string equation [32, (22)]. Hence Theorem 6.10 implies the corollary.

∎

##  7\. Degeneration formula

In this section, we adapt the constructions and arguments in [37, 46] to prove
the degeneration formulas in Theorem 1.15 for $K$-theoretic Quot scheme
invariants.

###  7.1. Expanded degenerations

We consider a simple degeneration of smooth projective curves. Let
$\mathcal{C}\rightarrow B$ be a projective morphism from a smooth variety
$\mathcal{C}$ to a smooth pointed curve $(B,0)$ such that the fibers outside 0
are smooth curves of genus $g$, and the fiber over 0 is a union of two smooth
irreducible components $C_{-}$ and $C_{+}$ along the node $x_{\rm node}$:

$C_{\rm deg}=C_{-}\coprod_{x_{\rm node}}C_{+}.$

Let $g_{-}$ and $g_{+}$ be the genera of $C_{-}$ and $C_{+}$, respectively.
Then $g_{-}+g_{+}=g$. Let $x_{-}$ and $x_{+}$ be the points on $C_{-}$ and
$C_{+}$ corresponding to the node $x_{\rm node}$.

Expanded degenerations were introduced in [34, 35]. We recall the setup in
[37]. Let $0:=[1:0]$ and $\infty:=[0:1]$ be two points on $\mathbb{P}^{1}$.
There is a universal family $\mathfrak{C}\rightarrow\mathfrak{B}$ of expanded
degenerations associated to the family $\mathcal{C}\rightarrow B$, whose
singular fibers are expansions of $C_{\rm deg}$ obtained by inserting a chain
of $\mathbb{P}^{1}$’s at the node $x_{\rm node}$. More precisely, we have

$C_{\rm
deg}[n]=C_{-}\coprod_{x_{-}=0}\underbrace{\mathbb{P}^{1}\coprod_{\infty=0}\dots\coprod_{\infty=0}\mathbb{P}^{1}}_{\text{$n$
copies}}\coprod_{\infty=x_{+}}C_{+},$

where $n\geq 0$ and we identify $x_{-}$ in $C_{-}$ with 0 in the first copy of
$\mathbb{P}^{1}$, identify $\infty$ in the $i$-th $\mathbb{P}^{1}$ with $0$ in
the $(i+1)$-th $\mathbb{P}^{1}$, and identify $\infty$ in the last
$\mathbb{P}^{1}$ with $x_{+}$ in $C_{+}$. We refer to this chain of
$\mathbb{P}^{1}$’s as a _rational bridge_ (of length $n$). The universal
family fits into the following commutative diagram

${\mathfrak{C}}$${\mathcal{C}}$${\mathfrak{B}}$${B}$

where the first horizontal map is an isomoprhism on smooth fibers and, on
singular fibers, contracts the chains of $\mathbb{P}^{1}$’s to the node
$x_{\rm node}$ in $C_{\rm deg}$.

######  Remark 7.1.

For detailed definitions of the stacks $\mathfrak{E}$ and $\mathfrak{B}$,
denoted as $\mathfrak{X}$ and $\mathfrak{C}$ respectively in [37, §2.1, §2.2],
we direct the reader to that source. Note that $\mathfrak{B}_{0}$, the fiber
of $\mathfrak{B}\rightarrow B$ over 0, is independent of $\mathfrak{B}$ and is
the same as $\mathfrak{T}_{0}$ in [1].

For the relative pairs $(C_{-},x_{-})$ and $(C_{+},x_{+})$, the universal
families of expanded degenerations (or expanded relative pairs) are
$(\mathfrak{C}_{-},\mathfrak{x}_{-})\rightarrow\mathcal{T}$ and
$(\mathfrak{C}_{+},\mathfrak{x}_{+})\rightarrow\mathcal{T}$, respectively.
Here, the base $\mathcal{T}$ will be explained in Remark 7.2. The fibers of
$(\mathfrak{C}_{\pm},\mathfrak{x}_{\pm})\rightarrow\mathcal{T}$ are expansions
of $(C_{\pm},x_{\pm})$ obtained by attaching chains of $\mathbb{P}^{1}$’s of
various lengths to $C_{\pm}$ at $x_{\pm}$:

$\displaystyle(C_{-}[n],x_{-}[n])$
$\displaystyle:=C_{-}\coprod_{x_{-}=0}\underbrace{\mathbb{P}^{1}\coprod_{\infty=0}\dots\coprod_{\infty=0}\mathbb{P}^{1}}_{\text{$n$
copies}},$ $\displaystyle(C_{+}[n],x_{+}[n])$
$\displaystyle:=\underbrace{\mathbb{P}^{1}\coprod_{\infty=0}\dots\coprod_{\infty=0}\mathbb{P}^{1}}_{\text{$n$
copies}}\coprod_{\infty=x_{+}}C_{+},$

We refer to these chains of $\mathbb{P}^{1}$’s as _rational tails_ , and the
above expansions are sometimes referred to as _accordions_. We also need the
family
$(\mathfrak{x}_{-},\mathfrak{C}_{\sim},\mathfrak{x}_{+})\rightarrow\mathcal{T}_{\sim}$
of nonrigid expanded degenerations associated to the node, whose fibers are
chains of $\mathbb{P}^{1}$’s of length $n$ for $n\geq 0$:

$(x_{-},\mathbb{P}^{1}[n]_{\sim},D_{+}):=\underbrace{\mathbb{P}^{1}\coprod_{\infty=0}\dots\coprod_{\infty=0}\mathbb{P}^{1}}_{\text{$n$
copies}}.$

######  Remark 7.2.

The stacks $\mathcal{T},\mathcal{T}_{\sim}$ of expanded degenerations are
independent of the choices of $(C_{\pm},x_{\pm})$ and the same as those in
[27, 1]. According to [27] (see also [1, Proposition 4.2]), the stack
$\mathcal{T}_{\sim}$ can be identified with the moduli stack
$\mathfrak{M}^{ss}_{0,2}$ of semistable, genus zero curves with two markings,
where nodes only separate these markings.

######  Remark 7.3.

Our notation $\mathcal{T}$ for expanded relative pairs aligns with that in
[46]; see Remark 4.3 therein for distinctions from the notation used in [37].

###  7.2. Good degeneration of Quot schemes

We specify the definition of _stable admissible quotients_ and their moduli
space from [37] to our setting. According to [37, Definition 3.1], for a
closed subscheme $D\subset W$ of a noetherian scheme, a coherent sheaf on $W$
is said to be _normal_ to $D$ if
$\mathrm{Tor}_{1}^{\mathcal{O}_{W}}(F,\mathcal{O}_{D})=0$. Let
$W=C_{1}\coprod_{p}C_{2}$ be the union of two smooth curves intersecting at
the node $p$. Let $Z$ be a scheme and let $\mathcal{F}$ be a $Z$-flat family
of coherent sheaves on $W\times Z$. Suppose for any $z\in Z$, the sheaf
$\mathcal{F}_{z}=\mathcal{F}\otimes_{\mathcal{O}_{Z}}\mathbb{C}(z)$ is normal
to $p$. After restricting to an affine neighborhood of $p$, the map (3.2) in
[37] is an isomorphism. According to [37, Proposition 3.5], there is an open
neighborhood $U\subset W\times Z$ of $p\times Z$ such that $\mathcal{F}|_{U}$
is flat over $U$. This implies that $\mathcal{F}|_{U}$ is torsion-free and,
therefore, locally free at $p\times Z$.

As the family $\mathfrak{C}\rightarrow\mathfrak{B}$ is represented by a
projective morphism, the relative Quot scheme of this family parametrizing
rank $N-r$ degree $d$ quotients of the rank-$N$ trivial sheaf is an algebraic
stack projective over $\mathfrak{B}$. In [37], the authors constructed an open
substack $\mathfrak{Quot}_{d}(\mathfrak{C}/\mathfrak{B},N,r)$ of the relative
Quot scheme that parametrizes _stable_ quotients. By [37, Definition 4.2], the
restriction of a quotient $\phi:\mathcal{O}^{\oplus
N}_{\mathfrak{C}}\rightarrow\mathcal{Q}$ on any smooth fiber of
$\mathfrak{C}\rightarrow\mathfrak{B}$ is automatically stable. Over the
component $C_{\rm deg}[n]$ of the singular fibers, the restriction of $\phi$
is said to be stable if its automorphism group, a subgroup of the automorphism
group of $C_{\rm deg}[n]$, is finite and it is _admissible_ in the sense that
$\mathcal{Q}$ is normal to all the nodes in $C_{\rm deg}[n]$. By [41, Lemma 1]
and the discussion in the previous paragraph, the stability condition on
$\phi$ introduced in [37, Definition 4.2] is equivalent to that
$\mathcal{Q}|_{C_{\rm deg}[n]}$ is locally free at all the nodes and the
restriction $\mathcal{Q}|_{\mathbb{P}^{1}}$ to any component of the rational
bridges is of positive degree. Hence, in our setting of expanded degeneration
of curves, we introduce the following equivalent definition:

######  Definition 7.4.

A $\mathfrak{B}$-flat family of coherent quotients $\mathcal{O}^{\oplus
N}\rightarrow\mathcal{Q}$ is stable if the restriction $\mathcal{Q}|_{C_{\rm
deg}[n]}$ is locally free at the nodes and has positive degrees over all
components of the rational bridge in $C_{\rm deg}[n]$.

Let $\mathfrak{Quot}_{d}(\mathfrak{C}/\mathfrak{B},N,r)$ be the moduli space
of flat families of stable quotients of rank $N-r$ and degree $d$ over the
universal family $\mathfrak{C}\rightarrow\mathfrak{B}$ of expanded
degenerations of curves. It is referred to as the _good degeneration_ of Quot
schemes in [37]. We refer the reader to §4.1 therein for the definition of
stable (admissible) quotients over general expanded degenerations.

Similarly, we define stable relative quotients on
$(\mathfrak{C}_{\pm},\mathfrak{x}_{\pm})\rightarrow\mathcal{T}$ and
$(\mathfrak{x}_{-},\mathfrak{C}_{\sim},\mathfrak{x}_{+})\rightarrow\mathcal{T}_{\sim}$
by requiring their restrictions to the fibers to be locally free at the nodes
and the markings $\mathfrak{x}_{\pm}$ and have positive degrees over all
$\mathbb{P}^{1}$’s in the rational tails or chains. Let
$\mathfrak{Quot}_{d}(\mathfrak{C}_{\pm}/\mathcal{T},N,r)$ and
$\mathfrak{Quot}_{d}(\mathfrak{C}_{\sim}/\mathcal{T}_{\sim},N,r)$ denote the
moduli spaces of flat families of stable relative quotients over the
corresponding universal families of expanded degenerations. According to [37,
Theorem 4.14, 4.15], all these stacks, including
$\mathfrak{Quot}_{d}(\mathfrak{C}/\mathfrak{B},N,r)$ are separate,
Deligne–Mumford stacks of finite type; the first three are proper, whereas
$\mathfrak{Quot}_{d}(\mathfrak{C}/\mathfrak{B},N,r)$ is proper over $B$. For
simplicity, we set
$\mathcal{M}^{d}:=\mathfrak{Quot}_{d}(\mathfrak{C}/\mathfrak{B},N,r)$,
$\mathcal{M}^{d}_{\pm}:=\mathfrak{Quot}_{d}(\mathfrak{C}_{\pm}/\mathcal{T},N,r)$,
and
$\mathcal{M}^{d}_{\sim}:=\mathfrak{Quot}_{d}(\mathfrak{C}_{\sim}/\mathcal{T}_{\sim},N,r)$.

