0}$, and let $D\in C^{(d)}(S)$ and $E\in C^{(e)}(S)$. Then we have:

$\mathcal{M}(\Sigma(D),\alpha_{\mathcal{L}}(\Sigma(E)))=\left(\mathrm{Norm}_{D/S}(\mathrm{id},b)^{*}\mathcal{L}\right)^{\otimes(1-e)}\otimes\mathrm{Norm}_{(D\times
E)/S}(\mathcal{L})\,.$

For $x\in C(S)$ we have

$T_{m{\cdot}\circ\alpha_{\mathcal{L}}}(j_{b}(x))=\mathcal{M}^{\times}(j_{b}(x),m{\cdot}\alpha_{\mathcal{L}}(j_{b}(x)))=\mathcal{L}^{\otimes
m}(x,x)^{\times}=(\mathbb{G}_{\mathrm{m}})_{S}\,.$

###### Proof.

We may and do assume (finite locally free base change on $S$) that we have
$x_{i}$ and $y_{j}$ in $C(S)$, such that $D=\sum_{i}x_{i}$ and
$E=\sum_{j}y_{j}$. Recall that, for $c\in C(S)$, $\beta_{\mathcal{L}}(c)$ in
$J(S)$ is $(\mathrm{id},c)^{*}\mathcal{L}$ on $C_{S}$, with its rigidification
at $b$. Then we have:

$None$ $\displaystyle\mathcal{M}(\Sigma(D),\alpha_{\mathcal{L}}(\Sigma(E)))$
$\displaystyle=\mathcal{M}(\alpha_{\mathcal{L}}(\Sigma(E)),\Sigma(D))$
$\displaystyle=\mathcal{M}\left(\beta_{\mathcal{L}}(b)+\sum_{j}(\beta_{\mathcal{L}}(y_{j})-\beta_{\mathcal{L}}(b)),\sum_{i}j_{b}(x_{i})\right)$
$\displaystyle=\left(\bigotimes_{i}\mathcal{L}(x_{i},b)^{\otimes(1-e)}\right)\otimes\bigotimes_{i,j}\mathcal{L}(x_{i},y_{j})\,.$

from which the desired equality follows.

Now we prove the second claim. Let $x$ be in $C(S)$. The first equality holds
by definition. Taking $D=E=x$ in what we just proved, gives the second
equality, and the third comes from the rigidification at $b$. ∎

Now let $\mathcal{L}$ be any extension of $\mathcal{L}$ with its
rigidification of $(b,\mathrm{id})^{*}\mathcal{L}$ from $(C\times
C)_{\mathbb{Z}[1/n]}$ to $C\times U$. For $q$ dividing $n$, let $W_{q}$ be the
valuation along $U_{\mathbb{F}_{q}}$ of the rational section $\ell$ of
$\mathrm{diag}^{*}\mathcal{L}$ on $U$. Then $\ell$, multiplied by the product,
over the primes $q$ dividing $n$, of $q^{-W_{q}}$, generates
$\mathrm{diag}^{*}\mathcal{L}$ on $U$:

$None$
$\left(\prod_{q|n}q^{-W_{q}}\right){\cdot}\ell\in(\mathrm{diag}^{*}\mathcal{L}^{\times})(U)\,.$

There is a unique divisor $V$ on $C\times U$ with support disjoint from
$(b,\mathrm{id})U$ and contained in the $(C\times U)_{\mathbb{F}_{q}}$ with
$q$ dividing $n$, such that

$None$ $\mathcal{L}^{m}:=\mathcal{L}^{\otimes m}(V)\quad\text{on $C\times U$}$

has multidegree $0$ on the fibres of $\mathrm{pr}_{2}\colon C\times U\to U$.
Then $\mathcal{L}^{m}$ is the pullback of $\mathcal{L}^{\mathrm{univ}}$ via
$\mathrm{id}\times(m{\cdot}\circ\alpha_{\mathcal{L}}\circ j_{b})\colon C\times
U\to C\times J^{0}$. Its restriction $\mathcal{L}^{m}|_{C^{\mathrm{sm}}\times
U}$ is then the pullback of $\mathcal{M}$ via
$j_{b}\times(m{\cdot}\circ\alpha_{\mathcal{L}}\circ j_{b})\colon
C^{\mathrm{sm}}\times U\to J\times J^{0}$, because on $C^{\mathrm{sm}}\times
J^{0}$ the restriction of $\mathcal{L}^{\mathrm{univ}}$ and
$(j_{b}\times\mathrm{id})^{*}\mathcal{M}$ are equal (both are rigidified after
$(b,\mathrm{id})^{*}$ and equal over $\mathbb{Z}[1/n]$; here we use that, for
all $q|n$, $J^{0}_{\mathbb{F}_{q}}$ is geometrically connected). Hence, on $U$
we have
$j_{b}^{*}T_{m{\cdot}\circ\alpha_{\mathcal{L}}}=\mathrm{diag}^{*}(\mathcal{L}^{\otimes
m}(V)^{\times})$, compatible with rigidifications at $b\in
U(\mathbb{Z}[1/n])$. Our trivialisation $\widetilde{j_{b}}$ on $U$ of
$T_{m{\cdot}\circ\alpha_{\mathcal{L}}}$ is therefore a generating section of
$\mathcal{L}^{\otimes m}$, multiplied by the product over the $q$ dividing
$n$, of the factors $q^{-V_{q}}$, where $V_{q}$ is the multiplicity in $V$ of
the prime divisor $(U\times U)_{\mathbb{F}_{q}}$. This means that we have
proved the following proposition.

###### 7.8 Proposition

For $x$ and $S$ as in Proposition 7.5, we have the following description of
$\widetilde{j_{b}}$:

$\widetilde{j_{b}}(x)=\left(\prod_{q|n}q^{-mW_{q}-V_{q}}\right){\cdot}\ell^{\otimes
m}\quad\text{in
$(T_{m{\cdot}\circ\alpha_{\mathcal{L}}}(j_{b}(x)))(S)=\mathcal{L}^{\otimes
m}(x,x)^{\times}(S)$.}$

##  8 An example with genus 2, rank 2, and 14 points

The example that we are going to treat is the quotient of the modular curve
$X_{0}(129)$ by the action of the group of order $4$ generated by the Atkin-
Lehner involutions $w_{3}$ and $w_{43}$. An equation for this quotient is
given in the table in [18], and Magma has shown that that equation and the
equations below give isomorphic curves over $\mathbb{Q}$.

Let $C_{0}$ be the curve over $\mathbb{Z}$ obtained from the following closed
subschemes of $\mathbb{A}^{2}_{\mathbb{Z}}$

$\displaystyle V_{1}:$ $\displaystyle\quad
y^{2}+y=x^{6}-3x^{5}+x^{4}+3x^{3}-x^{2}-x\,,$ $\displaystyle V_{2}:$
$\displaystyle\quad w^{2}+z^{3}w=1-3z+z^{2}+3z^{3}-z^{4}-z^{5}$

by glueing the open subset of $V_{1}$ where $x$ is invertible with the open
subset of $V_{2}$ where $z$ is invertible using the identifications $z=1/x$,
$w=y/x^{3}$. The scheme $C_{0}$ can be also described as a subscheme of the
line bundle $\mathcal{L}_{3}$ associated to the invertible
$\mathcal{O}$-module $\mathcal{O}_{\mathbb{P}_{\mathbb{Z}}^{1}}(3)$ on
$\mathbb{P}^{1}_{\mathbb{Z}}$ with homogeneous coordinates $X,Z$: the map
$\mathcal{O}_{\mathbb{P}_{\mathbb{Z}}^{1}}(3)\to\mathcal{O}_{\mathbb{P}_{\mathbb{Z}}^{1}}(6)$
sending a section $Y$ to $Y\otimes Y+Z^{3}\otimes Y$ induces a map $\varphi$
from $\mathcal{L}_{3}$ to the line bundle $\mathcal{L}_{6}$ associated to
$\mathcal{O}(6)$; then $C_{0}$ is isomorphic to the inverse image by $\varphi$
of the section
$s:=X^{6}{-}3X^{5}Z{+}X^{4}Z^{2}{+}3X^{3}Z^{3}{-}X^{2}Z^{4}{-}XZ^{5}$ of
$\mathcal{L}_{6}$ and since the map $\varphi$ is finite of degree $2$ then
$C_{0}$ is finite of degree $2$ over $\mathbb{P}^{1}_{\mathbb{Z}}$. Hence
$C_{0}$ is proper over $\mathbb{Z}$ and it is moreover smooth over
$\mathbb{Z}[1/n]$ with $n=3\cdot 43$. The generic fiber of $C_{0}$ is a curve
of genus $g=2$, labeled 5547.b.16641.1 on www.lmfdb.org. The only point where
$C_{0}$ is not regular is the point $P_{0}=(3,x-2,y-1)$ contained in $V_{1}$
and the blow up $C$ of $C_{0}$ in $P_{0}$ is regular.

In the rest of this section we apply our geometric method to the curve $C$ and
we prove that $C(\mathbb{Z})$ contains exactly $14$ elements. We use the same
notation as in Sections 2 and 4.

