\beta\in\mathcal{Q}(\mathbb{F}_{p})$ lying over
$\overline{b}=(b_{1},\ldots,b_{2g})\in(C^{2g})(\mathbb{F}_{p})$ and induces a
bijection between $\mathcal{Q}^{\times}({\mathbb{Z}/p^{2}})_{\overline{b}}$
and $\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{(\overline{D},\overline{E})}$.
In particular we can interpret $s_{D,E}(\lambda,\mu)$ as a section of
$\mathcal{Q}(b_{1,\lambda_{1}},\ldots b_{2g,\mu_{g}})$ and we can interpret
the parametrization in Lemma 6.6.8 as a parametrization of
$Q^{\times}({\mathbb{Z}/p^{2}})_{\xi\overline{s_{D,E}(0,0)}}$. It is then
enough to prove that the parametrization in Lemma 6.6.8 is the inverse of a
bijection given by parameters on $\mathcal{Q}^{\times}$. It comes from the
definition of $c_{\nu}$ for $c\in C({\mathbb{Z}/p^{2}})$ and
$\nu\in\mathbb{F}_{p}$, that the maps $\lambda_{i}\,\mu_{i}\colon
C^{2g}({\mathbb{Z}/p^{2}})_{\overline{b}}\to\mathbb{F}_{p}$ are given by
parameters in $\mathcal{O}_{C^{2g},\overline{b}}$ divided by $p$. In order to
see that also the coordinate
$\tau\colon\mathcal{Q}^{\times}({\mathbb{Z}/p^{2}})_{\xi
s_{D,E}(0,0)}\to\mathbb{F}_{p}$ is given by a parameter divided by $p$ it is
enough to prove that there is an open subset $U\subset C^{2g}$ containing
$\overline{b}$ and a section $s$ trivializing $\mathcal{Q}|_{U}$ such that
$s_{D,E}(\lambda,\mu)=s(b_{1,\lambda_{1}},\ldots,b_{2g,\mu_{g}})$. Remark
6.3.12 and ($None$) give that

$None$ $\displaystyle\mathcal{Q}=$
$\displaystyle\bigotimes_{i,j=1}^{g}\Big{(}(\pi_{i},\pi_{g+j})^{*}\mathcal{O}_{C\times
C}(\Delta)\Big{)}$
$\displaystyle\otimes\bigotimes_{i=1}^{g}\Big{(}\pi_{i}^{*}\mathcal{O}_{C}(E-(b_{g+1}+\cdots+b_{2g}))\otimes\pi_{g+i}^{*}\mathcal{O}_{C}(D-(b_{1}+\cdots+b_{g}))\Big{)}$
$\displaystyle\otimes\mathrm{Norm}_{E/{\mathbb{Z}/p^{2}}}(\mathcal{O}_{C}(D-(b_{1}+\cdots+b_{g})))\otimes\bigotimes_{i=1}^{g}b_{g+i}^{*}\mathcal{O}_{C}(D-(b_{1}+\cdots+b_{g}))^{-1}$

where $\Delta\subset C{\times}C$ is the diagonal and $\pi_{i}$ is the $i$-th
projection $C^{g}\times C^{g}\to C$. We can prove that there is an open subset
$U\subset C^{g}{\times}C^{g}$ containing $b$ and a section $s$ trivializing
$\mathcal{Q}|_{U}$ such that
$s_{D,E}(\lambda,\mu)=s(b_{1,\lambda_{1}},\ldots,b_{2g,\mu_{g}})$, by
trivializing each factor of the above tensor product in a neighborhood of $b$.
Let us see it, for example, for the pieces of the form
$(\pi_{i},\pi_{g+j})^{*}\mathcal{O}_{C\times C}(\Delta)$. Let
$\pi_{1},\pi_{2}$ be the two projections $C\times C\to C$ and let us consider
the divisor $\Delta$: for each pair of points $c_{1},c_{2}\in
C(\mathbb{F}_{p})$ the invertible $\mathcal{O}$-module $\mathcal{O}_{C\times
C}(-\Delta)$ is generated by the section $x_{\Delta,c_{1},c_{2}}:=1$ in a
neighborhood of $(c_{1},c_{2})$ if $c_{1}\neq c_{2}$, while it is generated by
the section
$x_{\Delta,c_{1},c_{2}}:=\pi_{1}^{*}x_{c_{1}}-\pi_{2}^{*}x_{c_{2}}$ in a
neighborhood of $(c_{1},c_{2})$ if $c_{1}=c_{2}$. If we now take
$c_{1}=b_{i},c_{2}=b_{g+j}\in C(\mathbb{F}_{p})$ we deduce there is a
neighborhood $U$ of $(b_{i},b_{g+j})$ such that
$x_{\Delta,b_{i},b_{g+j}}^{-1}$ generates $\mathcal{O}_{C\times
C}(\Delta)|_{U}$. For each $\lambda,\mu\in\mathbb{F}_{p}^{g}$ the point
$(b_{i,\lambda_{i}},b_{g+j,\mu_{j}})$ lies in $U({\mathbb{Z}/p^{2}})$ and the
canonical isomorphism
$(b_{i,\lambda_{i}},b_{g+j,\mu_{j}})^{*}\mathcal{O}_{C\times
C}(\Delta)=b_{g+j,\mu_{j}}^{*}\mathcal{O}(b_{i,\lambda_{i}})$ sends the
generating section
$(b_{i,\lambda_{i}},b_{j,\mu_{j}})^{*}x_{\Delta,c_{1},c_{2}}^{-1}$ to
$b_{j,\mu_{j}}^{*}x_{i}(b_{g+j},\lambda_{i})^{-1}$, which is a factor in
($None$). This gives a section $s_{i,j}$ trivializing
$\Big{(}(\pi_{i},\pi_{g+j})^{*}\mathcal{O}_{C\times C}(\Delta)\Big{)}$ in a
neighborhood of $b$. With similar choices we can find sections trivializing
the other factors in ($None$) in a neighborhood of $b$ and tensoring all such
sections we get a section $s$ such that
$s_{D,E}(\lambda,\mu)=s(b_{1,\lambda_{1}},\ldots,b_{2g,\mu_{g}})$. ∎