Now we identify $\mathcal{M}^{d}_{\pm}$ and $\mathcal{M}^{d}_{\sim}$ with the
quasimap moduli spaces introduced in Section 6. By Remark 7.2, we have
$\mathcal{T}_{\sim}\cong\mathfrak{M}_{0,2}^{ss}$. Note that both
$\mathcal{M}^{d}_{\sim}$ and the moduli space $Q^{\epsilon=0+}_{0,2}(X,d)$ of
$(\epsilon=0+)$-stable quasimaps, referred to as stable quotients in [41], are
defined over $\mathcal{T}_{\sim}$ with matching stability conditions (see
Definition 7.4 and [41, §2]). Hence, we have the isomorphism over
$\mathcal{T}_{\sim}$:

(30) $\mathcal{M}^{d}_{\sim}\cong Q^{\epsilon=0+}_{0,2}(X,d).$

Let $C$ be a smooth projective curve of genus $g$. Let
$\mathfrak{M}_{g,1}(C,1)$ denote the stack of maps of degree 1 from 1-pointed
prestable curves of genus $g$ to $C$. In other words,
$\mathfrak{M}_{g,1}(C,1)$ is the moduli stack of genus $g$, 1-pointed
prestable curves with one parametrized component isomorphic to $C$. We choose
a closed point $c\in C$. Let $\mathcal{T}_{\rm
maps}(C,c)\subset\mathfrak{M}_{g,1}(C,1)$ denote the locally closed substack
parametrizing curves that are obtained from the parametrized component $C$ by
gluing at $c$ a chain of rational curves, with the unique marking at the
extremal component (c.f. [1, Definition 3.1.2] in the case
$C=\mathbb{P}^{1}$). Let $\mathcal{T}_{naive}(C,c)$ denote the stack of _naive
expansions_ of the pair $(C,c)$ (see [1, Definition 2.1.3]). Then by using the
arguments in the proof of [1, Lemma 3.1.3], one can show that

$\mathcal{T}_{naive}(C,c)\cong\mathcal{T}_{\rm maps}(C,c).$

It follows from Lemma 3.2.2 and Proposition 3.2.3 in [1] that

$\mathcal{T}\cong\mathcal{T}_{\rm maps}(C,c).$

Let $Q^{0+}_{C_{\pm},1}(X,d)$ be the moduli stacks of $(0+)$-stable quasimaps
with one parametrized component as defined in Definition 6.2. Consider the
forgetful morphism $Q^{0+}_{C_{\pm},1}(X,d)\rightarrow\mathcal{T}_{\rm
maps}(C_{\pm},x_{\pm})$ forgetting the stable quotients. As a consequence of
Definition 7.4, we have an isomorphism over $\mathcal{T}$:

$\mathcal{M}_{\pm}^{d}\cong Q^{0+}_{C_{\pm},1}(X,d).$

For later applications, we fix a point $y_{-}\in C_{-}$ (resp. $y_{+}\in
C_{+}$) away from $x_{-}$ (resp. $x_{+}$). Let
$Q^{0+,0+}_{C_{\pm},1|1}(X,d_{\pm})$ be the moduli stacks of $(0+,0+)$-stable
quasimap with one parametrized component and $(1,1)$-weighted markings, as
defined in Definition 6.4. Here we assume the light marking is at $y_{\pm}$
and the heavy marking is contracted to $x_{\pm}$ via the parametrization map
from source curves to $C_{\pm}$. Since the point $y_{\pm}$ is fixed, adding a
light point on the parametrized component at $y_{\pm}$ does not change the
moduli space. Hence, we have:

(31) $\mathcal{M}_{\pm}^{d}\cong Q^{0+,0+}_{C_{\pm},1|1}(X,d).$

###  7.3. Decomposition of the central fiber

Let $\mathfrak{C}_{0},\mathfrak{B}_{0}$ and $\mathcal{M}^{d}_{0}$ denote the
fibers of $\mathfrak{C},\mathfrak{B}$ and $\mathcal{M}^{d}$ over $0\in B$,
respectively. In this section, we describe the structure of the central fiber
$\mathcal{M}^{d}_{0}$.

Set

$\mathfrak{B}_{0}[n]:=\mathcal{T}\times\underbrace{\mathcal{T}_{\sim}\times\dots\times\mathcal{T}_{\sim}}_{\text{$n$
copies}}\times\mathcal{T}.$

Then there is a natural map

$\mathrm{gl}_{n}:\mathfrak{B}_{0}[n]\rightarrow\mathfrak{B}_{0}$

induced by gluing the expanded relative pairs of $(C_{\pm},x_{\pm})$ and non-
rigid expanded degenerations associated to the node. By Remark 4.7 and the
proof of Theorem 4.9 in [46] (see also [37, Proposition 2.19] and [32, Lemma
3]), $\mathfrak{B}_{0}\subset\mathfrak{B}$ is a complete intersection
substack, $\mathfrak{B}_{0}[n]$ is the normalization of the codimension-$n$
strata in $\mathfrak{B}_{0}$, and we have

(32)
$\sum_{n=0}^{\infty}(-1)^{n}(\mathrm{gl}_{n})_{*}\mathcal{O}_{\mathfrak{B}_{0}[n]}=\mathcal{O}_{\mathfrak{B}_{0}}.$

There are evaluation maps

$\mathrm{ev}_{\pm}:\mathcal{M}^{d}_{\pm}\rightarrow
X\quad\text{and}\quad(\mathrm{ev}_{-}^{\sim},\mathrm{ev}_{+}^{\sim}):\mathcal{M}^{d}_{\sim}\rightarrow
X\times X$

defined by the restrictions of the universal quotients to
$\mathfrak{x}_{\pm}$. Given $d,n\geq 0$, we define

$\Lambda_{d}^{n}=\Big{\\{}\underline{d}=(d_{-},d_{1},\dots,d_{n},d_{+}):d_{-}+d_{+}+\sum_{i=1}^{n}d_{i}=d,d_{\pm}\geq
0,d_{i}>0\Big{\\}}.$

Given $\underline{d}\in\Lambda_{d}^{n}$, we set

$\mathcal{M}^{\underline{d}}_{0}[n]:=\mathcal{M}^{d_{-}}_{-}\times_{X}\mathcal{M}^{d_{1}}_{\sim}\times_{X}\dots\mathcal{M}^{d_{n}}_{\sim}\times_{X}\mathcal{M}^{d_{+}}_{+}.$

There is a gluing map
$\iota_{\underline{d}}:\mathcal{M}^{\underline{d}}_{0}[n]\rightarrow\mathcal{M}_{0}^{d}$
of stable quotients, which fits into the following cartesian diagram

(33)
${\coprod\limits_{\underline{d}\in\Lambda_{d}^{n}}\mathcal{M}_{0}^{\underline{d}}[n]}$${\mathcal{M}_{0}^{d}}$${\mathfrak{B}_{0}[n]}$${\mathfrak{B}_{0}}$$\scriptstyle{\mathrm{gl}_{n}}$

For $n\geq 0$, we think of
$\coprod_{\underline{d}\in\Lambda_{d}^{n}}\mathcal{M}_{0}^{\underline{d}}[n]$
as the normalization of the codimension-$n$ strata in $\mathcal{M}_{0}^{d}$.

###  7.4. Decomposition of the virtual structure sheaf

Let $\mathcal{S}_{\mathcal{M}_{0}^{d}}$ and
$\mathcal{Q}_{\mathcal{M}_{0}^{d}}$ be the universal subbundle and quotient
sheaf over the universal curve
$\pi_{d}:\mathcal{M}_{0}^{d}\times_{\mathfrak{B}_{0}}\mathfrak{C}_{0}\rightarrow\mathcal{M}_{0}^{d}$,
whose pullbacks to $\mathcal{M}_{0}^{\underline{d}}[n]$ via
$\iota_{\underline{d}}$ are denoted by
$\mathcal{S}_{\mathcal{M}_{0}^{\underline{d}}[n]}$ and
$\mathcal{Q}_{\mathcal{M}_{0}^{\underline{d}}[n]}$, respectively. We define
the relative POT of
$\mathcal{M}_{0}^{\underline{d}}[n]\rightarrow\mathfrak{B}_{0}[n]$ by pulling
back the relative POT
$(R\operatorname{\mathcal{H}om}_{\pi_{d}}(\mathcal{S}_{\mathcal{M}_{0}^{d}},\mathcal{Q}_{\mathcal{M}_{0}^{d}}))^{\vee}$
of $\mathcal{M}_{0}^{d}\rightarrow\mathfrak{B}_{0}$ via
$\iota_{\underline{d}}$:

$\left(R\operatorname{\mathcal{H}om}_{\pi_{\underline{d}}}(\mathcal{S}_{\mathcal{M}_{0}^{\underline{d}}[n]},\mathcal{Q}_{\mathcal{M}_{0}^{\underline{d}}[n]})\right)^{\vee},$

where
$\pi_{\underline{d}}:\mathfrak{C}_{0}[n]\rightarrow\mathcal{M}_{0}^{\underline{d}}[n]$
is the universal curve. These two relative POTs induce virtual pullbacks for
the two vertical maps in (33), and we define the virtual structure sheaves
$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{d}}$ and
$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{\underline{d}}[n]}$ as the
virtual pullbacks of $\mathcal{O}_{\mathfrak{B}_{0}}$ and
$\mathcal{O}_{\mathfrak{B}_{0}[n]}$, respectively. According to [46, Theorem
2.6], virtual pullbacks commute with Gysin pullbacks. Hence, we have

(34)
$(\mathrm{gl}_{n})^{!}\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{d}}=\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{\underline{d}}[n]}.$

If we consider the standard $T=(\mathbb{C}^{*})^{N}$-action on
$\mathcal{M}_{0}^{d}$ and $\mathcal{M}_{0}^{\underline{d}}[n]$, then the above
equation also holds in the $T$-equivariant $K$-group.

######  Theorem 7.5.

We have

$\sum_{n\geq
0}\sum_{\underline{d}\in\Lambda_{d}^{n}}(-1)^{n}(\iota_{\underline{d}})_{*}\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{\underline{d}}_{0}[n]}=\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{d}}$

in $K_{0}^{T}(\mathcal{M}_{0}^{d})$.