The fiber $C_{\mathbb{F}_{43}}$ is absolutely irreducible while
$C_{\mathbb{F}_{3}}$ is the union of two geometrically irreducible curves, a
curve of genus $0$ that lies above the point $P_{0}$ and that we call $K_{0}$,
and a curve of genus $1$ that we call $K_{1}$. We define $U_{0}:=C\setminus
K_{1}$ and $U_{1}:=C\setminus K_{0}$ so that
$C(\mathbb{Z})=C^{\mathrm{sm}}(\mathbb{Z})=U_{0}(\mathbb{Z})\cup
U_{1}(\mathbb{Z})$ and both $U_{0}$ and $U_{1}$ satisfy the hypothesis on $U$
in Section 2. We have $K_{0}\cdot K_{1}=2$ and consequently the self-
intersections of $K_{0}$ and $K_{1}$ are both equal to ${-}2$. We deduce that
all the fibers of $J$ over $\mathbb{Z}$ are connected except for
$J_{\mathbb{F}_{3}}$ which has group of connected components equal to
$\mathbb{Z}/2\mathbb{Z}$. Hence,

$None$ $m=2\,.$

The automorphism group of $C$ is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{2}$,
generated by the automorphisms $\iota$ and $\eta$ lifting the extension to
$C_{0}$ of

$\iota,\eta\colon V_{1}\longrightarrow
V_{1}\,,\quad\iota\colon(x,y)\longmapsto(x,-1-y)\,,\quad\eta\colon(x,y)\longmapsto(1-x,-1-y)\,.$

The quotients $E_{1}:=C_{\mathbb{Q}}/\eta$ and
$E_{2}:=C_{\mathbb{Q}}/(\iota\circ\eta)$ are curves of genus $1$ and the two
projections $C\to E_{i}$ induce an isogeny
$J\to\mathrm{Pic}^{0}(E_{1})\times\mathrm{Pic}^{0}(E_{2})$. The elliptic
curves $\mathrm{Pic}^{0}(E_{i})$ are not isogenous and $\rho=2$.

###  8.2 The torsor on the jacobian

Let $\infty,\infty_{-}\in C(\mathbb{Z})$ be the lifts of $(0,1),(0,-1)\in
V_{2}(\mathbb{Z})\subset C_{0}(\mathbb{Z})$ and let us fix the base point
$b=\infty$ in $C(\mathbb{Z})$. Following Section 7 we describe a
$\mathbb{G}_{\mathrm{m}}$-torsor $T\to J$ and maps $\widetilde{j_{b,i}}\colon
U_{i}\to T$ using invertible $\mathcal{O}$-modules on $C\times
C^{\mathrm{sm}}$. The torsor
$T=(\mathrm{id},m{\cdot}\circ\alpha)^{*}\mathcal{M}^{\times}$ only depends on
the scheme morphism $\alpha\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$, which, by
Proposition 7.2, is uniquely determined by an invertible $\mathcal{O}$-module
$\mathcal{L}$ 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.

We now look for a non-trivial $\mathcal{O}$-module $\mathcal{L}$ with these
properties using the homomorphism $\eta^{*}\colon J_{\mathbb{Q}}\to
J_{\mathbb{Q}}$, which does not belong to
$\mathbb{Z}\subset\mathrm{End}(J_{\mathbb{Q}})$. We can take $\alpha$ of the
form $\mathrm{tr}_{c}\circ(n_{1}{\cdot}\eta^{*}+n_{2}{\cdot}\mathrm{id})$,
where $\mathrm{id}\colon J_{\mathbb{Q}}\to J_{\mathbb{Q}}$ is the identity
map, $n_{i}$ are integers and $c$ lies in $J(\mathbb{Q})$. Using the map
$\alpha\mapsto\mathcal{L}_{\alpha}:=(j_{b}\times(j_{b}{\circ}\alpha))^{*}\mathcal{M}$
in Proposition 7.2, the $\mathcal{O}$-module $\mathcal{L}_{\mathrm{tr}_{c}}$
is isomorphic to $\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(\mathrm{pr}_{1}^{*}D)$, the $\mathcal{O}$-module
$\mathcal{L}_{\eta^{*}}$ is isomorphic to $\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(\Gamma_{\eta,\mathbb{Q}}-\mathrm{pr}_{1}^{*}\eta^{*}(b)-\mathrm{pr}_{2}^{*}\eta(b))$
and the $\mathcal{O}$-module $\mathcal{L}_{\mathrm{id}}$ is isomorphic to
$\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(\mathrm{diag}(C_{\mathbb{Q}})-\mathrm{pr}_{1}^{*}(b)-\mathrm{pr}_{2}^{*}(b))$,
where $D$ is a divisor on $C_{\mathbb{Q}}$ representing $c$, the maps
$\mathrm{pr}_{i}$ are the projections $C_{\mathbb{Q}}\times C_{\mathbb{Q}}\to
C_{\mathbb{Q}}$ and $\Gamma_{\eta}$ is the graph of the map $\eta\colon C\to
C$. Hence, we can take $\mathcal{L}$ of the form
$\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(n_{1}\Gamma_{\eta,\mathbb{Q}}+n_{2}\mathrm{diag}(C_{\mathbb{Q}})+\mathrm{pr}_{1}^{*}D_{1}+\mathrm{pr}_{2}^{*}D_{2})$
for some integers $n_{i}$ and some divisors $D_{i}$ on $C_{\mathbb{Q}}$. Among
the $\mathcal{O}$-modules of this form satisfying the needed properties, we
choose

$\mathcal{L}:=\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(\Gamma_{\eta,\mathbb{Q}}-\mathrm{pr}_{1}^{*}(\infty_{-})-\mathrm{pr}_{2}^{*}(\infty))=\mathcal{O}_{C_{\mathbb{Q}}\times
C_{\mathbb{Q}}}(\Gamma_{\eta,\mathbb{Q}}-\infty_{-}\times
C_{\mathbb{Q}}-C_{\mathbb{Q}}\times\infty)$

trivialised on $b\times C_{\mathbb{Q}}$ through the section

$l_{b}:=2\quad\text{ in
}((b,\mathrm{id})^{*}\mathcal{L})(C_{\mathbb{Q}})=\mathcal{O}_{C_{\mathbb{Q}}}(\eta(b)-b)(C_{\mathbb{Q}})=\mathcal{O}_{C_{\mathbb{Q}}}(C_{\mathbb{Q}})\,.$

For every $\overline{\mathbb{Q}}$-point $Q$ on $C_{\mathbb{Q}}$ the invertible
$\mathcal{O}_{C_{\overline{\mathbb{Q}}}}$-module
$(\mathrm{id},Q)^{*}\mathcal{L}$ is isomorphic to
$\mathcal{O}_{C_{\overline{\mathbb{Q}}}}(\eta(Q)-\infty_{-})$, hence

$\alpha_{\mathcal{L}}=\mathrm{tr}_{c}\circ f\,,\quad\text{with
}f=\eta_{*}\text{ and }c=[D_{0}]\,,D_{0}:=\infty-\infty_{-}\,.$

When restricted to the diagonal $\mathcal{L}$ is trivial since, compatibly
with the trivialisation at $(b,b)$,

$\mathrm{diag}^{*}\mathcal{L}=\mathcal{O}_{C_{\mathbb{Q}}}(\infty_{-}+\infty-\infty_{-}-\infty)=\mathcal{O}_{C_{\mathbb{Q}}}\,.$

In particular, the global section $l:=1$ of $\mathcal{O}_{C_{\mathbb{Q}}}$
gives a rigidification of $\mathrm{diag}^{*}\mathcal{L}$ that we write as

$\mathrm{diag}^{*}\mathcal{L}=l\cdot\mathcal{O}_{C_{\mathbb{Q}}}\,.$

Following Proposition 7.8 and the discussion preceding it, we choose the
extension of $\mathcal{L}$ over $C\times C^{\mathrm{sm}}$

$\mathcal{L}:=\mathcal{O}(\Gamma_{\eta}|_{C\times
C^{\mathrm{sm}}}-\infty_{-}\times C^{\mathrm{sm}}-C\times\infty)\,,$

trivialised along $b\times C^{\mathrm{sm}}$ through the section $l_{b}=2$ (the
points $\infty_{-}$ and $b$ have a simple intersection over the prime $2$). By
Proposition 7.5, the torsor $T:=T_{m{\cdot}\circ\alpha_{\mathcal{L}}}$ on $J$,
with $m=2$ as explained before Equation ($None$), satisfies, for $S$ a
$\mathbb{Z}[1/n]$-scheme and $x$ in $C(S)$, using the trivialisation given by
$l$ and $l_{b}$

$None$ $\displaystyle T(j_{b}(x))$
$\displaystyle=\mathcal{M}^{\times}(j_{b}(x),m{\cdot}\alpha_{\mathcal{L}}(j_{b}(x)))=\mathcal{M}^{\times}(j_{b}(x),(\mathrm{id},x)^{*}\mathcal{L}^{\otimes
m})$ $\displaystyle=x^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes
m,\times}\otimes b^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes-m,\times}$
$\displaystyle=\mathcal{L}^{\otimes m,\times}(x,x)\otimes\mathcal{L}^{\otimes
m,\times}(b,x)^{-1}=\mathcal{L}^{\otimes
m,\times}(x,x)=\mathcal{O}_{S}^{\times}\,.$

Using Proposition 7.8 we now compute $\widetilde{j_{b,0}}$ and
$\widetilde{j_{b,1}}$. Since $l$ generates $\mathrm{diag}^{*}(\mathcal{L})$ on
the whole $C^{\mathrm{sm}}$, we have $W_{3}=W_{43}=0$. The invertible
$\mathcal{O}$-module $\mathcal{L}^{\otimes m}$ has multidegree $0$ over all
the fibers $C\times U_{1}\to U_{1}$, hence in order to compute
$\widetilde{j_{b,1}}$ we must take $V=0$ in ($None$), giving $V_{3}=V_{43}=0$.
Hence for $S$ and $x$ as in ($None$), assuming moreover that $2$ is invertible
on $S$,

$None$ $\widetilde{j_{b,1}}(x)=l^{2}\otimes
l_{b}^{-2}=\frac{1}{4}(x^{*}1)\otimes(b^{*}1)^{-1}\quad\text{in }$
$T(j_{b}(x))=x^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes m,\times}\otimes
b^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes-m,\times}=x^{*}\mathcal{O}_{C_{S}}(\eta(x)-\infty_{-})^{\times}\otimes
b^{*}\mathcal{O}_{C_{S}}(\eta(x)-\infty_{-})^{\times}\,,$

where the last equality in ($None$) makes sense if the image of $x$ is
disjoint from $\infty,\infty_{-}$ in $C_{S}$.