###  6.7 Extension of the Poincaré biextension over Néron models

Let $C$ over $\mathbb{Z}$ be a curve as in Section 2. Let $q$ be a prime
number that divides $n$. We also write $C$ for $C_{\mathbb{Z}_{q}}$. Let $J$
be the Néron model over $\mathbb{Z}_{q}$ of
$\mathrm{Pic}^{0}_{C/\mathbb{Q}_{q}}$, and $J^{0}$ its fibre-wise connected
component of $0$. On $(J\times_{\mathbb{Z}_{q}}J)_{\mathbb{Q}_{q}}$ we have
$\mathcal{M}$ as in Proposition 6.3.2, rigidified at $0\times
J_{\mathbb{Q}_{q}}$ and at $J_{\mathbb{Q}_{q}}\times 0$.

###### 6.7.1 Proposition

The invertible $\mathcal{O}$-module $\mathcal{M}$ on
$(J\times_{\mathbb{Z}_{q}}J)_{\mathbb{Q}_{q}}$, with its rigidifications,
extends uniquely to an invertible $\mathcal{O}$-module
$\widetilde{\mathcal{M}}$ with rigidifications on
$J\times_{\mathbb{Z}_{q}}J^{0}$. The biextension structure on
$\mathcal{M}^{\times}$ extends uniquely to a biextension structure on
$\widetilde{\mathcal{M}}^{\times}$.

###### Proof.

First of all, $J\times_{\mathbb{Z}_{q}}J^{0}$ is regular, hence Weil divisors
and Cartier divisors are the same, and every invertible $\mathcal{O}$-module
on $(J\times_{\mathbb{Z}_{q}}J^{0})_{\mathbb{Q}_{q}}$ has an extension to an
invertible $\mathcal{O}$-module on $J\times_{\mathbb{Z}_{q}}J^{0}$. So let
$\mathcal{M}^{\prime}$ be an extension of $\mathcal{M}$. Any extension
$\mathcal{M}^{\prime\prime}$ of $\mathcal{M}$ is then of the form
$\mathcal{M}^{\prime}(D)$, with $D$ a divisor on
$J\times_{\mathbb{Z}_{q}}J^{0}$ with support in
$(J\times_{\mathbb{Z}_{q}}J^{0})_{\mathbb{F}_{q}}$. Such $D$ are
$\mathbb{Z}$-linear combinations of the irreducible components of the
$D_{i}\times_{\mathbb{F}_{q}}J^{0}_{\mathbb{F}_{q}}$, where the $D_{i}$ are
the irreducible components of $J_{\mathbb{F}_{q}}$. Now
$\mathcal{M}^{\prime}|_{J\times 0}$ extends
$\mathcal{M}|_{J_{\mathbb{Q}_{q}}\times 0}$, hence the rigidification of
$\mathcal{M}|_{J_{\mathbb{Q}_{q}}\times 0}$ is a rational section of
$\mathcal{M}^{\prime}|_{J\times 0}$ whose divisor is a $\mathbb{Z}$-linear
combination of the $D_{i}$. It follows that there is exactly one $D$ as above
such that the rigidification of $\mathcal{M}$ extends to a rigidification of
$\mathcal{M}^{\prime}(D)$ on $J\times 0$. That rigidification is compatible
with a unique rigidification of $\mathcal{M}^{\prime}(D)$ on $0\times J^{0}$.
We denote this extension $\mathcal{M}^{\prime}(D)$ of $\mathcal{M}$ to
$J\times_{\mathbb{Z}_{q}}J^{0}$ by $\widetilde{\mathcal{M}}$.

Let us now prove that the $\mathbb{G}_{\mathrm{m}}$-torsor
$\widetilde{\mathcal{M}}^{\times}$ on $J\times_{\mathbb{Z}_{q}}J^{0}$ has a
unique biextension structure, extending that of $\mathcal{M}^{\times}$. Over
$J\times_{\mathbb{Z}_{q}}J\times_{\mathbb{Z}_{q}}J^{0}$ we have the invertible
$\mathcal{O}$-modules whose fibres, at a point $(x,y,z)$ (with values in some
$\mathbb{Z}_{q}$-scheme) are $\widetilde{\mathcal{M}}(x+y,z)$ and
$\widetilde{\mathcal{M}}(x,z)\otimes\widetilde{\mathcal{M}}(y,z)$. The
biextension structure of $\mathcal{M}^{\times}$ gives an isomorphism between
the restrictions of these over $\mathbb{Q}_{q}$, that differs from an
isomorphism over $\mathbb{Z}_{q}$ by a divisor with support over
$\mathbb{F}_{q}$. The compatibility with the rigidification of
$\widetilde{\mathcal{M}}$ over $J\times_{\mathbb{Z}_{q}}0$ proves that this
divisor is zero. The other partial group law, and the required properties of
them follow in the same way. We have now shown that
$\widetilde{\mathcal{M}}^{\times}$ extends the biextension
$\mathcal{M}^{\times}$. ∎

###  6.8 Explicit description of the extended Poincaré bundle

Let $C$ over $\mathbb{Z}$ be a curve as in Section 2. Let $q$ be a prime
number that divides $n$. We also write $C$ for $C_{\mathbb{Z}_{q}}$. By [22],
Corollary 9.1.24, $C$ is cohomologically flat over $\mathbb{Z}_{q}$, which
means that for all $\mathbb{Z}_{q}$-algebras $A$, $\mathcal{O}(C_{A})=A$.
Another reference for this is [28], (6.1.4), (6.1.6) and (7.2.1).