###### Proof.

The theorem follows (32), (34), and the commutativity between virtual
pullbacks and proper pushforwards [46, Proposition 2.4]. ∎

To split invariants over $\mathcal{M}^{\underline{d}}_{0}[n]$, we first define
the relative POTs for $\mathcal{M}_{\pm}^{d_{\pm}}\rightarrow\mathcal{T}$ and
$\mathcal{M}_{\sim}^{d_{i}}\rightarrow\mathcal{T}_{\sim}$ using (13) (or
equivalently (12)). Note that the ($T$-equivariant) isomorphisms (30) and (31)
preserve these relative POTs. Let
$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{\pm}}_{\pm}}$ and
$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{i}}_{\sim}}$ be the virtual
pullbacks of the structure sheaves of $\mathcal{T}$ and $\mathcal{T}_{\sim}$,
respectively. Consider the cartesian diagram:

$\leavevmode\hbox to180.13pt{\vbox
to55.98pt{\pgfpicture\makeatletter\hbox{\hskip 90.0647pt\lower-28.03815pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-90.0647pt}{-27.93831pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
16.06784pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-11.7623pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{M}_{0}^{\underline{d}}[n]}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
16.06784pt\hfil&\hfil\hskip 72.53427pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-44.22876pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{M}^{d_{-}}_{-}\times\mathcal{M}^{d_{1}}_{\sim}\times\cdots\mathcal{M}^{d_{n}}_{\sim}\times\mathcal{M}^{d_{+}}_{+}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
48.5343pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
29.53041pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.22487pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathfrak{B}_{0}[n]\times X^{\times n}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
29.53041pt\hfil&\hfil\hskip 68.06506pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-39.75955pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathfrak{B}_{0}[n]\times(X\times X)^{\times n}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
44.0651pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-44.26645pt}{11.5145pt}\pgfsys@lineto{-7.60387pt}{11.5145pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-7.40388pt}{11.5145pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-60.53429pt}{2.65479pt}\pgfsys@lineto{-60.53429pt}{-15.21199pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-60.53429pt}{-15.41197pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{
{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{41.5304pt}{3.0548pt}\pgfsys@lineto{41.5304pt}{-14.5453pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{41.5304pt}{-14.74529pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-30.80388pt}{-25.43831pt}\pgfsys@lineto{-3.13466pt}{-25.43831pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.93468pt}{-25.43831pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.49985pt}{-21.71887pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\Delta_{X}^{\times n}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
Let $\mathfrak{C}^{\prime}_{0}[n]$ denote the pullback of the univeral curve
over
$\mathcal{M}^{d_{-}}_{-}\times\mathcal{M}^{d_{1}}_{\sim}\times\cdots\mathcal{M}^{d_{n}}_{\sim}\times\mathcal{M}^{d_{+}}_{+}$
to $\mathcal{M}_{0}^{\underline{d}}[n]$. Then we have a commutative diagram

${\mathfrak{C}^{\prime}_{0}[n]}$${\mathfrak{C}_{0}[n]}$${\mathfrak{X}}$$\scriptstyle{p}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$

where $f$ and $f^{\prime}$ are the universal maps and $p$ is the gluing map.
By the stability condition of the quotients,
$Lf^{*}\mathbb{T}_{\mathfrak{X}}=\operatorname{\mathcal{H}om}(\mathcal{S}_{\mathcal{M}_{0}^{\underline{d}}[n]},\mathcal{Q}_{\mathcal{M}_{0}^{\underline{d}}[n]})$
is locally free at the nodes of the universal curve. It follows from the
arguments in the proof of the cutting-edges axiom in [9] that the relative
POTs of $\mathcal{M}_{0}^{\underline{d}}[n]$ and
$\mathcal{M}^{d_{-}}_{-}\times\mathcal{M}^{d_{1}}_{\sim}\times\cdots\mathcal{M}^{d_{n}}\times\mathcal{M}^{d_{+}}_{+}$
are compatible with respect to $\Delta_{X}^{\times n}$ (see [12]). Then by the
functoriality of virtual pullbacks [46, Proposition 2.11], we have

(35)
$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}_{0}^{\underline{d}}[n]}=\left(\Delta_{X}^{\times
n}\right)^{!}\left(\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{-}}_{-}}\boxtimes\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{1}}_{\sim}}\boxtimes\dots\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{n}}_{\sim}}\boxtimes\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d_{+}}_{+}}\right)$

in $K_{0}^{T}(\mathcal{M}_{0}^{\underline{d}}[n])$.

###  7.5. Degeneration formula

We choose a point $y_{-}\in C_{-}$ (resp. $y_{+}\in C_{+}$) away from $x_{-}$
(resp. $x_{+}$). After shrinking $B$ if necessary, we extend these two
markings to sections $\mathfrak{y}_{\pm}:\mathfrak{B}\rightarrow\mathfrak{C}$
of the expanded degeneration of curves.

Consider the ‘stacky’ evaluation map

$\widetilde{\mathrm{ev}}_{y_{\pm}}:\mathcal{M}_{\pm}^{d_{\pm}}\rightarrow\mathfrak{X}$

induced by the restriciton of the universal subbundles at $y_{\pm}$. By abuse
of notation, we also use $\widetilde{\mathrm{ev}}_{y_{\pm}}$ to denote the
evaluation maps on $\mathcal{M}_{0}^{\underline{d}}[n]$ and
$\mathcal{M}^{d}_{0}$. For $\mathrm{V},\mathrm{W}\in K_{T}^{0}(\mathfrak{X})$,
we define

$\langle\\!\langle\mathrm{V},\mathrm{W}\rangle\\!\rangle^{T}_{\deg}:=\sum_{d\geq
0}q^{d}\chi^{\mathrm{vir},T}\left(\mathcal{M}_{0}^{d},\widetilde{\mathrm{ev}}_{y_{-}}^{*}\left(\mathrm{V}\right)\cdot\widetilde{\mathrm{ev}}_{y_{+}}^{*}\left(\mathrm{W}\right)\right).$

It follows from Theorem 7.5 that

$\displaystyle\langle\\!\langle\mathrm{V},\mathrm{W}\rangle\\!\rangle^{T}_{\deg}=\sum_{n\geq
0}(-1)^{n}\sum_{d\geq
0,\underline{d}\in\Lambda_{d}^{n}}q^{d}\chi^{\mathrm{vir},T}\left(\mathcal{M}_{0}^{\underline{d}}[n],\widetilde{\mathrm{ev}}_{y_{-}}^{*}\left(\mathrm{V}\right)\cdot\widetilde{\mathrm{ev}}_{y_{+}}^{*}\left(\mathrm{W}\right)\right).$

Define
$F^{\epsilon=0+}_{\alpha,\beta}=g_{\alpha,\beta}+F_{\alpha,\beta}^{\prime}$,
where
$g_{\alpha,\beta}:=\chi^{T}(X,\mathcal{O}_{\alpha}\cdot\mathcal{O}_{\beta})$
and

$F_{\alpha,\beta}^{\prime}=\sum_{d>0}q^{d}\langle\mathcal{O}_{\alpha},\mathcal{O}_{\beta}\rangle_{0,2,d}^{\epsilon=0+,T}.$

Using (35) and the isomorphisms (30), (31), we obtain

$\sum_{d\geq
0,\underline{d}\in\Lambda_{d}^{n}}q^{d}\chi^{\mathrm{vir},T}\left(\mathcal{M}_{0}^{\underline{d}}[n],\widetilde{\mathrm{ev}}_{y_{-}}^{*}\left(\mathrm{V}\right)\cdot\widetilde{\mathrm{ev}}_{y_{+}}^{*}\left(\mathrm{W}\right)\right)=\sum_{\alpha,\beta,a_{i},b_{i}\in\mathrm{P}_{r,k}}\langle\\!\langle\mathrm{V},\mathcal{O}_{\alpha}\rangle\\!\rangle_{C_{-},1|1}^{0+,T}\\\
\cdot g^{\alpha,a_{1}}F_{a_{1},b_{1}}^{\prime}g^{b_{1},a_{2}}\cdots
F_{a_{m},b_{m}}^{\prime}g^{b_{m},\beta}\cdot\langle\\!\langle\mathcal{O}_{\beta},\mathrm{W}\rangle\\!\rangle_{C_{+},1|1}^{0+,T}.$

Here $g^{\alpha,\beta}$ denote the coefficients of the inverse of
$(g_{\alpha,\beta})$. It was first observed in [22] (in quantum $K$-theory)
that the combinatorics involved in the above equation can be simplified using
the inverse matrix $F_{\epsilon=0+}^{\alpha,\beta}$, which equals to

$g^{\alpha,\beta}+\sum_{m\geq
1,a_{i},b_{i}\in\mathrm{P}_{r,k}}(-1)^{m}g^{\alpha,a_{1}}F_{a_{1},b_{1}}^{\prime}g^{b_{1},a_{2}}\cdots
F_{a_{m},b_{m}}^{\prime}g^{b_{m},\beta}.$

In sum, we obtain

$\langle\\!\langle\mathrm{V},\mathrm{W}\rangle\\!\rangle^{T}_{\deg}=\sum_{\alpha,\beta\in\mathrm{P}_{r,k}}\langle\\!\langle\mathrm{V},\mathcal{O}_{\alpha}\rangle\\!\rangle_{C_{-},1|1}^{0+,T}\cdot
F_{\epsilon=0+}^{\alpha,\beta}\cdot\langle\\!\langle\mathcal{O}_{\beta},\mathrm{W}\rangle\\!\rangle_{C_{+},1|1}^{0+,T}.$

Hence by using the wall-crossing results from Proposition 6.6 and Proposition
5.3, we have

(36)
$\langle\\!\langle\mathrm{V},\mathrm{W}\rangle\\!\rangle^{T}_{\deg}=\sum_{\alpha,\beta\in\mathrm{P}_{r,k}}\langle\\!\langle\mathrm{V},\mathrm{O}_{\alpha}\rangle\\!\rangle_{g_{-}}^{\operatorname{\mathsf{Quot}},T}\cdot
F^{\alpha,\beta}\cdot\langle\\!\langle\mathrm{O}_{\beta},\mathrm{W}\rangle\\!\rangle_{g_{+}}^{\operatorname{\mathsf{Quot}},T}.$

Let $q_{\mathfrak{B}}:\mathcal{M}^{d}\rightarrow\mathfrak{B}$ and
$q_{B}:\mathcal{M}^{d}\rightarrow B$ be the good degeneration of Quot schemes.
Let $i_{b}:b\hookrightarrow C$ be the regular embedding of a closed point.
Consider the cartesian diagram

${\mathcal{M}^{d}_{b}}$${\mathfrak{B}_{b}}$${b}$${\mathcal{M}^{d}}$${\mathfrak{B}}$${B}$$\scriptstyle{\iota_{b}}$$\scriptstyle{i_{b}}$$\scriptstyle{q_{\mathfrak{B}}}$

We define the virtual structure sheaf of the good degeneration by

$\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d}}:=q^{!}(\mathcal{O}_{\mathfrak{B}})\in
K_{0}^{T}(\mathcal{M}^{d}).$

Since the POT of $\mathcal{M}^{d}_{b}$ is the pullback of the relative POT of
$q$, we have

$i_{b}^{!}(\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d}})=\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d}_{b}}.$

Let $F\in K_{T}^{0}(\mathcal{M}^{d})$. Then

$\displaystyle\chi^{\mathrm{vir},T}\left(\mathcal{M}^{d}_{b},F|_{\mathcal{M}^{d}_{b}}\right)=\chi^{T}\left(\mathcal{M}^{d},F\cdot(\iota_{b})_{*}i_{b}^{!}(\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d}})\right)=\chi^{T}\left(B,i_{b}^{!}(q_{B})_{*}(F\cdot\mathcal{O}^{\mathrm{vir}}_{\mathcal{M}^{d}})\right)$

is independent of $b$. In particular, if we fix a point $b\neq 0$ in $B$, then

(37)
$\chi^{\mathrm{vir},T}\left(\mathcal{M}^{d}_{b},F|_{\mathcal{M}^{d}_{b}}\right)=\chi^{\mathrm{vir},T}\left(\mathcal{M}^{d}_{0},F|_{\mathcal{M}^{d}_{0}}\right).$

###### Proof of Theorem 1.15 (i).

Let $C=\mathfrak{C}_{b}$ be the fiber of $\mathfrak{C}\rightarrow B$ at $b\neq
0$. Then $C$ is a smooth curve of genus $g$, and we have