The restriction $\mathcal{L}^{\otimes m}$ to $C\times U_{0}$ has multidegree
$0$ over all the fibers $C\times U_{0}\to U_{0}$ of characteristic not $3$,
while if we consider a fiber of characteristic $3$ it has degree $2$ over
$K_{0}$ and degree ${-}2$ over $K_{1}$. Hence for computing
$\widetilde{j_{b,0}}$ we take $V=K_{0}\times(K_{0}\cap U_{0})$ in ($None$)
giving $V_{43}=0$, $V_{3}=1$. Hence for $S$ and $x$ as in ($None$), assuming
moreover that $2$ is invertible on $S$,

$None$ $\widetilde{j_{b,0}}(x)=\frac{1}{3}l^{2}\otimes
l_{b}^{-2}=\frac{1}{12}(x^{*}1)\otimes(b^{*}1)^{-1}\quad\text{in }$
$T(j_{b}(x))=x^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes m,\times}\otimes
b^{*}(\mathrm{id},x)^{*}\mathcal{L}^{\otimes-m,\times}=x^{*}\mathcal{O}_{C_{S}}(\eta(x)-\infty_{-})^{\times}\otimes
b^{*}\mathcal{O}_{C_{S}}(\eta(x)-\infty_{-})^{\times}\,,$

where the last equality in ($None$) makes sense if the image of $x$ is
disjoint from $\infty,\infty_{-}$ in $C_{S}$.

###  8.3 Some integral points on the biextension

On $C_{0}$ we have the following integral points that lift uniquely to
elements of $C(\mathbb{Z})$

$\displaystyle\infty=(0,1)\,,\quad\infty_{-}:=(0,-1)\quad\text{in
}V_{2}(\mathbb{Z})\,,$
$\displaystyle\alpha:=(1,0)\,,\quad\beta:=\eta(\alpha)=(0,-1)\,,\quad\gamma:=(2,1)\,,\quad\delta:=\eta(\gamma)=(-1,-2)\quad\text{in
}V_{1}(\mathbb{Z})\,.$

Computations in Magma confirm that $J(\mathbb{Z})$ is a free
$\mathbb{Z}$-module of rank $r=2$ generated by

$G_{1}:=\gamma-\alpha\,,\quad G_{2}:=\alpha+\infty_{-}-2\infty\,.$

The points in $T(\mathbb{Z})$ are a subset of points of
$\mathcal{M}^{\times}(\mathbb{Z})$ that can be constructed, using the two
group laws, from the points in
$\mathcal{M}^{\times}(G_{i},m{\cdot}f(G_{j}))(\mathbb{Z})$ and
$\mathcal{M}^{\times}(G_{i},m{\cdot}D_{0})(\mathbb{Z})$ for $i,j\in\\{1,2\\}$.
Let us compute in detail $\mathcal{M}^{\times}(G_{1},m\cdot
f(G_{1}))(\mathbb{Z})$. As explained in Proposition 6.9.3, we have

$\displaystyle\mathcal{M}(G_{1},m{\cdot}f(G_{1}))^{\times}$
$\displaystyle=\mathcal{M}^{\times}(\gamma-\alpha,2\delta-2\beta)$
$\displaystyle=3^{e_{3}}43^{e_{43}}\cdot\mathrm{Norm}_{(2\delta)/\mathbb{Z}}(\mathcal{O}_{C}(\gamma{-}\alpha))\otimes\mathrm{Norm}_{(2\beta)/\mathbb{Z}}(\mathcal{O}(\gamma{-}\alpha))^{-1}$
$\displaystyle=3^{e_{3}}43^{e_{43}}\cdot(2\delta-2\beta)^{*}\mathcal{O}_{C}(\gamma-\alpha)$

where, given a scheme $S$, an invertible $\mathcal{O}$-module $\mathcal{L}$ on
$C_{S}$ and a divisor $D_{+}{-}D_{-}=\sum_{i}n_{i}P_{i}$ on $C_{S}$ that is
sum of $S$-points, we define the invertible $\mathcal{O}_{S}$-module

$\left(\sum_{i}n_{i}P_{i}\right)^{*}\mathcal{L}:=\bigotimes_{i}P_{i}^{*}\mathcal{L}^{n_{i}}=\mathrm{Norm}_{D_{+}/S}(\mathcal{L})\otimes\mathrm{Norm}_{D_{-}/S}(\mathcal{L})^{-1}\,.$

Since $C_{\mathbb{F}_{43}}$ is irreducible then $2f(G_{1})$ has already
multidegree $0$ over $43$, hence $e_{43}=0$. If we look at
$C_{\mathbb{F}_{3}}$ then $2f(G_{1})$ does not have multidegree $0$, while
$2f(G_{1})+K_{0}$ has multidegree $0$; hence, by Proposition 6.9.3,

$e_{3}=-\deg_{\mathbb{F}_{3}}\mathcal{O}_{C}(\gamma-\alpha)|_{K_{0}}=-1\,.$

Notice that over $\mathbb{Z}[\frac{1}{2}]$ the divisor $G_{1}$ is disjoint
from $\beta$ and $\delta$ (to see that it is disjoint from $\delta=(-1,-2,1)$
over the prime $3$ one needs to look at local equations of the blow up) thus
$\beta^{*}\mathcal{O}_{C}(\gamma-\alpha)$ and
$\delta^{*}\mathcal{O}_{C}(\gamma-\alpha)$ are generated by $\beta^{*}1$ and
$\delta^{*}1$ over $\mathbb{Z}[\frac{1}{2}]$. Thus there are integers
$e_{\beta},e_{\delta}$ such that $\beta^{*}\mathcal{O}_{C}(\gamma-\alpha)$ and
$\delta^{*}\mathcal{O}_{C}(\gamma-\alpha)$ are generated by
$\beta^{*}2^{e_{\beta}}$ and $\delta^{*}2^{e_{\delta}}$ over $\mathbb{Z}$.
Looking at the intersections between $\beta,\gamma,\alpha$ and $\delta$ we
compute that $e_{\beta}=-1$ and $e_{\delta}=1$, hence

$\displaystyle\mathcal{M}(G_{1},m\cdot
f(G_{1}))=3^{-1}{\cdot}(\delta^{*}2)^{2}\otimes(\beta^{*}2^{-1})^{-2}\cdot\mathbb{Z}=2^{4}{\cdot}3^{-1}{\cdot}(\delta^{*}1)^{2}\otimes(\beta^{*}1)\cdot\mathbb{Z}\quad\text{and}$
$\displaystyle Q_{1,1}:=\pm
2^{4}{\cdot}3^{-1}{\cdot}(\delta^{*}1)^{2}\otimes(\beta^{*}1)^{-2}\in\mathcal{M}_{G_{1},m\cdot
f(G_{1})}^{\times}(\mathbb{Z})\,.$

With analogous computations we see that

$\displaystyle Q_{2,1}$
$\displaystyle:=2^{-2}{\cdot}(\delta^{*}1)^{2}{\otimes}(\beta^{*}1)^{-2}\quad$
$\displaystyle\text{generates }\mathcal{M}_{G_{2},m\cdot f(G_{1})}$
$\displaystyle Q_{1,2}$
$\displaystyle:=2^{-2}{\cdot}(\beta^{*}1)^{2}{\otimes}(\infty_{-}^{*}1)^{2}{\otimes}(\infty^{*}1)^{-4}\quad$
$\displaystyle\text{generates }\mathcal{M}_{G_{1},m\cdot f(G_{2})}$
$\displaystyle Q_{2,2}$
$\displaystyle:=2^{18}{\cdot}(\beta^{*}1)^{2}{\otimes}(\infty_{-}^{*}x)^{2}{\otimes}(\infty^{*}z^{2})^{-4}\quad$
$\displaystyle\text{generates }\mathcal{M}_{G_{2},m\cdot f(G_{2})}$
$\displaystyle Q_{1,2}$
$\displaystyle:=(\infty^{*}1)^{2}{\otimes}(\infty_{-}^{*}1)^{-2}\quad$
$\displaystyle\text{generates }\mathcal{M}_{G_{1},m\cdot D_{0}}$
$\displaystyle Q_{2,0}$
$\displaystyle:=2^{-12}{\cdot}(\infty^{*}z^{2})^{2}{\otimes}(\infty_{-}^{*}x)^{-2}\quad$
$\displaystyle\text{generates }\mathcal{M}_{G_{2},m\cdot D_{0}}\,.$

###  8.4 Some residue disks of the biextension

Let $p$ be a prime of good reduction for $C$. Given the divisors

$D:=\alpha-\infty\,,\quad
E:=2\beta-2\infty_{-}=(m{\cdot}\circ\mathrm{tr}_{c}\circ\eta_{*})(D)\quad\text{in
}\mathrm{Div}(C_{\mathbb{Z}/p^{2}})$

we use Lemma 6.6.8 to give parameters on the residue disks in
$\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{\overline{D},\overline{E}}$ and
$T({\mathbb{Z}/p^{2}})_{\overline{D}}$, with $\overline{D},\overline{E}$ the
images of $D,E$ in $\mathrm{Div}(C_{\mathbb{F}_{p}})$.