The relative Picard functor $\mathrm{Pic}_{C/\mathbb{Z}_{q}}$ sends a
$\mathbb{Z}_{q}$-scheme $T$ to the set of isomorphism classes of
$(\mathcal{L},\mathrm{rig})$ with $\mathcal{L}$ an invertible
$\mathcal{O}$-module on $C_{T}$ and $\mathrm{rig}$ a rigidification at $b$. By
cohomological flatness, such objects are rigid. But if the action of
$\mathrm{Gal}({\overline{\mathbb{F}}}_{q}/\mathbb{F}_{q})$ on the set of
irreducible components of $C_{{\overline{\mathbb{F}}}_{q}}$ is non-trivial,
then $\mathrm{Pic}_{C/\mathbb{Z}_{q}}$ is not representable by a
$\mathbb{Z}_{q}$-scheme, only by an algebraic space over $\mathbb{Z}_{q}$ (see
[28], Proposition 5.5). Therefore, to not be annoyed by such inconveniences,
we pass to $S:=\mathrm{Spec}(\mathbb{Z}_{q}^{\mathrm{unr}})$, the maximal
unramified extension of $\mathbb{Z}_{q}$. Then $\mathrm{Pic}_{C/S}$ is
represented by a smooth $S$-scheme, and on $C\times_{S}\mathrm{Pic}_{C/S}$
there is a universal pair $(\mathcal{L}^{\mathrm{univ}},\mathrm{rig})$ ([28],
Proposition 5.5, and Section 8.0). We note that $\mathrm{Pic}_{C/S}\to S$ is
separated if and only if $C_{{\overline{\mathbb{F}}}_{q}}$ is irreducible.

Let $\mathrm{Pic}^{[0]}_{C/S}$ be the open part of $\mathrm{Pic}_{C/S}$ where
$\mathcal{L}^{\mathrm{univ}}$ is of total degree zero on the fibres of $C\to
S$. It contains the open part $\mathrm{Pic}^{0}_{C/S}$ where
$\mathcal{L}^{\mathrm{univ}}$ has degree zero on all irreducible components of
$C_{{\overline{\mathbb{F}}}_{q}}$.

Let $E$ be the closure of the $0$-section of $\mathrm{Pic}_{C/S}$, as in [28].
It is contained in $\mathrm{Pic}^{[0]}_{C/S}$. By [28], Proposition 5.2, $E$
is represented by an $S$-group scheme, étale.

By [28], Theorem 8.1.4, or [9], Theorem 9.5.4, the tautological morphism
$\mathrm{Pic}^{[0]}_{C/S}\to J$ is surjective (for the étale topology) and its
kernel is $E$, and so $J=\mathrm{Pic}^{[0]}_{C/S}/E$. Also, the composition
$\mathrm{Pic}^{0}_{C/S}\to\mathrm{Pic}^{[0]}_{C/S}\to J$ induces an
isomorphism $\mathrm{Pic}^{0}_{C/S}\to J^{0}$.

Let $C_{i}$, $i\in I$, be the irreducible components of
$C_{{\overline{\mathbb{F}}}_{q}}$. Then, as divisors on $C$, we have

$None$ $C_{{\overline{\mathbb{F}}}_{q}}=\sum_{i\in I}m_{i}C_{i}\,.$

For $\mathcal{L}$ an invertible $\mathcal{O}$-module on
$C_{{\overline{\mathbb{F}}}_{q}}$, its multidegree is defined as

$None$ $\mathrm{mdeg}(\mathcal{L})\colon I\to\mathbb{Z},\quad
i\mapsto\deg_{C_{i}}(\mathcal{L}|_{C_{i}})\,,$

and its total degree is then

$None$ $\deg(\mathcal{L})=\sum_{i\in
I}m_{i}\deg_{C_{i}}(\mathcal{L}|_{C_{i}})\,.$

The multidegree induces a surjective morphism of groups

$None$ $\mathrm{mdeg}\colon\mathrm{Pic}_{C/S}(S)\to\mathbb{Z}^{I}\,.$

Now let $d\in\mathbb{Z}^{I}$ be a sufficiently large multidegree so that every
invertible $\mathcal{O}$-module $\mathcal{L}$ on
$C_{{\overline{\mathbb{F}}}_{q}}$ with $\mathrm{mdeg}(\mathcal{L})=d$
satisfies $\mathrm{H}^{1}(C_{{\overline{\mathbb{F}}}_{q}},\mathcal{L})=0$ and
has a global section whose divisor is finite. Let $\mathcal{L}_{0}$ be an
invertible $\mathcal{O}$-module on $C$, rigidified at $b$, with
$\mathrm{mdeg}(\mathcal{L}_{0})=d$. Then over $C\times_{S}J^{0}$ we have the
invertible $\mathcal{O}$-module
$\mathcal{L}^{\mathrm{univ}}\otimes\mathcal{L}_{0}$, and its pushforward
$\mathcal{E}$ to $J^{0}$. Then $\mathcal{E}$ is a locally free
$\mathcal{O}$-module on $J^{0}$. Let $E$ be the geometric vector bundle over
$J^{0}$ corresponding to $\mathcal{E}$. Then over $E$, $\mathcal{E}$ has its
universal section. Let $U\subset E$ be the open subscheme where the divisor of
this universal section is finite over $J^{0}$. The $J^{0}$-group scheme
$\mathbb{G}_{\mathrm{m}}$ acts freely on $U$. We define
$V:=U/\mathbb{G}_{\mathrm{m}}$. As the $\mathbb{G}_{\mathrm{m}}$-action
preserves the invertible $\mathcal{O}$-module and its rigidification, the
morphism $U\to J^{0}$ factors through $U\to V$ and gives a morphism
$\Sigma_{\mathcal{L}_{0}}\colon V\to J^{0}$. Then on $C\times_{S}V$ we have
the universal effective relative Cartier divisor $D^{\mathrm{univ}}$ on
$C\times_{S}V\to V$ of multidegree $d$, and
$\mathcal{L}^{\mathrm{univ}}\otimes\mathcal{L}_{0}$ together with its
rigidification at $b$ is (uniquely) isomorphic to
$\mathcal{O}_{C\times_{S}V}(D^{\mathrm{univ}})\otimes_{\mathcal{O}_{V}}b^{*}\mathcal{O}_{C\times_{S}V}(-D^{\mathrm{univ}})$
with its tautological rigidification at $b$, in a diagram:

$None$
${\mathcal{L}^{\mathrm{univ}}\otimes\mathcal{L}_{0}}$${\mathcal{O}_{C\times_{S}V}(D^{\mathrm{univ}})\otimes_{\mathcal{O}_{V}}b^{*}\mathcal{O}_{C\times_{S}V}(-D^{\mathrm{univ}})\,.}$

Then $\Sigma_{\mathcal{L}_{0}}$ sends, for $T$ an $S$-scheme, a $T$-point $D$
on $C_{T}$ to
$\mathcal{O}_{C_{T}}(D)\otimes_{\mathcal{O}_{T}}b^{*}\mathcal{O}_{C_{T}}(-D)\otimes_{\mathcal{O}_{C}}\mathcal{L}_{0}^{-1}$
with its rigidification at $b$. Let $s_{0}$ be in $\mathcal{L}_{0}(C)$ such
that its divisor $D_{0}$ is finite over $S$, and let $v_{0}\in V(S)$ be the
corresponding point.

On $\mathrm{Pic}^{[0]}_{C/S}\times_{S}V\times_{S}C$ we have the universal
$\mathcal{L}^{\mathrm{univ}}$ from $\mathrm{Pic}^{[0]}_{C/S}$ with
rigidification at $b$, and the universal divisor $D^{\mathrm{univ}}$. Then on
$\mathrm{Pic}^{[0]}_{C/S}\times_{S}V$ we have the invertible
$\mathcal{O}$-module $\mathcal{N}_{q,d}$ whose fibre at a $T$-point
$(\mathcal{L},\mathrm{rig},D)$ is
$\mathrm{Norm}_{D/T}(\mathcal{L})\otimes_{\mathcal{O}_{T}}\mathrm{Norm}_{D_{0}/T}(\mathcal{L})^{-1}$,
canonically trivial on $\mathrm{Pic}^{[0]}_{C/S}\times_{S}v_{0}$:

$None$
${\mathcal{N}_{q,d}\colon\left(\mathrm{Pic}^{[0]}_{C/S}\times_{S}V\right)(T)\ni(\mathcal{L},\mathrm{rig},D)}$${\mathrm{Norm}_{D/T}(\mathcal{L})\otimes_{\mathcal{O}_{T}}\mathrm{Norm}_{D_{0}/T}(\mathcal{L})^{-1}\,.}$

Any global regular function on the integral scheme
$\mathrm{Pic}^{[0]}_{C/S}\times_{S}V$ is constant on the generic fibre, hence
in $\mathbb{Q}_{q}^{\mathrm{unr}}$, and restricting it to $(0,v_{0})$ shows
that it is in $\mathbb{Z}_{q}^{\mathrm{unr}}$, and if it is $1$ on
$\mathrm{Pic}^{[0]}_{C/S}\times_{S}v_{0}$, it is equal to $1$. Therefore
trivialisations on $\mathrm{Pic}^{[0]}_{C/S}\times_{S}v_{0}$ rigidify
invertible $\mathcal{O}$-modules on $\mathrm{Pic}^{[0]}_{C/S}\times_{S}V$.

The next proposition generalises [25], Corollary 2.8.6 and Lemma 2.7.11.2:
there, $C\to S$ is nodal (but not necessarily regular), and the restriction of
$\mathcal{M}$ to $J^{0}\times_{S}J^{0}$ is described.

###### 6.8.7 Proposition

In the situation of Section 6.8, the pullback of the invertible
$\mathcal{O}$-module $\mathcal{M}$ on
$J\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}J^{0}$ to
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V$
by the product of the quotient map
$\mathrm{quot}\colon\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\to
J$ and the map $\Sigma_{\mathcal{L}_{0}}\colon V\to J^{0}$ is
$\mathcal{N}_{q,d}$, compatible with their rigidifications at $J\times 0$ and
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times v_{0}$. In a
diagram:

$None$
${P^{\times}}$${\mathcal{M}^{\times}}$${\mathcal{N}_{q,d}^{\times}}$${J\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}J^{\vee,0}}$${J\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}J^{0}}$${\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V\,.}$$\scriptstyle{\mathrm{id}\times
j_{b}^{*,-1}}$$\scriptstyle{\mathrm{quot}\times\Sigma_{\mathcal{L}_{0}}}$

For $T$ any $\mathbb{Z}_{q}^{\mathrm{unr}}$-scheme, for $x$ in $J(T)$ given by
an invertible $\mathcal{O}$-module $\mathcal{L}$ on $C_{T}$ rigidified at $b$,
and $y$ in $J^{0}(T)=\mathrm{Pic}^{0}_{C/\mathbb{Z}_{q}^{\mathrm{unr}}}(T)$
given by the difference $D=D^{+}-D^{-}$ of effective relative Cartier divisors
on $C_{T}$ of the same multidegree, we have

$P(x,j_{b}^{*,-1}(y))=\mathcal{M}(x,y)=\mathrm{Norm}_{D^{+}/T}(\mathcal{L})\otimes_{\mathcal{O}_{T}}\mathrm{Norm}_{D^{-}/T}(\mathcal{L})^{-1}\,.$

###### Proof.

The scheme
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V$
is smooth over $\mathbb{Z}_{q}^{\mathrm{unr}}$, hence regular, it is
connected, hence integral, and since $V_{{\overline{\mathbb{F}}}_{q}}$ is
irreducible, the irreducible components of
$(\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V)_{{\overline{\mathbb{F}}}_{q}}$
are the
$P^{i}\times_{{\overline{\mathbb{F}}}_{q}}V_{{\overline{\mathbb{F}}}_{q}}$,
with $P^{i}$ the irreducible components of
$(\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}})_{{\overline{\mathbb{F}}}_{q}}$,
with $i$ in
$\pi_{0}((\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}})_{{\overline{\mathbb{F}}}_{q}})$,
which, by the way, equals the kernel of $\mathbb{Z}^{I}\to\mathbb{Z}$,
$x\mapsto\sum_{j\in I}m_{j}x_{j}$.