$\mathcal{M}^{d}_{b}\cong\operatorname{\mathsf{Quot}}_{d}(C,N,r).$

Set
$F=\widetilde{\mathrm{ev}}_{\mathfrak{y}_{-}}^{*}\left(\mathrm{V}\right)\cdot\widetilde{\mathrm{ev}}_{\mathfrak{y}_{+}}^{*}\left(\mathrm{W}\right),$
where
$\widetilde{\mathrm{ev}}_{\mathfrak{y}_{\pm}}:\mathcal{M}^{d}\rightarrow\mathfrak{X}$
denote the stacky evaluation maps at $\mathfrak{y}_{\pm}$. Then the
degeneration formula (7) follows from the decomposition (36) and the
deformation invariance of virtual Euler characteristics (37). ∎

######  Remark 7.6.

The degeneration formula (8) in Theorem 1.15 (ii) can be proven similarly by
analyzing a simple degeneration $\mathcal{C}^{\prime}\rightarrow B$ over a
smooth pointed curve $(B,0)$. Here, the fibers outside 0 are smooth curves of
genus $g$, while the fiber at 0, $C^{\prime}_{\deg}$, is an irreducible curve
with one node. The associated stack of expanded degenerations,
$\mathfrak{C}^{\prime}\rightarrow\mathfrak{B}$, has central fibers formed by
inserting a chain of $\mathbb{P}^{1}$s (of various lengths) at the unique node
of $C^{\prime}_{\deg}$ (see [37, §2.2]). The good degeneration of Quot
schemes,
$\mathfrak{Quot}_{d}(\mathfrak{C}^{\prime}/\mathfrak{B},N,r)\rightarrow B$,
parametrizes flat families of stable quotients of rank $N-r$ and degree $d$
over the universal family $\mathfrak{C}^{\prime}\rightarrow\mathfrak{B}$.

Let $\tilde{C}$ be the normalization of $C^{\prime}_{\deg}$, with $x_{-}$ and
$x_{+}$ in $\tilde{C}$ representing the preimages of the unique node in
$C^{\prime}_{\deg}$. The central fiber of this degeneration of Quot schemes is
‘virtually’ a complete intersection substack, and its ‘normalization’ is

(38)
$\coprod\limits_{n,\underline{d}\in\tilde{\Lambda}_{d}^{n}}\tilde{\mathcal{M}}_{0}^{d_{0}}\times_{X}\mathcal{M}^{d_{1}}_{\sim}\times_{X}\dots\mathcal{M}^{d_{n}}_{\sim}\times_{X}\big{(}\tilde{\mathcal{M}}_{0}^{d_{0}}\big{)},$

where
$\tilde{\Lambda}_{d}^{n}=\\{\underline{d}=(d_{0},d_{1},\dots,d_{n}):d_{0}+\sum_{i=1}^{n}d_{i}=d\
\text{and}\ d_{i}>0\ \text{for}\ 1\leq i\leq n\\}$,

$\tilde{\mathcal{M}}_{0}^{d_{0}}\cong Q^{0+,0+}_{\tilde{C},1|2}(X,d_{0}),$

and we put the further right $\tilde{\mathcal{M}}_{0}^{d_{0}}$ in parenthesis
to indicate is is the same as the one on the left. Here, a source curve in
$Q^{0+,0+}_{\tilde{C},2|2}(X,d_{0})$ is formed by attaching rational tails to
$\tilde{C}$ at $x_{\pm}$ and placing heavy markings on the outermost
components of these rational tails. The inclusion of one light marking on the
parametrized component $\tilde{C}$ does not alter the moduli space but allows
us to define invariants with one insertion in $K_{T}^{0}(\mathfrak{X})$.
Similar to the proof of Theorem 1.15 (i), we apply the deformation invariance
of virtual Euler characteristics to relate
$\langle\\!\langle\mathrm{V}\rangle\\!\rangle^{\operatorname{\mathsf{Quot}},T}_{g}$
to the alternating sum of virtual Euler characteristics over the components in
(38). This sum can then be rewritten as the right side of the degeneration
formula (8) using the wall-crossing formulas in Proposition 6.6 and
Proposition 5.3.

## References

  * [1] Dan Abramovich, Charles Cadman, Barbara Fantechi, and Jonathan Wise.  Expanded degenerations and pairs.  Comm. Algebra, 41(6):2346–2386, 2013.
  * ACT [1] David Anderson, Linda Chen, and Hsian-Hua Tseng.  On the finiteness of quantum K-theory of a homogeneous space.  Int. Math. Res. Not. IMRN, (2):1313–1349, 2022.
  * ACT [2] David Anderson, Linda Chen, and Hsian-Hua Tseng.  On the finiteness of quantum K-theory of a homogeneous space.  Int. Math. Res. Not. IMRN, (2):1313–1349, 2022.
  * [4] Aaron Bertram, Ionuţ Ciocan-Fontanine, and Bumsig Kim.  Two proofs of a conjecture of Hori and Vafa.  Duke Math. J., 126(1):101–136, 2005.
  * BCMP [1] Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, and Nicolas Perrin.  Finiteness of cominuscule quantum $K$-theory.  Ann. Sci. Éc. Norm. Supér. (4), 46(3):477–494, 2013.
  * BCMP [2] Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, and Nicolas Perrin.  Rational connectedness implies finiteness of quantum $K$-theory.  Asian J. Math., 20(1):117–122, 2016.
  * BCMP [3] Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, and Nicolas Perrin.  A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties.  Algebr. Geom., 5(5):568–595, 2018.
  * [8] Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth.  Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians.  J. Amer. Math. Soc., 9(2):529–571, 1996.
  * [9] K. Behrend.  Gromov-Witten invariants in algebraic geometry.  Invent. Math., 127(3):601–617, 1997.
  * Ber [1] Aaron Bertram.  Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian.  Internat. J. Math., 5(6):811–825, 1994.
  * Ber [2] Aaron Bertram.  Quantum Schubert calculus.  Adv. Math., 128(2):289–305, 1997.
  * [12] K. Behrend and B. Fantechi.  The intrinsic normal cone.  Invent. Math., 128(1):45–88, 1997.
  * [13] Anders S. Buch and Leonardo C. Mihalcea.  Quantum $K$-theory of Grassmannians.  Duke Math. J., 156(3):501–538, 2011.
  * [14] Jim Bryan and Rahul Pandharipande.  Curves in Calabi-Yau threefolds and topological quantum field theory.  Duke Math. J., 126(2):369–396, 2005.
  * [15] Anders Skovsted Buch.  A Littlewood-Richardson rule for the $K$-theory of Grassmannians.  Acta Math., 189(1):37–78, 2002.
  * CF [1] Ionuţ Ciocan-Fontanine.  Quantum cohomology of flag varieties.  Internat. Math. Res. Notices, (6):263–277, 1995.
  * CF [2] Ionuţ Ciocan-Fontanine.  On quantum cohomology rings of partial flag varieties.  Duke Math. J., 98(3):485–524, 1999.
  * CF [3] Ionuţ Ciocan-Fontanine.  The quantum cohomology ring of flag varieties.  Trans. Amer. Math. Soc., 351(7):2695–2729, 1999.
  * [19] Ionuţ Ciocan-Fontanine, Bumsig Kim, and Davesh Maulik.  Stable quasimaps to GIT quotients.  J. Geom. Phys., 75:17–47, 2014.
  * [20] Linda Chen.  Quantum cohomology of flag manifolds.  Adv. Math., 174(1):1–34, 2003.
  * [21] William Fulton and Joe Harris.  Representation theory, volume 129 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1991.  A first course, Readings in Mathematics.
  * [22] Alexander Givental.  On the WDVV equation in quantum $K$-theory.  volume 48, pages 295–304. 2000.  Dedicated to William Fulton on the occasion of his 60th birthday.
  * [23] Wei Gu, Leonardo C. Mihalcea, Eric Sharpe, Weihong Xu, Hao Zhang, and Hao Zou.  Quantum K Whitney relations for partial flag varieties.  arXiv e-prints, page arXiv:2310.03826, October 2023.
  * GMSZ [1] Wei Gu, Leonardo Mihalcea, Eric Sharpe, and Hao Zou.  Quantum K theory of symplectic Grassmannians.  J. Geom. Phys., 177:Paper No. 104548, 38, 2022.
  * GMSZ [2] Wei Gu, Leonardo C. Mihalcea, Eric Sharpe, and Hao Zou.  Quantum K theory of Grassmannians, Wilson line operators, and Schur bundles.  arXiv e-prints, page arXiv:2208.01091, August 2022.
  * [26] Alexander Grothendieck.  Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert.  In Séminaire Bourbaki, Vol. 6, pages Exp. No. 221, 249–276. Soc. Math. France, Paris, 1995.
  * [27] Tom Graber and Ravi Vakil.  Relative virtual localization and vanishing of tautological classes on moduli spaces of curves.  Duke Math. J., 130(1):1–37, 2005.
  * [28] E. González and C. Woodward.  Quantum Kirwan for quantum K-theory.  In Facets of algebraic geometry. Vol. I, volume 472 of London Math. Soc. Lecture Note Ser., pages 265–332. Cambridge Univ. Press, Cambridge, 2022.
  * [29] M. M. Kapranov.  Derived category of coherent sheaves on Grassmann manifolds.  Izv. Akad. Nauk SSSR Ser. Mat., 48(1):192–202, 1984.
  * [30] Syu Kato.  Loop structure on equivariant $K$-theory of semi-infinite flag manifolds.  arXiv e-prints, page arXiv:1805.01718, May 2018.
  * [31] Bumsig Kim, Jeongseok Oh, Kazushi Ueda, and Yutaka Yoshida.  Residue mirror symmetry for Grassmannians.  In Schubert calculus and its applications in combinatorics and representation theory, volume 332 of Springer Proc. Math. Stat., pages 307–365. Springer, Singapore, [2020] ©2020.
  * [32] Y.-P. Lee.  Quantum $K$-theory. I. Foundations.  Duke Math. J., 121(3):389–424, 2004.
  * [33] Cristian Lenart.  Combinatorial aspects of the $K$-theory of Grassmannians.  Ann. Comb., 4(1):67–82, 2000.
  * Li [1] Jun Li.  Stable morphisms to singular schemes and relative stable morphisms.  J. Differential Geom., 57(3):509–578, 2001.
  * Li [2] Jun Li.  A degeneration formula of GW-invariants.  J. Differential Geom., 60(2):199–293, 2002.
  * [36] Jun Li and Gang Tian.  Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties.  J. Amer. Math. Soc., 11(1):119–174, 1998.
  * [37] Jun Li and Baosen Wu.  Good degeneration of Quot-schemes and coherent systems.  Comm. Anal. Geom., 23(4):841–921, 2015.
  * [38] I. G. Macdonald.  Symmetric functions and Hall polynomials.  Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015.  With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [ MR1354144].
  * MO [1] Alina Marian and Dragos Oprea.  Virtual intersections on the Quot scheme and Vafa-Intriligator formulas.  Duke Math. J., 136(1):81–113, 2007.
  * MO [2] Alina Marian and Dragos Oprea.  GL Verlinde numbers and the Grassmann TQFT.  Port. Math., 67(2):181–210, 2010.
  * [41] Alina Marian, Dragos Oprea, and Rahul Pandharipande.  The moduli space of stable quotients.  Geom. Topol., 15(3):1651–1706, 2011.
  * [42] D. Maulik, R. Pandharipande, and R. P. Thomas.  Curves on $K3$ surfaces and modular forms.  J. Topol., 3(4):937–996, 2010.  With an appendix by A. Pixton.
  * [43] Andrei Okounkov.  Lectures on K-theoretic computations in enumerative geometry.  In Geometry of moduli spaces and representation theory, volume 24 of IAS/Park City Math. Ser., pages 251–380. Amer. Math. Soc., Providence, RI, 2017.
  * [44] Dragos Oprea and Shubham Sinha.  Euler characteristics of tautological bundles over Quot schemes of curves.  Adv. Math., 418:Paper No. 108943, 45, 2023.
  * [45] Mihnea Popa and Mike Roth.  Stable maps and Quot schemes.  Invent. Math., 152(3):625–663, 2003.
  * [46] Feng Qu.  Virtual pullbacks in $K$-theory.  Ann. Inst. Fourier (Grenoble), 68(4):1609–1641, 2018.
  * RZ [1] Yongbin Ruan and Ming Zhang.  The level structure in quantum K-theory and mock theta functions.  arXiv e-prints, page arXiv:1804.06552, Apr 2018.
  * RZ [2] Yongbin Ruan and Ming Zhang.  Verlinde/Grassmannian Correspondence and Rank 2 $\delta$-Wall-Crossing.  Peking Math. J., 2022.
  * Sin [1] Shubham Sinha.  Numerical Invariants of Quot Schemes of Curves.  ProQuest LLC, Ann Arbor, MI, 2023.  Thesis (Ph.D.)–University of California, San Diego.
  * Sin [2] Shubham Sinha.  The virtual intersection theory of isotropic quot schemes.  J. Geom. Phys., 198:Paper No. 105128, 39, 2024.
  * [51] Stein Arild Strømme.  On parametrized rational curves in Grassmann varieties.  In Space curves (Rocca di Papa, 1985), volume 1266 of Lecture Notes in Math., pages 251–272. Springer, Berlin, 1987.
  * [52] Shubham Sinha and Ming Zhang.  Quantum K-invariants via quot schemes II.
  * [53] Yukinobu Toda.  Moduli spaces of stable quotients and wall-crossing phenomena.  Compos. Math., 147(5):1479–1518, 2011.
  * [54] Ravi Vakil.  A geometric Littlewood-Richardson rule.  Ann. of Math. (2), 164(2):371–421, 2006.  Appendix A written with A. Knutson.
  * [55] Ming Zhang and Yang Zhou.  $K$-theoretic quasimap wall-crossing.  arXiv e-prints, page arXiv:2012.01401, December 2020.