We choose the “base points” $b_{1}=\alpha,b_{2}=\infty$,
$b_{3}=\beta,b_{4}=\infty$, so that $b_{1}\neq b_{2}$, $b_{3}\neq b_{4}$ and
$h^{0}(C_{\mathbb{F}_{p}},b_{1}+b_{2})=h^{0}(C_{\mathbb{F}_{p}},b_{3}+b_{4})=1$.
As in Equation ($None$), we define $x_{\alpha}=x{-}1$, $x_{\infty}=z$,
$x_{\beta}=x$ and $x_{D,\beta}=x_{D,\infty_{-}}=1$, $x_{D,\infty}=z^{-1}$. For
$Q$ in $\\{\infty,\beta,\alpha\\}$ and $a\in\mathbb{F}_{p}$ let $Q_{a}$ be the
unique ${\mathbb{Z}/p^{2}}$-point of $C$ that is congruent to $Q$ modulo $p$
and such that $x_{Q}(Q_{a})=ap\in{\mathbb{Z}/p^{2}}$. We have the bijections

$\displaystyle\mathbb{F}_{p}^{2}$ $\displaystyle\longrightarrow
J({\mathbb{Z}/p^{2}})_{\overline{D}}\,,\quad\lambda\longmapsto
D_{\lambda}:=D+\alpha_{\lambda_{1}}-\alpha+\infty_{\lambda_{2}}-\infty=\alpha_{\lambda_{1}}+\infty_{\lambda_{2}}-2\infty$
$\displaystyle\mathbb{F}_{p}^{2}$ $\displaystyle\longrightarrow
J({\mathbb{Z}/p^{2}})_{\overline{E}}\,,\quad\mu\longmapsto
E_{\mu}:=E+\beta_{\mu_{1}}-\beta+\infty_{\mu_{2}}-\infty=\beta+\beta_{\mu_{1}}+\infty_{\mu_{2}}-\infty-2\infty_{-}\,.$

Following ($None$) for $\lambda,\mu\in\mathbb{F}_{p}^{2}$ we define

$s_{D,E}(\lambda,\mu):=(\beta^{*}1)\otimes(\beta_{\mu_{1}}^{*}1)\otimes(\infty_{\mu_{2}}^{*}\frac{z^{2}}{z-\lambda_{2}p})\otimes(\infty^{*}\frac{z^{2}}{z-\lambda_{2}p})^{-1}\otimes(\infty_{-}^{*}1)^{-2}$

that, by Proposition 6.3.2 and Remark 6.3.12, generates
$E_{\mu}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{\lambda})=\mathcal{M}_{D_{\lambda},E_{\mu}}$.
The points in $\mathcal{M}^{\times}(\mathbb{F}_{p})$ projecting to
$(\overline{D},\overline{E})$ are in bijection with the elements $\xi$ in
$\mathbb{F}_{p}^{\times}$ and are exactly the points $\xi\cdot{s_{D,E}(0,0)}$.
Using
$({\mathbb{Z}/p^{2}})^{\times}=\mathbb{F}_{p}^{\times}\times(1{+}p\mathbb{F}_{p})$,
for each $\xi\in\mathbb{F}_{p}^{\times}$ we parametrise the residue disk of
$\xi\cdot{s_{D,E}(0,0)}$ using bijection in Lemma 6.6.8

$\mathbb{F}_{p}^{5}\longrightarrow\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})_{\xi\cdot{s_{D,E}(0,0)}}\,,\quad(\lambda_{1},\lambda_{2},\mu_{1},\mu_{2},\tau)\longmapsto(1+p\tau)\xi{\cdot}s_{D,E}((\lambda_{1},\lambda_{2}),(\mu_{1},\mu_{2}))\,.$

Since $(m{\cdot}\circ\mathrm{tr}_{c}\circ f)(D_{\lambda})=E_{-2\lambda}$ then
we have

$T({\mathbb{Z}/p^{2}})_{\overline{D}}=\bigcup_{\lambda\in\mathbb{F}_{p}^{2}}T_{D_{\lambda}}({\mathbb{Z}/p^{2}})=\bigcup_{\lambda\in\mathbb{F}_{p}^{2}}\mathcal{M}_{D_{\lambda},E_{-2\lambda}}^{\times}({\mathbb{Z}/p^{2}})\,.$

As $\xi$ varies in $\mathbb{F}_{p}^{\times}$ the point
$\xi{\cdot}s_{D,E}(0,0)$ varies in all the points in
$\mathcal{M}^{\times}(\mathbb{F}_{p})$ projecting to
$(\overline{D},\overline{E})$ and we have the following bijection induced by
parameters in $\xi{\cdot}s_{D,E}(0,0)$

$None$ $\mathbb{F}_{p}^{3}\longrightarrow T(\mathbb{Z}_{p})_{\xi
s_{D,E}(0,0)}\,,\quad(\lambda_{1},\lambda_{2},\tau)\longmapsto(1+\tau
p){\cdot}\xi{\cdot}s_{D,E}((\lambda_{1},\lambda_{2}),(-2\lambda_{1},-2\lambda_{2}))\,.$

If we apply ($None$) and ($None$) to $Q=\alpha_{\lambda}$ and we use the
symmetry of the Poincaré torsor explained in Proposition 6.3.2 and made
explicit in Lemma 6.5.4 we obtain the following description of
$\widetilde{j_{b,i}}$ on $C({\mathbb{Z}/p^{2}})_{\alpha_{\mathbb{F}_{p}}}$
when $p\neq 2$

$\widetilde{j_{b,1}}(\alpha_{\lambda})=(1/4)\cdot
s_{D,E}((\lambda,0),(-2\lambda,0))\,,\quad\widetilde{j_{b,0}}(Q)=(1/12)\cdot
s_{D,E}((\lambda,0),(-2\lambda,0))\,.$

If $p=5$ then $18$ and ${-}1$ are $(p-1)$-th roots of unity in
$({\mathbb{Z}/p^{2}})^{\times}$, thus $1/4=(-1)(1+p)$ and $1/12=3(1+2p)$ in
$({\mathbb{Z}/p^{2}})^{\times}=\mathbb{F}_{p}^{\times}\times(1{+}p\mathbb{F}_{p})$,
hence

$None$ $\widetilde{j_{b,1}}(\alpha_{\lambda})=-(1+p)\cdot
s_{D,E}((\lambda,0),(-2\lambda,0))\,,\quad\widetilde{j_{b,0}}(Q)=18\cdot(1+2p)\cdot
s_{D,E}((\lambda,0),(-2\lambda,0))\,.$

Since it is useful for computing the map $\kappa_{\mathbb{Z}}$ in the residue
disks of $T({\mathbb{Z}/p^{2}})$ projecting to $\overline{D}$, we also apply
Lemma 6.6.8 to the residue disks of $\mathcal{M}^{\times}({\mathbb{Z}/p^{2}})$
lying over $(\overline{D},0)$, $(0,\overline{E})$ and $(0,0)$. Hence for
$\lambda,\mu\in\mathbb{F}_{p}^{2}$ we define the divisors on
$C_{{\mathbb{Z}/p^{2}}}$