We prove the first claim. Both $\mathcal{N}_{q,d}$ and the pullback of
$\mathcal{M}$ are rigidified on
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times v_{0}$. Below we
will give, after inverting $q$, an isomorphism $\alpha$ from
$\mathcal{N}_{q,d}$ to the pullback of $\mathcal{M}$ that is compatible with
the rigidifications. Then there is a unique divisor $D_{\alpha}$ on
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V$,
supported on
$(\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V)_{{\overline{\mathbb{F}}}_{q}}$,
such that $\alpha$ is an isomorphism from $\mathcal{N}_{q,d}(D_{\alpha})$ to
the pullback of $\mathcal{M}$. Let $i$ be in
$\pi_{0}((\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}})_{{\overline{\mathbb{F}}}_{q}})$,
and let $x$ be in
$\mathrm{Pic}^{[0]}_{C/{\mathbb{Z}_{q}^{\mathrm{unr}}}}(\mathbb{Z}_{q}^{\mathrm{unr}})$
specialising to an ${\overline{\mathbb{F}}}_{q}$-point of $P^{i}$, then
restricting $\alpha$ to $(x_{i},v_{0})$ and using the compatibility of
$\alpha$ (over $\mathbb{Q}_{q}^{\mathrm{unr}}$) with the rigidifications,
gives that the multiplicity of $P^{i}\times V_{{\overline{\mathbb{F}}}_{q}}$
in $D_{\alpha}$ is zero. Hence $D_{\alpha}$ is zero.

Let us now give, over
$(\mathrm{Pic}^{[0]}_{C/\mathbb{Z}_{q}^{\mathrm{unr}}}\times_{\mathbb{Z}_{q}^{\mathrm{unr}}}V)_{\mathbb{Q}_{q}^{\mathrm{unr}}}$,
an isomorphism $\alpha$ from $\mathcal{N}_{q,d}$ to the pullback of
$\mathcal{M}$. Note that
$(\mathrm{Pic}^{[0]}_{C/\mathbb{Z}_{q}^{\mathrm{unr}}})_{\mathbb{Q}_{q}^{\mathrm{unr}}}=J_{\mathbb{Q}_{q}^{\mathrm{unr}}}$,
and that
$V_{\mathbb{Q}_{q}^{\mathrm{unr}}}=C_{\mathbb{Q}_{q}^{\mathrm{unr}}}^{(|d|)}$,
where $|d|=\sum_{i}m_{i}d_{i}$ is the total degree given by the multidgree
$d$. For $T$ a $\mathbb{Q}_{q}^{\mathrm{unr}}$-scheme, $x\in J(T)$ given by
$\mathcal{L}$ an invertible $\mathcal{O}_{C_{T}}$-module rigidified at $b$,
and $v\in V(T)$ given by a relative Cartier divisor $D$ of degree $|d|$ on
$C_{T}$, we have, using Proposition 6.3.2 and ($None$), the following
isomorphisms (functorial in $T$), respecting the rigidifications at $v=v_{0}$:

$None$ $\displaystyle\mathcal{M}(x,\Sigma_{\mathcal{L}_{0}}(v))$
$\displaystyle=\mathcal{M}(x,\Sigma(v)-\Sigma(v_{0}))=\mathcal{M}(x,\Sigma(v))\otimes\mathcal{M}(x,\Sigma(v_{0}))^{-1}$
$\displaystyle=\mathrm{Norm}_{D/T}(\mathcal{L})\otimes_{\mathcal{O}_{T}}\mathrm{Norm}_{D_{0}/T}(\mathcal{L})^{-1}=\mathcal{N}_{q,d}(x,v)\,.$

This finishes the proof of the first claim of the Proposition. The second
claim follows directly from the definition of $\mathcal{N}_{q,d}$, plus the
compatibility at the end of Proposition 6.3.2. ∎

###  6.9 Integral points of the extended Poincaré torsor

Let $C$ over $\mathbb{Z}$ be a curve as in Section 2. Given a point
$(x,y)\in(J\times J^{0})(\mathbb{Z})$ we want to describe explicitly the free
$\mathbb{Z}$-module $\mathcal{M}(x,y)$ when $x$ is given by an invertible
$\mathcal{O}$-module $\mathcal{L}$ of total degree $0$ on $C$ rigidified at
$b$ and $y$ is given as a relative Cartier divisor $D$ on $C$ of total degree
$0$ with the property that there exists a unique divisor $V$ whose support is
disjoint from $b$ and contained in the bad fibres of
$C\to\mathrm{Spec}(\mathbb{Z})$ such that $\mathcal{O}(D{+}V)$ has degree zero
when restricted to every irreducible component of any fibre of
$C\to\mathrm{Spec}(\mathbb{Z})$. Since $\mathcal{M}(x,y)$ is a free
$\mathbb{Z}$-module of rank $1$ then it is a submodule of
$\mathcal{M}(x,y)[1/n]$ and writing $D=D^{+}-D^{-}$ as a difference of
relative effective Cartier divisors, Proposition 6.3.2, with
$S=\mathrm{Spec}(\mathbb{Z}[1/n])$, gives