# Integrality of $\mathfrak{p}$-adic $L$-functions at Eisenstein Primes

Matthew Verheul

###### Abstract.

Let $f$ be a normalized, ordinary newform of weight $\geq 2$. For each prime
$\mathfrak{p}$ of $F=\mathbb{Q}(a_{n})_{n\in\mathbb{N}}$, there is an
associated $\mathfrak{p}$-adic $L$-function
$\mathcal{L}_{\mathfrak{p}}(f)\in\Lambda\otimes\mathbb{Q}$ interpolating
special values of the classical $L$-function. If $f$ is not congruent modulo
$\mathfrak{p}$ to an Eisenstein series, one knows
$\mathcal{L}_{\mathfrak{p}}(f)\in\Lambda$. In this paper, we show, under mild
hypotheses on the ramification of $f$, that this integrality result holds when
$f$ is congruent to an Eisenstein series. Moreover, we also obtain a
divisibility in the main conjecture for $\mathcal{L}_{\mathfrak{p}}(f)$. As an
application, we show that the integrality result and the divisibility hold in
particular when $f$ is of weight $2$.

1\. Introduction

Let $f$ be a normalized newform of level $N$ and weight $k\geq 2$. Let $F$ be
the number field obtained by adjoining the Fourier coefficients of $f$ to
$\mathbb{Q}$. Throughout the paper, we fix a prime $\mathfrak{p}$ of $F$, and
let $p$ denote the rational prime it lies over. We shall also assume that
$p>2$. Let $K$ be the completion of $F$ at $\mathfrak{p}$. We denote the ring
of integers of $K$ by $\mathcal{O}$, and we fix a uniformizer $\varpi$ of the
maximal ideal of $\mathcal{O}$. Let $\rho$ denote the $\mathfrak{p}$-adic
Galois representation
$\rho:\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\text{Gl}(V)$ associated
to $f$, where $V$ is a two-dimensional vector space over $K$. We say that
$\mathfrak{p}$ is an Eisenstein prime for $f$ if it is congruent modulo
$\mathfrak{p}$ to an Eisenstein series. Equivalently, $\mathfrak{p}$ is
Eisenstein if for some (hence any) choice of Galois-stable lattice $T\subseteq
V$, $T/\varpi$ is reducible as a Galois-representation over
$\mathbb{F}_{q}:=\mathcal{O}/\varpi$. If $\mathfrak{p}$ is Eisenstein, then
$f$ is ordinary at $\mathfrak{p}$.

Finally, we write $\Lambda$ to denote the Iwasawa algebra:

$\Lambda:=\varprojlim_{n}\mathcal{O}[\text{Gal}(\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q})]$

Associated to $f$ is a $\mathfrak{p}$-adic $L$-function
$\mathcal{L}_{\mathfrak{p}}$. This is constructed, for example in [4, Theorem
16.6], as an element of $\Lambda\otimes\mathbb{Q}$ when $f$ is ordinary. If
one assumes further that $\mathfrak{p}$ is not an Eisenstein prime for $f$,
Kato shows [4, Theorem 17.4] that $\mathcal{L}_{\mathfrak{p}}\in\Lambda$.

Outline. In section 2, we show the integrality of Kato’s zeta element under
mild hypotheses. In section 3, we use this integrality to adapt the methods of
Wuthrich in [9] to our situation to obtain a divisibility in the main
conjecture. As a consequence, we derive the integrality of
$\mathcal{L}_{\mathfrak{p}}$. In section 4, we show that $f$ satisfies the
hypothesis imposed in section 2 if it has weight $2$; in particular, for any
abelian variety $A$ of $Gl_{2}$-type, $\mathcal{L}_{\mathfrak{p}}(A)$ is
integral.

2\. Freeness and Integrality

In this section, we derive criteria for the integrality of Kato’s zeta element
$z_{\gamma}^{(p)}$, which is defined in terms of some $\gamma\in V$ in [4,
Theorem 12.5]. For any Galois-stable lattice $T\subseteq V$, we define:

$\mathbf{H}^{1}(T):=H^{1}(\text{Gal}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)$

A priori, $z_{\gamma}^{(p)}\in\mathbf{H}^{1}(T)\otimes\mathbb{Q}$. However,
Kato has shown in [4, 13.14] that if $T$ is the integral lattice
$V_{\mathcal{O}}(f)$ and $\mathbf{H}^{1}(T)$ is free, then
$z_{\gamma}^{(p)}\in\mathbf{H}^{1}(T)$ whenever $\gamma\in T$.

We will show under a mild hypothesis that $\mathbf{H}^{1}(T)$ is free for some
lattice $T$. We then show that integrality of $z_{\gamma}^{(p)}$ follows even
if $T\neq V_{\mathcal{O}}(f)$.

The following result, whose proof will be important to us, shows freeness in
the non-Eisenstein case.

######  Theorem 1 (Kato [4]).

Suppose that some character $\psi:G\to\mathbb{F}_{q}$, not equal to a twist
$\mathbb{F}_{q}(r)$ of the trivial character, occurs in
$\overline{\rho}^{ss}$. Then we may find a lattice $T$ for which
$\mathbf{H}^{1}(T)$ is free over $\Lambda$.

Proof.  It is known that we may choose a Galois-stable lattice $T$ of $V$
whose residual representation fits into a non-split short exact sequence:

$0\to\psi\to T/\varpi\to\psi^{\prime}\to 0$

where $\psi^{\prime}$ is the other character occuring in the
semisimplification.

Let $(x,\varpi)$ be a maximal ideal of $\Lambda$. Kato shows that
multiplication by $x$ is injective for $\mathbf{H}^{1}(T)$. So it suffices to
show that
$\varpi:\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)\to\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)$
is injective.

Recalling that
$\mathbf{H}^{1}(T)=H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)$, the short
exact sequence:

${0}$${T\otimes\Lambda}$${T\otimes\Lambda}$${T\otimes\Lambda/x}$${0}$$\scriptstyle{\cdot
x}$

yields the following inclusion upon taking cohomology:

$\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)\subseteq
H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$

It now suffices to show that $H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$
is $\varpi$-torsion free. The exact sequence:

${0}$${T\otimes\Lambda/x}$${T\otimes\Lambda/x}$${T\otimes\Lambda/(x,\varpi)}$${0}$$\scriptstyle{\cdot\varpi}$

shows that the kernel of multiplication by $\varpi$ on
$H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$ comes from
$H^{0}(T\otimes\Lambda/(x,\varpi))$. But we know that
$\Lambda/(x,\varpi)\cong\mathbb{F}_{q}(r)$ for some $r$, and hence

$T\otimes\Lambda/(x,\varpi)\cong(T/\varpi)(r)$

By our hypothesis on $\psi$ and $T/\varpi$, this has no fixed points. $\Box$

Therefore, throughout the rest of this section, we may assume
$\overline{\rho}^{ss}=\mathbb{F}_{q}\oplus\mathbb{F}_{q}(r)$. We shall extend
Kato’s result to a general ordinary newform under a mild hypothesis. In order
to state the hypothesis, we need to recall the following result due
independently to Carayol and Livné.

######  Theorem 2 (Carayol [2], Livné [5]).

Let $\ell\neq p$ be a prime, and suppose the exponent of $\ell$ in the (Artin)
conductor of $\overline{\rho}^{ss}$ is less than the exponent of $\ell$ in the
conductor of $\rho$. Then the restriction $\rho|_{D_{\ell}}$ of $\rho$ to the
decomposition group at $\ell$ takes one of the following forms:

  1. (1)

Decomposable: $\rho|_{D_{\ell}}\cong\chi_{1}\oplus\chi_{2}$;

  2. (2)

Special: $\rho|_{D_{\ell}}\cong\chi\otimes\text{Sp}(2)$, where $\chi$ is a
possibly ramified character, whose reduction modulo $\varpi$ is trivial, and
$\text{Sp}(2)$ is isomorphic to a ramified extension of unramified characters;

  3. (3)

Cuspidal:
$\rho|_{D_{\ell}}=\text{ind}(\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell}),\text{Gal}(\overline{\mathbb{Q}}_{\ell}/K),\chi)$,
where $K$ is the unramified quadratic extension of $\mathbb{Q}_{\ell}$, and
$\chi$ is a character of $\overline{\mathbb{Q}}_{\ell}/K$

######  Remark 3.

The roles of $\ell$ and $p$ are opposite in the original papers of Carayol and
Livné. One can also state more precise conditions on the characters occurring
in cases one and three; we will not need this.

We will show that Kato’s result holds whenever case $2$ of the above
classification occurs. In particular, we must assume that $\rho$ is ramified.
If the second case occurs for some prime $\ell\neq p$, we will say that $\ell$
is special for $f$. We will see later that this holds whenever $f$ corresponds
to an abelian variety. Assuming there exists some $\ell$ special for $f$, one
knows that $\rho|_{D_{\ell}}$ can be written as an upper-triangular matrix in
some basis. One also knows that $\overline{\rho}$ can be written as an upper-
triangular matrix in some basis. The following lemma shows, in some sense,
that we can choose a single basis compatible with both of these descriptions.