$D_{\lambda}^{0}:=\alpha_{\lambda_{1}}-\alpha+\infty_{\lambda_{2}}-\infty\,,\quad
E^{0}_{\mu}:=\beta_{\mu_{1}}-\beta+\infty_{\mu_{2}}-\infty$

and the sections

$\displaystyle s_{D,0}(\lambda,\mu)$
$\displaystyle:=(\beta_{\mu_{1}}^{*}1){\otimes}(\infty_{\mu_{2}}^{*}\frac{z^{2}}{z{-}\lambda_{2}p}){\otimes}(\beta^{*}1)^{-1}{\otimes}(\infty^{*}\frac{z^{2}}{z{-}\lambda_{2}p})^{-1}$
$\displaystyle\text{in
}\mathcal{M}^{\times}(D_{\lambda},E_{\mu}^{0})({\mathbb{Z}/p^{2}})$
$\displaystyle s_{0,E}(\lambda,\mu)$
$\displaystyle:=(\beta^{*}1){\otimes}(\beta_{\mu_{1}}^{*}1){\otimes}(\infty_{\mu_{2}}^{*}\frac{z}{z{-}\lambda_{2}p}){\otimes}(\infty^{*}\frac{z}{z{-}\lambda_{2}p})^{-1}{\otimes}(\infty_{-}^{*}1)^{-2}$
$\displaystyle\text{in
}\mathcal{M}^{\times}(D_{\lambda}^{0},E_{\mu})({\mathbb{Z}/p^{2}})$
$\displaystyle s_{0,0}(\lambda,\mu)$
$\displaystyle:=(\beta_{\mu_{1}}^{*}1){\otimes}(\infty_{\mu_{2}}^{*}\frac{z}{z{-}\lambda_{2}p}){\otimes}(\beta^{*}1)^{-1}{\otimes}(\infty^{*}\frac{z}{z{-}\lambda_{2}p})^{-1}$
$\displaystyle\text{in
}\mathcal{M}^{\times}(D_{\lambda}^{0},E_{\mu}^{0})({\mathbb{Z}/p^{2}})\,.$

###  8.5 Geometry mod $p^{2}$ of integral points

From now on $p=5$. Let $\overline{\alpha}\in C({\mathbb{Z}/p^{2}})$ be the
image of $\alpha\in C(\mathbb{Z})$. In this subsection we compute the
composition $\overline{\kappa}\colon\mathbb{Z}^{2}\to
T({\mathbb{Z}/p^{2}})_{\widetilde{j_{b,1}}(\overline{\alpha})}$ of the map
$\kappa_{\mathbb{Z}}\colon\mathbb{Z}^{2}\to
T(\mathbb{Z}_{p})_{\widetilde{j_{b,1}}(\overline{\alpha})}$ in ($None$) and
the reduction map
$T(\mathbb{Z}_{p})_{\widetilde{j_{b,1}}(\overline{\alpha})}\to
T({\mathbb{Z}/p^{2}})_{\widetilde{j_{b,1}}(\overline{\alpha})}$. With a
suitable choice of parameters in
$\mathcal{O}_{T,\widetilde{j_{b,1}}(\overline{\alpha})}$, the map
$\kappa_{\mathbb{Z}}$ is described by integral convergent power series
$\kappa_{1},\kappa_{2},\kappa_{3}\in\mathbb{Z}_{p}\langle z_{1},z_{2}\rangle$
and $\overline{\kappa}$, composed with the inverse of the parametrization
($None$), is given by the images
$\overline{\kappa_{1}},\overline{\kappa_{2}},\overline{\kappa_{3}}$ of
$\kappa_{1},\kappa_{2},\kappa_{3}$ in $\mathbb{F}_{p}[z_{1},z_{2}]$.

The divisor $j_{b}(\overline{\alpha})$ is equal to the image of

$\widetilde{G_{t}}:=e_{0,1}G_{1}+e_{0,2}G_{2}\text{ with
}e_{0,1}:=6\,,e_{0,2}:=3$

in $J(\mathbb{F}_{p})$ and

$\tilde{t}:=Q_{1,0}^{6}\otimes Q_{2,0}^{3}\otimes Q_{1,1}^{6{\cdot}6}\otimes
Q_{1,2}^{6{\cdot}3}\otimes Q_{2,1}^{3{\cdot}6}\otimes
Q_{2,2}^{3{\cdot}3}\quad\text{in
}\mathcal{M}^{\times}(\widetilde{D_{1}}\,,m{\cdot}(D_{0}+\eta_{*}\widetilde{G_{t}}))(\mathbb{Z})$

is a lift of $\widetilde{j_{b,1}}(\overline{\alpha})$. The kernel of
$J(\mathbb{Z})\to J(\mathbb{F}_{p})$ is a free $\mathbb{Z}$-module generated
by

$\widetilde{G_{1}}:=e_{1,1}G_{1}+e_{1,2}G_{2}\,,\quad\widetilde{G_{2}}:=e_{2,1}G_{1}+e_{2,2}G_{2}\,,\text{
with }e_{1,1}:=16\,,e_{1,2}:=2\,,e_{2,1}:=0\,,e_{2,2}:=5\,.$

Let $\widetilde{G_{t,2}}$ be the divisor
$m(D_{0}{+}\eta_{*}(\widetilde{G_{t}}))$ representing
$(m{\cdot}{\circ}\mathrm{tr}_{c}{\circ}f)(\widetilde{G_{t}})\in
J^{0}(\mathbb{Z})$. Following ($None$) for $i,j\in\\{1,2\\}$ we define

${\displaystyle
P_{i,j}:=\bigotimes_{m,l=1}^{2}Q_{l,m}^{e_{i,l}{\cdot}e_{j,m}}}$${\displaystyle
R_{i,\tilde{t}}:=\bigotimes_{l=1}^{2}Q_{l,0}^{e_{i,l}}\otimes\bigotimes_{m,l=1}^{2}Q_{l,m}^{e_{i,l}{\cdot}e_{0,m}}}$${\displaystyle
S_{\tilde{t},j}:=\bigotimes_{m,l=1}^{2}Q_{l,m}^{e_{0,l}{\cdot}e_{j,m}}}$${(\widetilde{G_{i}},f(m\widetilde{G_{j}}))}$${(\widetilde{G_{i}},\widetilde{G_{t,2}}))}$${(\widetilde{G_{t}},f(m\widetilde{G_{j}}))\,.}$

Computations in $C_{{\mathbb{Z}/p^{2}}}$ show the following linear
equivalences of divisors

$\widetilde{G_{t}}\sim D_{0,3}\,,\quad\widetilde{G_{1}}\sim
D^{0}_{4,0}\,,\quad\widetilde{G_{2}}\sim D^{0}_{0,3}$

and applying Lemma 6.4.8 and the functoriality of the norm we compute

$None$ $\displaystyle P_{1,1}$
$\displaystyle=(1+4p){\cdot}s_{0,0}((4,0),(2,0))\quad$ $\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{1}},\widetilde{G_{1}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{4,0}^{0},E_{2,0}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle P_{1,2}$ $\displaystyle=(1+4p){\cdot}s_{0,0}((4,0),(0,4))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{1}},\widetilde{G_{2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{4,0}^{0},E_{0,4}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle P_{2,1}$ $\displaystyle=(1+4p){\cdot}s_{0,0}((0,3),(2,0))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{2}},\widetilde{G_{1}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3}^{0},E_{2,0}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle P_{2,2}$
$\displaystyle=(-1){\cdot}(1+2p){\cdot}s_{0,0}((0,3),(0,4))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{2}},\widetilde{G_{2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3}^{0},E_{0,4}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle R_{1,\tilde{t}}$ $\displaystyle=s_{0,E}((4,0),(0,4))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{1}},\widetilde{G_{t,2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{4,0}^{0},E_{0,4})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle R_{2,\tilde{t}}$
$\displaystyle=(1+4p){\cdot}s_{0,E}((0,3),(0,4))\quad$ $\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{2}},\widetilde{G_{t,2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3}^{0},E_{0,4})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle S_{\tilde{t},1}$ $\displaystyle=s_{D,0}((0,3),(2,0))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{t}},\widetilde{G_{1}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3},E_{2,0}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle S_{\tilde{t},2}$
$\displaystyle=(-1)(1+4p){\cdot}s_{D,0}((0,3),(0,4))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{t}},\widetilde{G_{2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3},E_{0,4}^{0})({\mathbb{Z}/p^{2}})\,,$
$\displaystyle\tilde{t}$
$\displaystyle=(-1){\cdot}(1+2p){\cdot}s_{D,E}((0,3),(0,4))\quad$
$\displaystyle\text{in
}\mathcal{M}^{\times}(\widetilde{G_{t}},\widetilde{G_{t,2}})({\mathbb{Z}/p^{2}})=\mathcal{M}^{\times}(D_{0,3},E_{0,4})({\mathbb{Z}/p^{2}})\,.$

We now show these computations in the cases of $\widetilde{G_{t}}$ and
$\tilde{t}$. The Riemann-Roch space relative to the divisor
$\widetilde{G_{t}}{+}\infty{+}\alpha{-}D$ on $C_{\mathbb{Z}/p^{2}}$ is
generated by the inverse of the rational function

$h_{1}:=\frac{x^{9}{-}5x^{8}{-}2x^{7}{+}7x^{6}{-}9x^{5}{-}5x^{4}{+}14x^{3}{+}7x^{2}{+}13x{+}1{+}(x^{6}{+}9x^{5}{-}5x^{4}{+}15x^{3}{-}5x^{2}{+}4x{+}14)y}{15x^{5}-x^{4}+4x^{3}+19x^{2}+4x+9}$

and indeed

$\mathrm{div}(h_{1})=\widetilde{G_{t}}-D_{0,3}=(6\gamma+3\infty_{-}-3\alpha-6\infty)-(\alpha+\infty_{3}-2\infty)\quad\text{in
}\mathrm{Div}(C_{\mathbb{Z}/p^{2}})\,.$

Hence multiplication by $h_{1}$ gives an isomorphism
$\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(\widetilde{G_{t}})\to\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})$
and by functoriality of the norm we get