$None$
$\mathcal{M}(x,y)[1/n]=\left(\mathrm{Norm}_{D^{+}/\mathbb{Z}}(\mathcal{L})\otimes_{\mathbb{Z}}\mathrm{Norm}_{D^{-}/\mathbb{Z}}(\mathcal{L})^{-1}\right)[1/n]$

and consequently there exist unique integers $e_{q}$, for $q$ varying among
the primes dividing $n$, such that, as submodules of
$\left(\mathrm{Norm}_{D^{+}/\mathbb{Z}}(\mathcal{L})\otimes_{\mathbb{Z}}\mathrm{Norm}_{D^{-}/\mathbb{Z}}(\mathcal{L})^{-1}\right)[1/n]$,

$None$
$\mathcal{M}(x,y)=\left(\prod_{q|n}q^{e_{q}}\right)\cdot\left(\mathrm{Norm}_{D^{+}/\mathbb{Z}}(\mathcal{L})\otimes_{\mathbb{Z}}\mathrm{Norm}_{D^{-}/\mathbb{Z}}(\mathcal{L})^{-1}\right)\,.$

We write $V=\sum_{q|n}V_{q}$ where $V_{q}$ is a divisor supported on
$C_{\mathbb{F}_{q}}$. For every prime $q$ dividing $n$ let $C_{i,q},i\in
I_{q}$ the irreducible components of $C_{\mathbb{F}_{q}}$ with multiplicity
$m_{i,q}$ and let $V_{i,q}$ be the integers so that $V_{q}=\sum_{i\in
I_{q}}V_{i,q}C_{i,q}$.

###### 6.9.3 Proposition

The integers in ($None$) are given by

$e_{q}=-\sum_{i\in
I_{q}}V_{i,q}\deg_{\mathbb{F}_{q}}(\mathcal{L}|_{C_{i,q}})\,.$

###### Proof.

For every $q$ dividing $n$ let $H_{q}$ be an effective relative Cartier
divisor on $C_{\mathbb{Z}_{q}}$ whose complement $U_{q}$ is affine (recall
that $C$ is projective over $\mathbb{Z}$, take a high degree embedding and a
hyperplane section that avoids chosen closed points $c_{i,q}$ on the
$C_{i,q}$). The Chinese remainder theorem, applied to the
$\mathcal{O}_{C}(U_{q})$-module $(\mathcal{O}_{C}(D+V))(U_{q})$ and the
(distinct) closed points $c_{i,q}$, provides an element $f_{q}$ of
$(\mathcal{O}_{C}(D+V))(U_{q})$ that generates $\mathcal{O}_{C}(D+V)$ at all
$c_{i,q}$. Let $D_{q}=D_{q}^{+}-D_{q}^{-}$ be the divisor of $f_{q}$ as
rational section of $\mathcal{O}_{C}(D+V)$. Then $D_{q}^{+}$ and $D_{q}^{-}$
are finite over $\mathbb{Z}_{q}$, and $f_{q}$ is a rational function on
$C_{\mathbb{Z}_{q}}$ with

$None$
$\mathrm{div}(f_{q})=(D_{q}^{+}-D_{q}^{-})-(D+V)=(D_{q}^{+}+D^{-})-(D^{+}+D_{q}^{-})-V\,.$

This linear equivalence, restricted to $\mathbb{Q}_{q}$, gives, via the
definition in ($None$), the isomorphism

$None$
$\varphi\colon\mathrm{Norm}_{(D^{+}+D_{q}^{-})/\mathbb{Q}_{q}}(\mathcal{L})\longrightarrow\mathrm{Norm}_{(D_{q}^{+}+D^{-})/\mathbb{Q}_{q}}(\mathcal{L})\,.$

Tensoring with
$\mathrm{Norm}_{(D^{-}+D^{-}_{q})/\mathbb{Q}_{q}}(\mathcal{L})^{-1}$ we obtain
the isomorphism

$None$
${\varphi\otimes\mathrm{id}\colon\mathrm{Norm}_{D^{+}/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D^{-}/\mathbb{Q}_{q}}(\mathcal{L})^{-1}}$${\mathrm{Norm}_{D_{q}^{+}/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D^{-}_{q}/\mathbb{Q}_{q}}(\mathcal{L})^{-1}}$

using the identifications

$None$
$\displaystyle\mathrm{Norm}_{D^{+}/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D^{-}/\mathbb{Q}_{q}}(\mathcal{L})^{-1}$
$\displaystyle=\mathrm{Norm}_{(D^{+}+D_{q}^{-})/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{(D^{-}+D^{-}_{q})/\mathbb{Q}_{q}}(\mathcal{L})^{-1}$
$\displaystyle\mathrm{Norm}_{D_{q}^{+}/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D^{-}_{q}/\mathbb{Q}_{q}}(\mathcal{L})^{-1}$
$\displaystyle=\mathrm{Norm}_{(D_{q}^{+}+D^{-})/\mathbb{Q}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{(D^{-}+D^{-}_{q})/\mathbb{Q}_{q}}(\mathcal{L})^{-1}\,.$

Using the same method as for getting the rational section $f_{q}$ of
$\mathcal{O}_{C}(D+V)$, we get a rational section $l$ of $\mathcal{L}$ with
the support of $\mathrm{div}(l)$ finite over $\mathbb{Z}_{q}$ and disjoint
from the supports of $D$ and $D_{q}$, and from the intersections of different
$C_{i,q}$ and $C_{j,q}$. By Proposition 6.8.7, and the choice of $l$,