######  Lemma 4.

Let $\ell$ be special for $f$. There exists an ordered pair $(v_{1},v_{2})$ in
$V$, such that:

  * •

the $\mathcal{O}$-span of $v_{1},v_{2}$ is a $G_{\Sigma}(\mathbb{Q})$-stable
lattice $T^{\prime}$ of $V$;

  * •

the action of $D_{\ell}$ on $T^{\prime}$ is upper-triangluar with respect to
the ordered basis $(v_{1},v_{2})$;

  * •

the action of $G_{\Sigma}(\mathbb{Q})$ on $T^{\prime}/\varpi$ is upper-
triangular with respect to the ordered basis
$(\overline{v}_{1},\overline{v}_{2})$.

Moreover, we may also insist that the lattice $T$ spanned by $(v_{1},\varpi
v_{2})$ is also $G_{\Sigma}(\mathbb{Q})$-stable, and the residual
representation of $T$ is not semisimple.

Proof.  Consider the set $S$ of equivalence classes of lattices (not
necessarily Galois-stable) in $V$, up to homothety. We give $S$ the structure
of a graph by drawing an edge between distinct nodes $[A],[A^{\prime}]$ if,
for some choice of representatives, we have

$\varpi A\subset A^{\prime}\subset A$

Let $X$ be the subgraph of $S$ consisting of those classes of lattices which
are stable under $G_{\Sigma}(\mathbb{Q})$. Likewise, let $X_{\ell}$ be the the
subgraph of $S$ consisting of classes stable under $D_{\ell}$. Since the
determinant of the residual representation is odd, the characters occuring in
the semisimplification must be distinct. Hence, one knows (see [1]) that
$X_{\ell}$ is a ray, and that $X$ appears as a bounded subgraph of $X_{\ell}$.
We may represent this situation pictorially:

$\overbrace{[A_{0}]\to[A_{1}]\to\cdots\to[A_{i-1}]\to\underbrace{[A_{i}]\to\cdots\to[A_{i+k}]}_{G_{\Sigma}(\mathbb{Q})\text{-stable
segment}}\to[A_{i+k+1}]\to\cdots}^{D_{\ell}\text{-stable ray}}$

Renaming representatives if necessary, we will assume we have inclusions
$A_{j}\supseteq A_{j+1}$. Notice in particular that since the residual
representation of $\rho$ is reducible, the $G_{\Sigma}(\mathbb{Q})$-stable
line segment has more than one node. Let $(b_{1},b_{2})$ be an ordered basis
of $A_{0}$, for which the action of $D_{\ell}$ is upper-triangular. Then
$(b_{1},\varpi^{i+k-1}b_{2})$ is a basis of $A_{i+k-1}$. We set
$(v_{1},v_{2})=(b_{1},\varpi^{i+k-1}b_{2})$, so $T^{\prime}=A_{i+k-1}$. By our
choice of basis for $A_{0}$, the $\mathcal{O}$-span of $b_{1}$ is
$D_{\ell}$-stable. It follows that in the basis $(b_{1},\varpi^{i+k-1}b_{2})$,
the action of $D_{\ell}$ on $T^{\prime}$ is upper-triangular. Since
$A_{i+k-1}$ is in the $G_{\Sigma}(\mathbb{Q})$-stable segment, this shows the
first two bullet points.

As for the last point, notice that the inclusion $A_{i+k}\subset A_{i+k-1}$
yields a short exact sequence of $G_{\Sigma}(\mathbb{Q})$-modules:

$0\to A_{i+k}\to A_{i+k-1}\to\psi\to 0$

where $\psi$ is some representation of $G_{\Sigma}(\mathbb{Q})$ on
$\mathbb{F}_{q}$. Notice that $v_{1}$ is in the kernel of the projection
$A_{i+k-1}\to\psi$. In turn, this projection factors through
$A_{i+k-1}=T^{\prime}\to T^{\prime}/\varpi\to\psi$. Thus $\overline{v}_{1}$ is
in the kernel of $T^{\prime}/\varpi\to\psi$. Counting
$\mathbb{F}_{q}$-dimensions shows the span of $\overline{v}_{1}$ is precisely
equal to the kernel, hence the span is $G_{\Sigma}(\mathbb{Q})$-stable. In
other words, the Galois-action on $T^{\prime}/\varpi$ is upper-triangular in
the basis $(\overline{v}_{1},\overline{v}_{2})$. Finally, we take $T=A_{i+k}$,
it is still in the $G_{\Sigma}(\mathbb{Q})$-stable segment, and since it is on
the boundary of this segment, its residual representation cannot be
semisimple. $\Box$ Fix $T,T^{\prime}$ as in the lemma above. Let $\psi$ be the
character occuring as a sub-$G_{\Sigma}(\mathbb{Q})$-module of $T/\varpi$. The
proof of theorem $1$ works for every maximal ideal of $\Lambda$, except the
unique maximal ideal $(x,\varpi)$ for which
$\psi\otimes\Lambda/(x,\varpi)=\mathbb{F}_{q}$. With such $(x,\varpi)$, the
image in
$\mathbf{H}^{1}(\psi)=H^{1}(G_{\Sigma}(\mathbb{Q}),\psi\otimes\Lambda)$ under
the connecting homomorphism for

${0}$${\psi\otimes\Lambda}$${\psi\otimes\Lambda}$${\psi\otimes\Lambda/x\cong\psi\otimes\Lambda/(\varpi,x)\cong\mathbb{F}_{q}}$${0}$$\scriptstyle{\cdot
x}$

is characterized by the following property:

######  Proposition 5.

An element $\\{\gamma_{n}\\}_{n\geq 0}\in\mathbf{H}^{1}(\psi)$ is in the image
of the connecting homomorphism if and only if, for every $n\geq 1$, we have
$\text{res}(\gamma_{n})=0$, where res denotes the restriction from
$G_{\Sigma}(\mathbb{Q}(\zeta_{p^{n}}))$ to
$G_{\Sigma}(\mathbb{Q}(\zeta_{p^{n+1}}))$.

Proof.  In fact, we know that the image lies in $\mathbf{H}^{1}(\psi)[x]$. For
$n\geq 1$, by [7, Corollary 1.5.7] we have:

$\text{res}\circ\text{cor}(\gamma_{n+1})=\text{res}(\gamma_{n})=N(\gamma_{n+1})$

where $N$ is the norm map:

$N(\gamma_{n+1})=\sum_{\sigma\in G}\sigma(\gamma_{n+1})$

where $G=\text{Gal}(\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q}(\zeta_{p^{n}}))$.
But since $(\gamma_{n})_{n\in\mathbb{N}}$ is $x$-torsion, and since $n\geq 1$,
we know $\gamma_{n}$ is fixed by the various $\sigma\in G$. Thus, the norm map
is just multiplication by $p$. It follows that $\text{res}(\gamma_{n})=0$ for
$n\geq 1$.

It remains to see that every element of $\mathbf{H}^{1}(\psi)$ satisfying this
condition is in the image. First, notice the connecting homomorphism is
injective. Indeed, its kernel is isomorphic to a quotient of
$\mathbf{H}^{0}(\psi)$, but the correstriction map on $0$-th cohomology is
multiplication by $p$, so this group is trivial. It follows that the image has
size $q$, and it suffices to show that no more than $q$ elements of
$\mathbf{H}^{1}(\psi)$ satisfy the conditions of the proposition.

This follows from inflation-restriction. First, notice that $\psi$ is either
$\mathbb{F}_{q}$, or $\mathbb{F}_{q}(r)$, which are both trivial after
restricting to $\mathbb{Q}(\zeta_{p^{n}})$. Hence, any $\gamma_{n}$ whose
restriction vanishes must come from the inflation of a class in
$H^{1}(\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q}(\zeta_{p^{n}}),\mathbb{F}_{q})$.
Since this group is cyclic, there are only $q$ such cohomology classes. $\Box$
In particular, the connecting homomorphism for that exact sequence is
injective. Before stating the next proposition, we note that $\psi$ fits into
the exact sequence:

$0\to T\to T^{\prime}\to\psi\to 0$

Indeed, if this were not the case then $\psi$ would be forced to occur as a
quotient representation of $T/\varpi$ instead of a subrepresentation. As
$T/\varpi$ is a nonsplit extension of distinct characters, this is impossible.

######  Proposition 6.

Let $\\{\gamma_{n}\\}_{n\geq 0}$ be a nontrivial element of
$\mathbf{H}^{1}(\psi)$ satisfying the property from the previous proposition.
Then its image in
$\mathbf{H}^{2}(T)=H^{2}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)$ under the
connecting homomorphism from

$0\to T\otimes\Lambda\to T^{\prime}\otimes\Lambda\to\psi\otimes\Lambda\to 0$

is nontrivial.

Proof.  First, notice that [7, Proposition 1.4.2] allows us to twist by the
character $\chi^{-1}$ as in the second case of the classification. Therefore,
we will assume without loss of generality that $T^{\prime}|_{D_{\ell}}$ is a
ramified extension of two unramified characters. In fact (see [8]), we may
twist so that the characters occuring are the trivial character, and the
cyclotomic character.

Since $T^{\prime}$ is equal to $\varprojlim_{k}T^{\prime}/\varpi^{k}$, and
$T^{\prime}$ is ramified at $\ell$, we are allowed to choose $k$ minimal such
that $T^{\prime}/\varpi^{k}$ is ramified at $\ell$.

Recall that there are only finitely many primes of
$\mathbb{Q}(\zeta_{p^{\infty}})$ above $\ell$. Thus, we may choose some $n>k$
large enough that the primes of $\mathbb{Q}(\zeta_{p^{n}})$ above $\ell$
remain inert in the extension
$\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q}(\zeta_{p^{n}})$. Fix such an $n$.
Since the action of $G_{\Sigma}(\mathbb{Q})$ on $T^{\prime}/\varpi^{k}$ is
ramified at $\ell$, and since $\mathbb{Q}(\zeta_{p^{n}})/\mathbb{Q}$ is
unramified away from $p$, we may choose a prime $\mathfrak{l}$ of
$\mathbb{Q}(\zeta_{p^{n}})$ above $\ell$ such that the action of
$G_{\Sigma}(\mathbb{Q}(\zeta_{p^{n}}))$ on $T^{\prime}/\varpi^{k}$ is ramified
at $\mathfrak{l}$.

To prove the proposition, it suffices to project onto the $n$-th factor of
$\mathbf{H}^{1}(\psi)$ and to restrict to the decomposition group at
$\mathfrak{l}$, then prove that the image of $\gamma_{n}$ under the connecting
homomorphism for

$0\to T\to T^{\prime}\to\psi\to 0$

is nontrivial in $H^{2}(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}},T)$, where
the exact sequence is regarded as a sequence of
$G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}})$-modules. We have a projection
$T^{\prime}/\varpi^{k}\to\psi$. Write $K$ for the kernel of this projection.
We have the following diagram (of
$G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}})$-modules):

${0}$${T}$${T^{\prime}}$${\psi}$${0}$${0}$${K}$${T^{\prime}/\varpi^{k}}$${\psi}$${0}$$\scriptstyle{=}$

Clearly, to show that the image of $\gamma_{n}$ in
$H^{2}(G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}),T)$ is nonzero, it suffices
to show that its image under
$H^{2}(G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}),T)\to
H^{2}(G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}),K)$ is nonzero. By
functoriality of $\delta$-morphisms, this is the same as just applying the
connecting homomorphism from the exact sequence in the bottom row.