$\displaystyle\delta^{*}\mathcal{O}_{C}(\widetilde{G_{t}})$
$\displaystyle\to\delta^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})\,,\quad$
$\displaystyle\delta^{*}1\mapsto\delta^{*}(h_{1})=h_{1}(\delta){\cdot}\delta^{*}1=12{\cdot}\delta^{*}1\,,$
$\displaystyle\beta^{*}\mathcal{O}_{C}(\widetilde{G_{t}})$
$\displaystyle\to\beta^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})\,,\quad$
$\displaystyle\beta^{*}1\mapsto\beta^{*}(h_{1})=h_{1}(\beta){\cdot}\beta^{*}1=18{\cdot}\beta^{*}1\,,$
$\displaystyle\infty^{*}\mathcal{O}_{C}(\widetilde{G_{t}})$
$\displaystyle\to\infty^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})\,,\quad$
$\displaystyle\infty^{*}z^{6}\mapsto\infty^{*}(z^{6}h_{1})=13{\cdot}\infty^{*}\frac{z^{2}}{z-3p}\,,$
$\displaystyle\infty_{-}^{*}\mathcal{O}_{C}(\widetilde{G_{t}})$
$\displaystyle\to\infty_{-}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})\,,\quad$
$\displaystyle\infty_{-}^{*}z^{-3}\mapsto\infty_{-}^{*}(z^{-3}h_{1})=(z^{-3}h_{1})(\infty_{-}){\cdot}\infty_{-}^{*}1=6{\cdot}\infty_{-}^{*}1\,.$

Since $\widetilde{G_{t,2}}=12\delta+4\infty_{-}-6\beta-10\infty$, the above
isomorphisms, tensored with the exponents, give the canonical isomorphism

$None$
$\mathcal{M}(\widetilde{G_{t}},\widetilde{G_{t,2}})=\widetilde{G_{t,2}}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(\widetilde{G_{t}})\to\widetilde{G_{t,2}}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})=\mathcal{M}(D_{0,3},\widetilde{G_{t,2}})$
$\tilde{t}{=}14{\cdot}(\delta^{*}1)^{12}{\otimes}(\beta^{*}1)^{-6}{\otimes}(\infty^{*}z^{6})^{-10}{\otimes}(\infty_{-}^{*}z^{-3})^{4}\mapsto
14{\cdot}(\delta^{*}1)^{12}{\otimes}(\beta^{*}1)^{-6}{\otimes}(\infty^{*}\frac{z^{2}}{z{-}3p})^{-10}{\otimes}(\infty_{-}^{*}1)^{4}\,.$

The Riemann-Roch space relative to the divisor
$\widetilde{G_{t,2}}{+}\infty{+}\alpha{-}E$ on $C_{\mathbb{Z}/p^{2}}$ is
generated by the inverse of the rational function

$\displaystyle
h_{2}{:}{=}\frac{x^{17}{-}8x^{16}{+}x^{15}{-}4x^{14}{+}7x^{13}{+}4x^{12}{+}12x^{11}{+}x^{10}{+}2x^{9}-5x^{8}{+}x^{7}{+}3x^{6}{+}12x^{5}{-}6x^{4}{-}6x^{3}{+}4x^{2}{-}6}{20x^{8}-6x^{7}}$
$\displaystyle{+}\frac{10x^{2}{+}(x^{15}{+}6x^{14}{-}5x^{13}{-}x^{12}{-}2x^{11}{+}14x^{10}{-}4x^{9}{+}14x^{8}{+}3x^{7}{+}8x^{6}{-}6x^{5}{-}3x^{4}{+}4x^{3}{+}13x^{2}{-}x{-}7)y}{20x^{9}-6x^{8}}$

and indeed

$\mathrm{div}(h_{2})=\widetilde{G_{t,2}}-E_{0,4}=(12\delta+4\infty_{-}-6\beta-10\infty)-(2\beta+\infty_{4}-\infty-\infty_{-})\quad\text{in
}\mathrm{Div}(C_{\mathbb{Z}/p^{2}})\,.$

Following the recipe in Section 6.4 that describes the map ($None$), we
consider the following rational section of
$\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})$

$l:=\frac{10x^{4}+x^{3}+17x+14+(15x+9)y}{10x^{4}+16x^{3}+7x^{2}+7x+10}\,.$

since it generates $\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})$ in a
neighborhood of the supports of $\widetilde{G_{t,2}}$ and $E_{0,4}$. Then

$\mathrm{div}(l)=3{\cdot}(-1,1){+}(17,23){+}(15,10){-}2{\cdot}(12,23){-}2{\cdot}(5,20){-}(0,1)\text{
in
}\mathrm{Div}(V_{1,{\mathbb{Z}/p^{2}}}){\subset}\mathrm{Div}(C_{{\mathbb{Z}/p^{2}}})\,.$

Hence by Lemma 6.4.8 the canonical isomorphism

$\mathcal{M}(D_{0,3},\widetilde{G_{t,2}})=\widetilde{G_{t,2}}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})\longrightarrow
E_{0,4}^{*}\mathcal{O}_{C_{\mathbb{Z}/p^{2}}}(D_{0,3})=\mathcal{M}(D_{0,3},E_{0,4})$

described in Equation ($None$) sends

$None$ $\widetilde{G_{t,2}}^{*}l\longmapsto h_{2}(\mathrm{div}(l))\cdot
E_{0,4}^{*}l=14\cdot E_{0,4}^{*}l\,.$

where

$\displaystyle\widetilde{G_{t,2}}^{*}l$
$\displaystyle:=(\delta^{*}l)^{12}{\otimes}(\beta^{*}l)^{-6}{\otimes}(\infty^{*}l)^{-10}{\otimes}(\infty_{-}^{*}l)^{4}=-(\delta^{*}1)^{12}{\otimes}(\beta^{*}1)^{-6}{\otimes}(\infty^{*}\frac{z^{2}}{z{-}3p})^{-10}{\otimes}(\infty_{-}^{*}1)^{4}\,,$
$\displaystyle E_{0,4}^{*}l$
$\displaystyle:=(\beta^{*}l)^{2}{\otimes}(\infty_{4}^{*}l){\otimes}(\infty^{*}l)^{-1}{\otimes}(\infty^{*}_{-}l)^{-2}=16{\cdot}(\beta^{*}1)^{2}{\otimes}(\infty_{4}^{*}\frac{z^{2}}{z{-}3p}){\otimes}(\infty^{*}\frac{z^{2}}{z{-}3p})^{-1}{\otimes}(\infty_{-}^{*}1)^{-2}\,.$

Equations ($None$) and ($None$) imply that
$\tilde{t}=-(1+2p){\cdot}s_{D,E}((0,3),(0,4))$.

Let $\overline{A_{\tilde{t}}},\overline{B_{\tilde{t}}},\overline{C}$ and
$\overline{D_{\tilde{t}}}$ be the compositions of the reduction map
$\mathcal{M}^{\times}(\mathbb{Z}_{p})\to\mathcal{M}({\mathbb{Z}/p^{2}})$ and
respectively $A_{\tilde{t}},B_{\tilde{t}},C$ and $D_{\tilde{t}}$, defined in
($None$), ($None$) and ($None$). Using ($None$) and ($None$) we get, for $n$
in $\mathbb{Z}^{2}$,

$None$ $\displaystyle\overline{A_{\tilde{t}}}(n)$
$\displaystyle=(-1)^{n_{2}}(1+(4n_{2})t)\cdot
s_{D,0}((0,3),(2n_{1},4n_{2}))\,,$ $\displaystyle\overline{B_{\tilde{t}}}(n)$
$\displaystyle=(1+(4n_{2})p)s_{0,E}((4n_{1},3n_{2}),(0,4))\,,$
$\displaystyle\overline{C}(n)$
$\displaystyle=(-1)^{n_{2}^{2}}(1+(4n_{1}^{2}+(4+4)n_{1}n_{2}+2n_{2}^{2})p)\cdot
s_{0,0}((4n_{1},3n_{2}),(2n_{1},4n_{2}))\,,$
$\displaystyle\overline{D_{\tilde{t}}}(n)$
$\displaystyle=-(1+(4n_{1}^{2}+3n_{1}n_{2}+2n_{2}^{2}+3n_{2}+2)p)\cdot
s_{D,E}((4n_{1},3+3n_{2}),(2n_{1},4+4n_{2}))\,,$
$\displaystyle\overline{\kappa}(n)$
$\displaystyle=-(1+(4n_{1}^{2}+3n_{1}n_{2}+2n_{2}^{2}+2n_{2}+2)p)\cdot
s_{D,E}((n_{1},3+2n_{2}),(3n_{1},4+n_{2}))\,,$

hence, using the bijection ($None$),

$None$
$\overline{\kappa_{1}}=z_{1}\,,\quad\overline{\kappa_{2}}=3+2z_{2}\,,\quad\overline{\kappa_{3}}=4z_{1}^{2}+3z_{1}z_{2}+2z_{2}^{2}+2z_{2}+2\,.$

###  8.6 The rational points with a specific image mod 5.

By ($None$) the image in $T(\mathbb{F}_{p})$ of a point
$\pm\overline{D_{\tilde{t}}}(n)$ for $n\in\mathbb{Z}^{2}$ is always of the
form $\pm s_{D,E}(0,0)$, hence, looking at ($None$) we see that there is no
point $T(\mathbb{Z})$ with reduction
$\widetilde{j_{b,0}}(\overline{\alpha})\in T(\mathbb{F}_{p})$. Hence
$C(\mathbb{Z})_{\overline{\alpha}}=U_{1}(\mathbb{Z})_{\overline{\alpha}}$.