$None$
$\mathcal{M}(x,y)_{\mathbb{Z}_{q}}=\mathrm{Norm}_{D_{q}^{+}/\mathbb{Z}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D_{q}^{-}/\mathbb{Z}_{q}}(\mathcal{L})^{-1}=\mathbb{Z}_{q}{\cdot}\mathrm{Norm}_{D_{q}^{+}/\mathbb{Z}_{q}}(l)\otimes\mathrm{Norm}_{D_{q}^{-}/\mathbb{Z}_{q}}(l)^{-1}\,,$

and

$None$
$\mathrm{Norm}_{D^{+}/\mathbb{Z}_{q}}(\mathcal{L})\otimes\mathrm{Norm}_{D^{-}/\mathbb{Z}_{q}}(\mathcal{L})^{-1}=\mathbb{Z}_{q}{\cdot}\mathrm{Norm}_{D^{+}/\mathbb{Z}_{q}}(l)\otimes\mathrm{Norm}_{D^{-}/\mathbb{Z}_{q}}(l)^{-1}\,.$

By ($None$), we have that $\varphi\otimes\mathrm{id}$ maps

$\mathrm{Norm}_{D^{+}/\mathbb{Q}_{q}}(l)\otimes\mathrm{Norm}_{D^{-}/\mathbb{Q}_{q}}(l)^{-1}$

to

$None$
$f_{q}(\mathrm{div}(l))^{-1}\cdot\mathrm{Norm}_{D_{q}^{+}/\mathbb{Q}_{q}}(l)\otimes\mathrm{Norm}_{D_{q}^{-}/\mathbb{Q}_{q}}(l)^{-1}\,.$

Comparing with ($None$), we conclude that

$None$ $e_{q}=v_{q}(f_{q}(\mathrm{div}(l)))\,.$

We write $\mathrm{div}(l)=\sum_{j}n_{j}D_{j}$ as a sum of prime divisors.
These $D_{j}$ are finite over $\mathbb{Z}_{q}$, disjoint from the support of
the horizontal part of $\mathrm{div}(f_{q})$, that is of $D_{q}-D$, and each
of them meets only one of the $C_{i,q}$, say $C_{s(j),q}$. Then, for each $j$,
$f_{q}^{m_{s(j),q}}$ and $q^{-V_{s(j),q}}$ have the same multiplicity along
$C_{s(j),q}$, and consequently they differ multiplicatively by a unit on a
neighborhood of $D_{j}$. Then we have

$None$ $\displaystyle v_{q}(f_{q}(D_{j}))$
$\displaystyle=\frac{v_{q}(f_{q}^{m_{s(j),q}}(D_{j}))}{m_{s(j),q}}=\frac{v_{q}(q^{-V_{s(j),q}}(D_{j}))}{m_{s(j),q}}=\frac{v_{q}\left(\mathrm{Norm}_{D_{j}/\mathbb{Z}_{q}}(q^{-V_{s(j),q}})\right)}{m_{s(j),q}}$
$\displaystyle=\frac{-V_{s(j),q}\deg_{\mathbb{Z}_{q}}(D_{j})}{m_{s(j),q}}=\frac{-V_{s(j),q}{\cdot}(D_{j}\cdot
C_{\mathbb{F}_{q}})}{m_{s(j),q}}=\frac{-V_{s(j),q}{\cdot}(D_{j}\cdot
m_{s(j),q}C_{s(j),q})}{m_{s(j),q}}$ $\displaystyle=-V_{s(j),q}(D_{j}\cdot
C_{s(j)})=-V_{q}\cdot D_{j}\,.$

We get

$None$ $\displaystyle e_{q}$
$\displaystyle=v_{q}(f_{q}(\mathrm{div}(l)))=-V_{q}\cdot\mathrm{div}(l)=-\sum_{i\in
I_{q}}V_{i,q}(C_{i}\cdot\mathrm{div}(l))$ $\displaystyle=-\sum_{i\in
I_{q}}V_{i,q}\deg_{\mathbb{F}_{q}}(\mathcal{L}|_{C_{i,q}})\,.$

∎

##  7 Description of the map from the curve to the torsor

The situation is as in Section 2. The aim of this section is to give
descriptions of all morphisms in the diagram ($None$), in terms of invertible
$\mathcal{O}$-modules on $(C\times C)_{\mathbb{Q}}$ and extensions of them
over $C\times U$, to be used for doing computations when applying Theorem
4.12. The main point is that each $\mathrm{tr}_{c_{i}}\circ f_{i}$ is
described in ($None$) as a morphism (of schemes)
$\alpha_{\mathcal{L}_{i}}\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ with
$\mathcal{L}_{i}$ an invertible $\mathcal{O}$-module on $C\times U$, and that
Proposition 7.8 describes $(\widetilde{j_{b}})_{i}\colon
C_{\mathbb{Z}[1/n]}\to T_{i}$. For finding the line required line bundles, see
[12].

We describe the morphism $\widetilde{j_{b}}\colon U\to T$ in terms of
invertible $\mathcal{O}$-modules on $C\times C^{\mathrm{sm}}$. Since $T$ is
the product, over $J$, of the $\mathbb{G}_{\mathrm{m}}$-torsors
$T_{i}:=(\mathrm{id},m{\cdot}\circ\mathrm{tr}_{c_{i}}\circ
f_{i})^{*}P^{\times}$ this amounts to describing, for each $i$, the morphism
$(\widetilde{j_{b}})_{i}\colon U\to T_{i}$. Note that
$\mathrm{tr}_{c_{i}}\circ f_{i}\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ is a
morphism of groupschemes composed with a translation, and that all morphisms
of schemes $\alpha\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ are of this form.
From now on we fix one such $i$ and omit it from our notation.