Now, lemma $1$ shows the existence of the following commutative diagram:

${0}$${\mathcal{O}/\varpi^{k}}$${T^{\prime}/\varpi^{k}}$${\mathcal{O}/\varpi^{k}}$${0}$${0}$${K}$${T^{\prime}/\varpi^{k}}$${\mathbb{F}_{q}}$${0}$$\scriptstyle{=}$

Here, we use the fact that since $n>k$, the restriction to
$G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}})$ of the cyclotomic character is
trivial modulo $\varpi^{k}$. Since $\mathfrak{l}$ is inert in
$\mathbb{Q}(\zeta_{p^{n+1}})/\mathbb{Q}(\zeta_{p^{n}})$, the extension
$\mathbb{Q}(\zeta_{p^{n+1}})_{\mathfrak{l}}/\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}$
is an unramified, cyclic extension of degree $p$. Choose a generator $\sigma$
for the corresponding Galois group. An element of
$H^{1}(\text{Gal}(\mathbb{Q}(\zeta_{p^{n+1}})_{\mathfrak{l}}/\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}),\mathbb{F}_{q})$
is determined by where it maps this generator. Let us say that
$\text{res}_{\mathfrak{l}}(\gamma_{n})$ is the inflation of the cohomology
class which maps $\sigma$ to $m\in\mathbb{F}_{q}$. Then, choosing some
$\tilde{m}$ which maps to $m$ under the projection
$\mathcal{O}/\varpi^{k}\to\mathbb{F}_{q}$, we let $\tilde{\gamma}_{n}\in
H^{1}(G(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}}),\mathcal{O}/\varpi^{k})$ be
the inflation of the class mapping $\sigma$ to $\tilde{m}$.

Now, to calculate the image of $\gamma_{n}$ under the connecting homomorphism
from the bottom row, it suffices to use the connecting homomorphism from the
top row on $\tilde{\gamma}_{n}$, then map to $K$. But since
$T^{\prime}/\varpi^{k}$ is a ramified extension of two copies of
$\mathcal{O}/\varpi^{k}$, this corresponds to taking the cup product
$\tilde{\gamma}_{n}$ with some ramified $\alpha\in
H^{1}(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}},\mathcal{O}/\varpi^{k})$. Now,
all unramified classes in $H^{2}$ of a local field are trivial, so
$(\tilde{\gamma}_{n}\cup-)$ annihilates any unramified class. Since the cup
product is nondegenerate, and $\tilde{\gamma}_{n}$ is not a multiple of
$\varpi$, we find that its cup product with any ramified class is nontrivial.

It remains to show that, given a nontrivial class in
$H^{2}(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}},\mathcal{O}/\varpi^{k})$, its
image in $H^{2}(\mathbb{Q}(\zeta_{p^{n}})_{\mathfrak{l}},K)$ is nontrivial.
Since the class is ramified, its restriction to inertia is nontrivial.
Denoting the inertia group by $I_{\mathfrak{l}}$, it suffices to show
$H^{2}(I_{\mathfrak{l}},\mathcal{O}/\varpi^{k})\to H^{2}(I_{\mathfrak{l}},K)$
has trivial kernel.

Recall that $k$ was chosen to be minimal such that the action of inertia on
$T^{\prime}/\varpi^{k}$ is ramified. Therefore, in the basis $(v_{1},v_{2})$,
the inertia group acts via

$\begin{bmatrix}\chi_{1}&\varpi^{k-1}*\\\ 0&\chi_{2}\end{bmatrix}$

Writing $\overline{v}_{1},\overline{v}_{2}$ for the reductions modulo
$\varpi^{k}$, $K$ is the submodule of $T^{\prime}/\varpi^{k}$ spanned by
$\overline{v}_{1},\varpi\overline{v}_{2}$. In this form, it is clear that
$K|_{I_{\mathfrak{l}}}$ is a split extension of $\mathcal{O}/\varpi^{k}$ and
$\mathcal{O}/\varpi^{k-1}$. This shows that
$H^{2}(I_{\mathfrak{l}},\mathcal{O}/\varpi^{k})\to H^{2}(I_{\mathfrak{l}},K)$
has trivial kernel as desired. $\Box$

######  Theorem 7.

Let $T$ be as above. Then $\mathbf{H}^{1}(T)$ is free over $\Lambda$.

Proof.  Let $(x,\varpi)$ be the maximal ideal of $\Lambda$ such that
$\psi\otimes\Lambda/(x,\varpi)\cong\mathbb{F}_{q}$. We must show that
$\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)$ is $\varpi$-torsion free. Recall that
we have an embedding $\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)\subseteq
H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$ which comes from the exact
sequence

${0}$${T\otimes\Lambda}$${T\otimes\Lambda}$${T\otimes\Lambda/x}$${0}$$\scriptstyle{\cdot
x}$

We see therefore, that the image of $\mathbf{H}^{1}(T)/x\mathbf{H}^{1}(T)$ in
$H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$ corresponds with the kernel
of the map $H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)\to
H^{2}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)$. Since the $\varpi$-torsion
inside $H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)$ must come from
$H^{0}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/(x,\varpi))$, it suffices to
show that the composition

$H^{0}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/(x,\varpi))\to
H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)\to
H^{2}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)$

is nonzero. Indeed,
$H^{0}(T\otimes\Lambda/(x,\varpi))\cong\mathcal{O}/\varpi$, and all maps are
$\mathcal{O}$-linear, so to show the composition has trivial kernel it
suffices to find just one element which is not annihilated.

Notice we have the following anticommutative diagram:

${H^{0}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/(x,\varpi))}$${H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/x)}$${H^{1}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda/\varpi)}$${H^{2}(G_{\Sigma}(\mathbb{Q}),T\otimes\Lambda)}$$\scriptstyle{a}$$\scriptstyle{b}$

therefore, it suffices to show that the map $b\circ a$ is nonzero. Writing
$T\otimes\Lambda/(x,\varpi)\cong T/\varpi\otimes\Lambda/x$, we see that $a$ is
the connecting homomorphism from the bottom row of the following diagram,
where the vertical maps are induced from the inclusion $\psi\to T/\varpi$.

${0}$${\psi\otimes\Lambda}$${\psi\otimes\Lambda}$${\psi\otimes\Lambda/x}$${0}$${0}$${T/\varpi\otimes\Lambda}$${T/\varpi\otimes\Lambda}$${T/\varpi\otimes\Lambda/x}$${0}$$\scriptstyle{\cdot
x}$$\scriptstyle{c}$$\scriptstyle{\cdot x}$

By abuse of notation, we will also write $c$ for the morphism it induces on
cohomology. Write $\delta_{1}$ for the connecting homomorphism from the top
row. Then $a=c\circ\delta_{1}$. Next, notice that $b$ is the connecting
homomorphism from the bottom row of the following diagram:

${0}$${T\otimes\Lambda}$${T^{\prime}\otimes\Lambda}$${\psi\otimes\Lambda}$${0}$${0}$${T\otimes\Lambda}$${T\otimes\Lambda}$${T/\varpi\otimes\Lambda}$${0}$$\scriptstyle{=}$$\scriptstyle{c}$$\scriptstyle{\cdot\varpi}$

Now, letting $\delta_{2}$ denote the connecting homomorphism from the top row
of this diagram, we see $b\circ c=\delta_{2}$. Therefore, $b\circ a=b\circ
c\circ\delta_{1}=\delta_{2}\circ\delta_{1}$. But proposition $6$ shows that
this composition is nonzero. $\Box$

Next, we show that this freeness result implies the integrality of
$z_{\gamma}^{(p)}$. For $T=V_{\mathcal{O}}(f)$, Kato shows this in [4, 13.14]
by showing that
$Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}$ for any
height $1$ prime $\mathfrak{q}$ of $\Lambda$, where $Z(f,T)$ is the module
generated by $z_{\gamma}^{(p)}$ for $\gamma\in T$. Kato then uses freeness to
derive that $Z(f,T)\subseteq\mathbf{H}^{1}(T)$. We will show that the
inclusion of local modules
$Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}$ holds for any
lattice $T$.

######  Theorem 8.

Suppose that for some lattice $T$, one knows
$Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}$ where
$\mathfrak{q}$ is a height $1$ prime of $\Lambda$. If $T^{\prime}$ is adjacent
to $T$ in the sense that $\varpi T^{\prime}\subseteq T\subseteq T$, then
$Z(f,T^{\prime})_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T^{\prime})_{\mathfrak{q}}$

Proof.  Let $\chi_{1}$ be the character occurring in

$0\to T\to T^{\prime}\to\chi_{1}\to 0$

and let $\chi_{2}$ be the character occurring in

$0\to\varpi T^{\prime}\to T\to\chi_{2}\to 0$

We know one of the characters is odd, and the other is even. Assume without
loss of generality that $\chi_{1}$ is odd and $\chi_{2}$ is even.

Recall that the connected components of $\text{spec}(\Lambda)$ correspond to
Galois-characters of $G$ acting on $\mathbb{F}_{q}$. We will say that
$\mathfrak{q}$ is odd if the Galois-character corresponding to its connected
component is odd, and even otherwise.

First, suppose $\mathfrak{q}$ is odd. Then we know that for any $\gamma\in V$,
$z_{\gamma}=z_{\gamma^{+}}$ inside the localization
$\mathbf{H}^{1}(V)_{\mathfrak{q}}$. See chapter $13$ of [4].

Let $a,b$ be such that $(a,b)$ generates $T^{\prime}$, and that $(a,\varpi b)$
generates $T$. Then the image of $b$ in $\chi_{1}$ is negated by complex
conjugation since $\chi_{1}$ is odd. Hence the image of $b^{+}$ in $\chi$ is
$0$, and so $b^{+}\in T$. Therefore, $z_{b}\in Z(f,T)_{\mathfrak{q}}$. It
follows now that

$Z(f,T^{\prime})_{\mathfrak{q}}=Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T^{\prime})_{\mathfrak{q}}$

Now assume $\mathfrak{q}$ is even. We will now work with the character
$\chi_{2}$:

$0\to\varpi T^{\prime}\to T\to\chi_{2}\to 0$

Recall that $(a,\varpi b)$ generates $T$, and $(\varpi a,\varpi b)$ generates
$\varpi T^{\prime}$. The image of $a$ in $\chi_{2}$ is fixed by conjugation,
and hence $a^{-}$ must lie in $\varpi T$. But this time, since $\mathfrak{q}$
is even, we have $z_{a}=z_{a^{-}}$ in the localization. Hence
$Z(f,T^{\prime})_{\mathfrak{q}}=Z(f,\varpi T)_{\mathfrak{q}}$ Now, the
cokernel of $\mathbf{H}^{1}(\varpi T^{\prime})\to\mathbf{H}^{1}(T)$ embeds
into
$\mathbf{H}^{1}(\chi_{2})=\mathbb{F}_{q}(\chi_{2})\oplus\Lambda^{-}/\varpi$.
Here $\Lambda^{-}$ denotes just the Teichmuller-eigenspaces of $\Lambda$
corresponding to odd Galois-characters.

After localizing at the even prime $\mathfrak{q}$, the finite part disappears
since it is pseudonull, and the odd Teichmuller-eigenspaces vanish since we
localize at an even prime. Therefore, the cokernel is trivial.