Let $f_{1},f_{2}\in\mathcal{O}(\widetilde{T}_{t}^{p})^{\wedge_{p}}$ be
generators of the kernel of
$\widetilde{j_{b,1}}^{*}\colon\mathcal{O}(\widetilde{T}_{t}^{p})^{\wedge_{p}}\to\mathcal{O}(\widetilde{U}_{u}^{p})^{\wedge_{p}}$
as in Section 4. The bijection ($None$) gives an isomorphism
$\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{T}_{t}^{p})=\mathbb{F}_{p}[\lambda_{1},\lambda_{2},\tau]$
and since the images $\overline{f_{1}},\overline{f_{2}}$ of $f_{1},f_{2}$ in
$\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{T}_{t}^{p})$ are generators of
the kernel of
$\widetilde{j_{b,1}}^{*}\colon\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{T}_{t}^{p})^{\wedge_{p}}\to\mathbb{F}_{p}\otimes\mathcal{O}(\widetilde{U}_{u}^{p})^{\wedge_{p}}$
we can suppose that

$\overline{f_{1}}=\lambda_{2}\,,\quad\overline{f_{2}}=\tau-1\,.$

By ($None$) we have

$\kappa^{*}\overline{f_{1}}=\overline{\kappa_{2}}=3+2\\\
z_{2}\,,\quad\kappa^{*}\overline{f_{2}}=\overline{\kappa_{3}}-1=4z_{1}^{2}+3z_{1}z_{2}+2z_{2}^{2}+2z_{2}+1\,.$

Let $A$ be $\mathbb{Z}_{p}\langle
z_{1},z_{2}\rangle/(\kappa^{*}f_{1},\kappa^{*}f_{2})$. Then the ring

$None$
$\overline{A}:=A/pA=\mathbb{F}_{p}[z_{1},z_{2}]/(\kappa^{*}\overline{f_{1}},\kappa^{*}\overline{f_{2}})=\mathbb{F}_{p}[z_{1},z_{2}]/(z_{2}-1\,,4z_{1}^{2}+3z_{1})$

has dimension $2$ over $\mathbb{F}_{p}$, hence by Theorem 4.12
$U(\mathbb{Z})_{\overline{\alpha}}$ contains at most $2$ points. Since both

$\alpha\quad\text{and}\quad(12/7,20/7)\in V_{1}(\mathbb{Z}[1/7])$

reduce to $\overline{\alpha}$ we deduce that
$C(\mathbb{Z})_{\overline{\alpha}}=U_{1}(\mathbb{Z})_{\overline{\alpha}}$ is
made of the these two points.

###  8.7 Determination of all rational points

Denoting $(3,-1)\in V_{1}(\mathbb{F}_{p})\subset C(\mathbb{F}_{p})$ as
$\varepsilon$ we have

$C(\mathbb{F}_{p})=\\{\overline{\infty}\,,\overline{\infty_{-}}\,,\overline{\alpha}\,,\iota(\overline{\alpha})\,,\eta(\overline{\alpha})\,,(\iota\circ\eta)(\overline{\alpha})\,,\overline{\gamma}\,,\iota(\overline{\gamma})\,,\eta(\overline{\gamma})\,,(\iota\circ\eta)(\overline{\gamma})\,,\varepsilon\,,\iota(\varepsilon)\\}\,.$

Using that for any point $Q$ in $C(\mathbb{F}_{p})$ the condition
$T(\mathbb{Z})_{\widetilde{j_{b,i}}(Q)}=\emptyset$ implies
$U_{i}(\mathbb{Z})_{Q}=\emptyset$ we get

$U_{0}(\mathbb{Z})_{\overline{\infty}}=U_{0}(\mathbb{Z})_{\overline{\infty_{-}}}=U_{1}(\mathbb{Z})_{\varepsilon}=U_{1}(\mathbb{Z})_{\iota(\varepsilon)}=U_{1}(\mathbb{Z})_{\overline{\gamma}}=U_{1}(\mathbb{Z})_{\eta(\overline{\gamma})}=U_{1}(\mathbb{Z})_{\eta(\overline{\gamma})}=U_{1}(\mathbb{Z})_{\iota\eta(\overline{\gamma})}=\emptyset\,.$

Applying our method to $\overline{\infty}$ we discover that
$U_{1}(\mathbb{Z})_{\overline{\infty}}$ contains at most $2$ points and the
same holds for $U_{1}(\mathbb{Z})_{\overline{\infty_{-}}}$. Moreover the
action of $\langle\eta,\iota\rangle$ on $C(\mathbb{Z})$ tells that
$U_{1}(\mathbb{Z})_{\iota(\overline{\alpha})}$,
$U_{1}(\mathbb{Z})_{\eta(\overline{\alpha})}$ and
$U_{1}(\mathbb{Z})_{\eta\iota(\overline{\alpha})}$ are sets containing exactly
$2$ elements. Hence

$U_{1}(\mathbb{Z})=U_{1}(\mathbb{Z})_{\overline{\alpha}}\cup
U_{1}(\mathbb{Z})_{\iota(\overline{\alpha})}\cup
U_{1}(\mathbb{Z})_{\eta(\overline{\alpha})}\cup
U_{1}(\mathbb{Z})_{\eta\iota(\overline{\alpha})}\cup
U_{1}(\mathbb{Z})_{\overline{\infty_{-}}}\cup
U_{1}(\mathbb{Z})_{\overline{\infty}}$

contains at most $12$ elements. Looking at the orbits of the action of
$\langle\eta,\iota\rangle$ on $U_{1}(\mathbb{Z})$ we see that
$\\#U_{1}(\mathbb{Z})\equiv 2\pmod{4}$, hence $\\#U_{1}(\mathbb{Z})\leq 10$.
Since $U_{1}(\mathbb{Z})$ contains $\infty,\infty_{-}$ and all the images by
$\langle\eta,\iota\rangle$ of $U_{1}(\mathbb{Z})_{\overline{\alpha}}$ we
conclude that $\\#U_{1}(\mathbb{Z})=10$.

Applying our method to the point $\overline{\gamma}$ we see that
$U_{0}(\mathbb{Z})_{\overline{\gamma}}$ contains at most two points, one of
them being $\gamma$. Moreover solving the equations
$\kappa^{*}\overline{f_{i}}=0$ we see that if there is another point
$\gamma^{\prime}$ in $U_{0}(\mathbb{Z})_{\overline{\gamma}}$ then there exist
$n_{1},n_{2}\in\mathbb{Z}$ such that

$j_{b}(\gamma^{\prime})=39G_{1}+17G_{2}+5n_{1}\widetilde{G_{1}}+5n_{2}\widetilde{G_{2}}\,.$

Using the Mordell-Weil sieve (see [27]) we derive a contradiction: for all
integers $n_{1},n_{2}$, the image in $J(\mathbb{F}_{7})$ of
$39G_{1}{+}17G_{2}{+}5n_{1}\widetilde{G_{1}}{+}5n_{2}\widetilde{G_{2}}$ is not
contained in $j_{b}(C(\mathbb{F}_{7}))$. We deduce that

$U_{0}(\mathbb{Z})_{\overline{\gamma}}=\\{\gamma\\}\,.$

Applying our method to to $\varepsilon$ we see that
$U_{0}(\mathbb{Z})_{\varepsilon}$ contains at most $2$ points corresponding to
two different solutions to the equations $\kappa^{*}\overline{f_{i}}=0$. We
can see that one of the two solutions does not lift to a point in
$U_{0}(\mathbb{Z})_{\varepsilon}$ in the same way we excluded the existence of
$\gamma^{\prime}\in U_{0}(\mathbb{Z})_{\overline{\gamma}}$. Hence
$U_{0}(\mathbb{Z})_{\varepsilon}$ has cardinality at most $1$. Using that for
every $Q\in C(\mathbb{F}_{p})$ and every automorphism $\omega$ of $C$ we have
$\\#U_{0}(\mathbb{Z})_{Q}=\\#U_{0}(\mathbb{Z})_{\omega(Q)}$, we deduce that

$U_{0}(\mathbb{Z})=U_{0}(\mathbb{Z})_{\overline{\gamma}}\cup
U_{0}(\mathbb{Z})_{\iota(\overline{\gamma})}\cup
U_{0}(\mathbb{Z})_{\eta(\overline{\gamma})}\cup
U_{0}(\mathbb{Z})_{\eta\iota(\overline{\gamma})}\cup
U_{0}(\mathbb{Z})_{\varepsilon}\cup U_{0}(\mathbb{Z})_{\iota(\varepsilon)}$

contains at most $6$ points. Looking at the orbits of the action of
$\langle\eta,\iota\rangle$ on $U_{0}(\mathbb{Z})$ we see that
$\\#U_{0}(\mathbb{Z})\equiv 0\pmod{4}$, hence $\\#U_{4}(\mathbb{Z})\leq 4$,
and since $U_{0}(\mathbb{Z})$ contains the orbit of $\gamma$ we conclude that
$\\#U_{0}(\mathbb{Z})=4$. Finally