Let $\alpha\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ be a morphism of schemes,
let $\mathcal{L}_{\alpha}$ be the pullback of $\mathcal{M}$ (see ($None$)) to
$C_{\mathbb{Q}}\times C_{\mathbb{Q}}$ via $j_{b}\times(\alpha\circ j_{b})$,
and let $T_{\alpha}:=(\mathrm{id},\alpha)^{*}\mathcal{M}^{\times}$ on
$J_{\mathbb{Q}}$:

$None$
${T_{\alpha}}$${\mathcal{M}^{\times}}$${C_{\mathbb{Q}}}$${J_{\mathbb{Q}}}$${(J\times
J)_{\mathbb{Q}}}$${(C\times C)_{\mathbb{Q}}}$${(C\times
J)_{\mathbb{Q}}}$${(C\times
J)_{\mathbb{Q}}}$${\mathcal{L}_{\alpha}^{\times}}$${\mathcal{L}^{\mathrm{univ},\times}\,.}$$\scriptstyle{j_{b}}$$\scriptstyle{\mathrm{diag}}$$\scriptstyle{(\mathrm{id},\alpha)}$$\scriptstyle{\mathrm{id}\times
j_{b}}$$\scriptstyle{\mathrm{id}\times\alpha}$$\scriptstyle{j_{b}\times\mathrm{id}}$

Then $(b,\mathrm{id})^{*}\mathcal{L}_{\alpha}=\mathcal{O}_{C_{\mathbb{Q}}}$,
$\mathcal{L}_{\alpha}$ is of degree zero on the fibres of
$\mathrm{pr}_{2}\colon(C\times C)_{\mathbb{Q}}\to C_{\mathbb{Q}}$, and:
$j_{b}^{*}T_{\alpha}$ is trivial if and only if
$\mathrm{diag}^{*}\mathcal{L}_{\alpha}$ is trivial. Note that diagram ($None$)
without the $\mathbb{G}_{\mathrm{m}}$-torsors is commutative.

Conversely, let $\mathcal{L}$ be an invertible $\mathcal{O}$-module on
$(C\times C)_{\mathbb{Q}}$, rigidified on $\\{b\\}\times C_{\mathbb{Q}}$, and
of degree $0$ on the fibres of $\mathrm{pr}_{2}\colon(C\times
C)_{\mathbb{Q}}\to C_{\mathbb{Q}}$. The universal property of
$\mathcal{L}^{\mathrm{univ}}$ gives a unique $\beta_{\mathcal{L}}\colon
C_{\mathbb{Q}}\to J_{\mathbb{Q}}$ such that
$(\mathrm{id}\times\beta_{\mathcal{L}})^{*}\mathcal{L}^{\mathrm{univ}}=\mathcal{L}$
(compatible with rigidification at $b$). The Albanese property of $j_{b}\colon
C_{\mathbb{Q}}\to J_{\mathbb{Q}}$ then gives that $\beta_{\mathcal{L}}$
extends to a unique $\alpha_{\mathcal{L}}\colon J_{\mathbb{Q}}\to
J_{\mathbb{Q}}$ such that $\alpha_{\mathcal{L}}\circ
j_{b}=\beta_{\mathcal{L}}$. Then $j_{b}^{*}T_{\alpha_{\mathcal{L}}}$ is
trivial if and only if $\mathrm{diag}^{*}\mathcal{L}$ is trivial. We have
proved the following proposition.

###### 7.2 Proposition

In the situation of Section 2, the above maps
$\alpha\mapsto\mathcal{L}_{\alpha}$ and
$\mathcal{L}\mapsto\alpha_{\mathcal{L}}$ are inverse maps between the sets

> {scheme morphisms $\alpha\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ such that
> $j_{b}^{*}(\mathrm{id},\alpha)^{*}\mathcal{M}$ is trivial}

and

> {invertible $\mathcal{O}$-modules $\mathcal{L}$ on $(C\times
> C)_{\mathbb{Q}}$, rigidified on $\\{b\\}\times C_{\mathbb{Q}}$, of degree
> $0$ on the fibres of $\mathrm{pr}_{2}\colon(C\times C)_{\mathbb{Q}}\to
> C_{\mathbb{Q}}$, and such that $\mathrm{diag}^{*}\mathcal{L}$ is trivial}.

Now let $\mathcal{L}$ be in the second set of Proposition 7.2. Then
$\mathrm{diag}^{*}\mathcal{L}=\mathcal{O}_{C_{\mathbb{Q}}}$, compatible with
rigidifications at $b$. Let

$None$ $\ell\in(\mathrm{diag}^{*}\mathcal{L}^{\times})(C_{\mathbb{Q}})$

correspond to $1$. Then $m{\cdot}\circ\alpha_{\mathcal{L}}$ extends over
$\mathbb{Z}$ to $m{\cdot}\circ\alpha_{\mathcal{L}}\colon J\to J^{0}$, and the
restriction of $j_{b}^{*}(m{\cdot}\circ\alpha_{\mathcal{L}})^{*}\mathcal{M}$
on $C^{\mathrm{sm}}$ to $U$ is trivial, giving a lift $\widetilde{j_{b}}$,
unique up to sign:

$None$
${T_{m{\cdot}\circ\alpha_{\mathcal{L}}}}$${\mathcal{M}^{\times}}$${U}$${C^{\mathrm{sm}}}$${J}$${J\times
J^{0}\,.}$$\scriptstyle{\widetilde{j_{b}}}$$\scriptstyle{j_{b}}$$\scriptstyle{(\mathrm{id},m{\cdot}\circ\alpha_{\mathcal{L}})}$

The invertible $\mathcal{O}$-module $\mathcal{L}$ on $(C\times
C)_{\mathbb{Q}}$ with its rigidification of $(b,\mathrm{id})^{*}\mathcal{L}$,
extends uniquely to an invertible $\mathcal{O}$-module on $(C\times
C)_{\mathbb{Z}[1/n]}$, still denoted $\mathcal{L}$.

###### 7.5 Proposition

Let $S$ be a $\mathbb{Z}[1/n]$-scheme, let $d$ and $e$ be in $\mathbb{Z}_{\geq