We now conclude:

$Z(f,\varpi
T^{\prime})_{\mathfrak{q}}=Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}=\mathbf{H}^{1}(\varpi
T^{\prime})_{\mathfrak{q}}$ $Z(f,\varpi
T^{\prime})_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(\varpi
T^{\prime})_{\mathfrak{q}}$

Recall that the morphism $\gamma\mapsto z_{\gamma}$ is actually linear, so we
can conclude that

$Z(f,T^{\prime})_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T^{\prime})_{\mathfrak{q}}$

$\Box$ In particular, since
$Z(f,T)_{\mathfrak{q}}\subseteq\mathbf{H}^{1}(T)_{\mathfrak{q}}$ for all
height $1$ primes $\mathfrak{q}$ for $T=V_{\mathcal{O}}(f)$, and since the
graph defined in lemma $4$ is connected, we see that
$\mathbf{H}^{1}(T)_{\mathfrak{q}}$ for any lattice.

Therefore, if $T$ is such that $\mathbf{H}^{1}(T)$ is free over $\Lambda$,
then $z_{\gamma}^{(p)}\in\mathbf{H}^{1}(T)$ for $\gamma\in T$.

3\. A Divisibility in the Main Conjecture

We will show that this integrality result for $z_{\gamma}^{(p)}$ yields a
divisibility in the main conjecture. Fix a Galois-stable lattice $T$ of $V$.
Let $X$ be the Pontryagin dual of Selmer group associated to $T$ (see section
$14$ of [4]). Let $\text{char}_{\Lambda}(X)$ be the characteristic ideal of
$X$. For $\mathfrak{p}$ good, the main conjecture predicts that

$\text{char}_{\Lambda}(X)=(\mathcal{L}_{\mathfrak{p}}(f))$

where $\mathcal{L}_{\mathfrak{p}}(f)$ is the $\mathfrak{p}$-adic $L$-function
associated to $f$. We shall show that, if some $\ell$ is special for $f$, then
the characteristic ideal divides the $\mathfrak{p}$-adic $L$-function inside
$\Lambda$. Note that the definition of $\mathcal{L}_{\mathfrak{p}}$ depends on
certain parameters $\gamma,\omega$; however, if these are “good” in the sense
of Kato [4, 17.5], then the main conjecture is independent of these choices.
In particular, it suffices to prove the main conjecture for just one lattice
$T$; we fix $T$ such that $\mathbf{H}^{1}(T)$ is free.

Kato’s original strategy for proving the divisibility in the main conjecture
relies on a strong hypothesis concerning the image of the Galois
representation $\rho$. In the case of elliptic curves, Wuthrich shows [9,
Theorem 16] that the divisibility follows without such a hypothesis in the
Eisenstein case, so long as the $z_{\gamma}^{(p)}$ are integral. Wuthrich’s
argument works equally well in our case, replacing isogenies with inclusions
of lattices where appropriate. We need only adapt a result of Coates-Sujatha
[3, Corollary 3.6] concerning the fine Selmer group of an elliptic curve.

Let $T^{\vee}$ denote the Cartier dual of $T$. For a number field
$L/\mathbb{Q}$, we define the fine Selmer group to be the kernel of the map

$H^{1}(L,T^{\vee})\to\bigoplus_{w|v,v\in\Sigma}H^{1}(L_{w},T^{\vee})$

Let $Y(T/L)$ be the Pontryagin dual of the fine Selmer group. For
$\mathcal{L}/L$ a pro-$p$ extension of $L$, we let $Y(T/\mathcal{L})$ be the
Pontryagin dual of the limit over finite subextensions $L^{\prime}/L$. Also,
for any $L$-module $A$, we let $L(A)$ denote the field extension of $L$ fixed
by the kernel of $\rho_{A}:\text{Gal}(\overline{L}/L)\to\text{Aut}(A)$.

######  Lemma 9.

Let $\chi_{1},\chi_{2}$ be the characters occuring in the semisimplification
of $T/\varpi$. Let $H_{i}$ be the fixed field of the kernel of $\chi_{i}$, and
let $H=H_{1}H_{2}(\zeta_{p})$. Then $Y(T/H)$ is a finitely generated
$\mathcal{O}$-module.

Proof.  First, the definition of the fine Selmer group in [3], and the proof
of [3, Theorem 3.4] can be carried out purely formally in the language of
Galois modules. Notice $H/\mathbb{Q}$ is a subfield of a cyclotomic field,
hence abelian. In particular, the Iwasawa $\mu$-invariant conjecture holds for
$H^{\text{cyc}}$. Thus, upon replacing $E_{p^{\infty}}$ with $T^{\vee}$, and
using coefficients $\mathcal{O}$ rather than $\mathbb{Z}_{p}$, in the proof of
[3, Theorem 3.4] it suffices to show that $H(T^{\vee})/H$ is pro-$p$.

Notice that $H(T^{\vee})/H(T^{\vee}[\varpi])$ is pro-$p$ if and only if
$H(T)/H(T/\varpi)$ is pro-$p$. But, via the defining representation of $T$,
the Galois group of the latter extension embeds into the kernel of the
reduction map
$\text{Gl}_{2}(\mathcal{O})\to\text{Gl}_{2}(\mathcal{O}/\varpi)$, which is
pro-$p$. It remains to show that $H(T^{\vee}[\varpi])/H$ is pro-$p$. Since $H$
contains $\zeta_{p}$, this is equivalent to showing $H(T/\varpi)/H$ is
pro-$p$. After restricting to $H$, the Galois action on $T/\varpi$ takes the
form

$\begin{bmatrix}1&\psi\\\ 0&1\end{bmatrix}$

Hence $\psi$ is an additive character
$\psi:\text{Gal}(\overline{H}/H)\to\mathbb{F}_{q}$. It follows that
$H(T/\varpi)/H$ is pro-$p$. $\Box$

######  Theorem 10 (Wuthrich [9]).

Suppose $f$ has good reduction at $\mathfrak{p}$ in the sense of Kato. Assume
there exists some prime $\ell\neq p$ which is special for $f$. Let
$\text{char}_{\Lambda}(X(T))$ denote the characteristic ideal of $X(T)$. Then
$\text{char}_{\Lambda}(X(T))$ divides $(\mathcal{L}_{\mathfrak{p}})$.

Proof.  The statement is invariant under isogeny, so assume that $T$ is as in
Lemma $1$. Our assumption that $\mathfrak{p}$ is good allows us to use $17.11$
in [4]. Thus, with the preceding lemma, the proof of [9, Theorem 16] works in
our situation, mutatis mutandis. $\Box$ In particular, combining this theorem
with the main theorem of the previous section, we conclude that
$\mathcal{L}_{\mathfrak{p}}$ is integral whenever $\mathfrak{p}$ is good, and
there exists $\ell\neq p$ which is special for $f$.

4\. Application to Abelian Varieties

To prove the freeness result, our proofs relied on the existence of some prime
$\ell\neq p$ which is special for $f$. Here, we show that such prime exists
when $f$ is of weight $2$, and therefore corresponds to an abelian variety. As
before, the proof of freeness only becomes an issue when the characters
$\mathbb{F}_{q},\mathbb{F}_{q}(r)$ occur in the semisimplification.

######  Lemma 11.

Suppose $\overline{\rho}^{ss}=\mathbb{F}_{q}\oplus\mathbb{F}_{q}(r)$. Then for
any $\ell\mid N$, the cuspidal case of Carayol and Livne’s classification
cannot occur.

Proof.  Suppose otherwise. Let $\Omega/\mathbb{Q}_{\ell}$ be the unique
unramified, quadratic extension of $\mathbb{Q}_{\ell}$. Let $\xi$ be a
character of $\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\Omega)$ as in the
classification. Then $\rho|_{D_{\ell}}$, the restriction of $\rho$ to the
decomposition group at $\ell$, is the representation of
$\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ induced from this
character. We know that, for any choice of lattice, the reduction
$\overline{\rho}|_{D_{\ell}}$ must be reducible. By Mackey’s criterion, we
must have that $\overline{\xi}=\overline{\xi}^{\sigma}$, where $\sigma$ is the
unique automorphism of $\Omega/\mathbb{Q}_{\ell}$. It now follows that:

$\overline{\rho}^{ss}|_{\text{Gal}(\overline{\mathbb{Q}}_{\ell}/\Omega)}=\overline{\xi}\oplus\overline{\xi}$

But this is impossible since $\text{det}(\overline{\rho})$ must be odd. $\Box$

######  Lemma 12.

For $f$ an ordinary, weight $2$ newform, there exists some prime $\ell\neq p$
which is special for $f$.

Proof.  Indeed, if no such $\ell$ exists then by the previous lemma,
$\rho|_{D_{\ell}}$ is decomposable at each bad prime $\ell\neq p$, according
to the classification. One knows (see [8, Proposition 4.4]) that for such
$\ell$, we may twist $f$ by some dirichlet character of conductor $\ell$, such
that the associated newform has level dividing $N/\ell$, and such that the
residual representations coincide. In particular, since $f$ is ordinary, the
exponent of $p$ in the level is $0$ or $1$. Thus, by repeating this process,
we strip away the primes away from $p$ and obtain a weight $2$ newform of
level $N=1$, or $N=p$. Moreover, we know $f$ is Eisenstein at the prime
$\mathfrak{p}|p$. The first case cannot occur, as all weight $2$ forms have
level $>1$. In the second case, Mazur’s study of the Eisenstein ideal for
$N=p$ in [6] shows that $\mathfrak{p}$ cannot be Eisenstein. Indeed,
$\mathfrak{p}$ would need to divide the numerator of $(p-1)/12$ which is
impossible. $\Box$

######  Corollary 13.

Suppose $\mathfrak{p}$ is Eisenstein for $f$ a weight $2$ newform. Then
$\mathcal{L}_{\mathfrak{p}}\in\Lambda$, and $\text{char}_{\Lambda}(X)$ divides
$\mathcal{L}_{\mathfrak{p}}$

Proof.  Combine Theorem $10$ with the previous two lemmas. $\Box$ In
particular, the integrality and the divisibility hold when $f$ corresponds to
an abelian variety of $\text{Gl}_{2}$-type.

## References

  * [1] Joel Bellaïche.  Ribet’s lemma, generalizations, and pseudocharacters.  2009\.
  * [2] Henri Carayol.  Sur les représentations galoisiennes modulo l attachées aux formes modulaires.  Duke Math. J, 59(3):785–801, 1989.
  * [3] John Coates and Ramdorai Sujatha.  Fine selmer groups of elliptic curves over p-adic lie extensions.  Mathematische Annalen, 331:809–839, 2005.
  * [4] Kazuya Kato et al.  p-adic hodge theory and values of zeta functions of modular forms.  Astérisque, 295:117–290, 2004.
  * [5] Ron Livné.  On the conductors of mod/galois representations coming from modular forms.  Journal of Number Theory, 31(2):133–141, 1989.
  * [6] Barry Mazur.  Modular curves and the eisenstein ideal.  Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 47(1):33–186, 1977.
  * [7] Jürgen Neukirch, Alexander Schmidt, and Kay Wingberg.  Cohomology of number fields, volume 323.  Springer Science & Business Media, 2013.
  * [8] Kenneth A Ribet.  Report on mod l representations of gal (q/q).  In Proceedings of Symposia in Pure Mathematics, volume 55, pages 639–676, 1994.
  * [9] Christian Wuthrich.  On the integrality of modular symbols and kato’s euler system for elliptic curves.  2014\.