$\\#C(\mathbb{Z})=\\#U_{0}(\mathbb{Z})+\\#U_{1}(\mathbb{Z})=4+10=14\,.$

##  9 Some further remarks

###  9.1 Complex uniformisations of some of the objects involved

Let $C$ be a projective curve over $\mathbb{Q}$, smooth, and geometrically
irreducible, and let $g$ be its genus. The universal cover of
$P^{\times}(\mathbb{C})$ is described in [6], Propositions 4.5 and 4.6. The
covering space, denoted $D_{\tau}$, is
$\mathrm{M}_{1,g}(\mathbb{C})\times\mathrm{M}_{g,1}(\mathbb{C})\times\mathbb{C}$,
hence a $\mathbb{C}$-vector space of dimension $2g+1$. The biextension
structure on
$\mathrm{M}_{1,g}(\mathbb{C})\times\mathrm{M}_{g,1}(\mathbb{C})\times\mathbb{C}$
is trivial, that is, for all $x$, $x_{1}$, $x_{2}$ in
$\mathrm{M}_{1,g}(\mathbb{C})$, all $y$, $y_{1}$, $y_{2}$ in
$\mathrm{M}_{g,1}(\mathbb{C})$, and all $z_{1}$, $z_{2}$ in $\mathbb{C}$, we
have:

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

The fundamental group $\pi_{1}(P^{\times}(\mathbb{C}),1)$ is

$None$ $Q^{u}(\mathbb{Z}):=\left\\{\begin{pmatrix}1&x&z\\\ 0&1_{2g}&y\\\
0&0&1\end{pmatrix}:\text{$x\in\mathrm{M}_{1,2g}(\mathbb{Z})$,
$y\in\mathrm{M}_{2g,1}(\mathbb{Z})$, $z\in\mathbb{Z}$}\right\\},$

also known as a Heisenberg group. Its action on $D_{\tau}$ is given in [6,
(4.5.3)].

Now recall the definition of $T$ in ($None$). As
$\mathrm{M}_{2g,1}(\mathbb{Z})$ is the lattice of $J(\mathbb{C})$, and
$\mathrm{M}_{1,2g}(\mathbb{Z})$ the lattice of $J^{\vee}(\mathbb{C})$, each
$f_{i}$ is given by an antisymmetric matrix $f_{i,\mathbb{Z}}$ in
$\mathrm{M}_{2g,2g}(\mathbb{Z})$ such that for all $y$ in
$\mathrm{M}_{2g,1}(\mathbb{Z})$ we have
$f_{i}(y)=y^{t}{\cdot}f_{i,\mathbb{Z}}$, and by a complex matrix
$f_{i,\mathbb{C}}$ in $\mathrm{M}_{g,g}(\mathbb{C})$ such that for all $v$ in
$\mathrm{M}_{g,1}(\mathbb{C})$, for each $i$ we have
$f_{i}(v)=v^{t}{\cdot}f_{i,\mathbb{C}}$ in $\mathrm{M}_{1,g}(\mathbb{C})$. For
more details about this description of the $f_{i}$ see the beginning of [6,
§4.7]. Then we have

$None$
$\pi_{1}(T(\mathbb{C}))=\left\\{\begin{pmatrix}1_{\rho-1}&m{\cdot}f(y)&z\\\
0&1_{2g}&y\\\ 0&0&1\end{pmatrix}:\text{$y\in\mathrm{M}_{2g,1}(\mathbb{Z})$,
$z\in\mathrm{M}_{\rho-1,1}(\mathbb{Z})$}\right\\},$

with $m{\cdot}f(y)\in\mathrm{M}_{\rho-1,2g}(\mathbb{Z})$ with rows the
$m{\cdot}y^{t}{\cdot}f_{i,\mathbb{Z}}$. So, $\pi_{1}(T(\mathbb{C}))$ is a
central extension of $\mathrm{M}_{2g,1}(\mathbb{Z})$ by
$\mathrm{M}_{\rho-1,1}(\mathbb{Z})$, with commutator pairing sending
$(y,y^{\prime})$ to $(2my^{t}{\cdot}f_{i,\mathbb{Z}}{\cdot}y^{\prime})_{i}$.

The universal covering $\widetilde{T(\mathbb{C})}$ is given by

$None$ $\displaystyle\widetilde{T(\mathbb{C})}$
$\displaystyle=\\{(m{\cdot}(c+f(v)),v,w):\text{$v\in\mathrm{M}_{g,1}(\mathbb{C})$,
$w\in\mathrm{M}_{\rho-1,1}(\mathbb{C})$}\\}$
$\displaystyle\subset\mathrm{M}_{\rho-1,g}(\mathbb{C})\times\mathrm{M}_{1,g}(\mathbb{C})\times\mathrm{M}_{\rho-1,1}(\mathbb{C})\,,$

with $m{\cdot}(c+f(v))\in\mathrm{M}_{\rho-1,g}(\mathbb{C})$ with rows the
$m{\cdot}(\widetilde{c_{i}}+v^{t}{\cdot}f_{i,\mathbb{C}})$ with
$\widetilde{c_{i}}$ a lift of $c_{i}$ in $\mathrm{M}_{1,g}(\mathbb{C})$. The
action of $\pi_{1}(T(\mathbb{C}),1)$ on $\widetilde{T(\mathbb{C})}$ is given
again, with the necessary changes, by [6, (4.5.3)].

Now that we know $\pi_{1}(T(\mathbb{C}),1)$ we investigate which quotient of
$\pi_{1}(C(\mathbb{C}),b)$ it is, via $\widetilde{j_{b}}\colon
C(\mathbb{C})\to T(\mathbb{C})$. We consider the long exact sequence of
homotopy groups induced by the $\mathbb{C}^{\times,\rho-1}$-torsor
$T(\mathbb{C})\to J(\mathbb{C})$, taking into account that
$\mathbb{C}^{\times,\rho-1}$ is connected and that $\pi_{2}(J(\mathbb{C}))=0$:

$None$
${\pi_{1}(\mathbb{C}^{\times,\rho-1},1)}$${\pi_{1}(T(\mathbb{C}),1)}$${\pi_{1}(J(\mathbb{C}),0)\,.}$

Again we see that $\pi_{1}(T(\mathbb{C}),1)$ is a central extension of the
free abelian group $\pi_{1}(J(\mathbb{C}),0)$ by $\mathbb{Z}^{\rho-1}$, and
from the matrix description we know that the $i$th coordinate of the
commutator pairing is given by
$mf_{i}\colon\mathrm{H}_{1}(J(\mathbb{C}),\mathbb{Z})\to\mathrm{H}_{1}(J^{\vee}(\mathbb{C}),\mathbb{Z})=\mathrm{H}_{1}(J(\mathbb{C}),\mathbb{Z})^{\vee}$.
The $\mathbb{Z}$-module of antisymmetric $\mathbb{Z}$-valued pairings on
$\mathrm{H}_{1}(J^{\vee}(\mathbb{C}),\mathbb{Z})$ is
$\bigwedge^{2}\mathrm{H}^{1}(J(\mathbb{C}),\mathbb{Z})=\mathrm{H}^{2}(J(\mathbb{C}),\mathbb{Z})$,
and $mf_{i}$ is the cohomology class (first Chern class) of the
$\mathbb{C}^{\times}$-torsor $T_{i}$:

$None$ $mf_{i}=c_{1}(T_{i})\quad\text{in
$\mathrm{H}^{2}(J(\mathbb{C}),\mathbb{Z})$.}$

There is a central extension

$None$
${\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})}$${E}$${\pi_{1}(J(\mathbb{C}),0)}$

that is universal in the sense that every central extension of
$\pi_{1}(J(\mathbb{C}),0)$ by a free abelian group arises by pushout from
$\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})$. We denote

$None$ $G:=\pi_{1}(C(\mathbb{C}),b)\,.$

The map $j_{b}\colon C\to J$ gives $G\to\pi_{1}(J(\mathbb{C}),0)$, and this is
the maximal abelian quotient. The second quotient in the descending central
series of $G$ gives the central extension:

$None$
${{}[G,G]/[G,[G,G]]}$${G/[G,[G,G]]}$${G/[G,G]=G^{\mathrm{ab}}=\pi_{1}(J(\mathbb{C}),0)\,.}$

This extension ($None$) arises from ($None$) by pushout via a morphism from
$\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})$ to $[G,G]/[G,[G,G]]$:

$None$
${\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})}$${E}$${G^{\mathrm{ab}}}$${{}[G,G]/[G,[G,G]]}$${G/[G,[G,G]]}$${G^{\mathrm{ab}}\,.}$

The left vertical arrow is surjective because commutators of lifts in $E$ of
elements of $G^{\mathrm{ab}}$ are mapped to the commutators of lifts in
$G/[G,[G,G]]$, and so give generators of $[G,G]/[G,[G,G]]$.

From the usual presentation of $G$ with generators
$\alpha_{1},\beta_{1},\ldots,\alpha_{g},\beta_{g}$, with the only relation
$[\alpha_{1},\beta_{1}]\cdots[\alpha_{g},\beta_{g}]=1$, we see that the
obstruction in lifting $G\to G^{\mathrm{ab}}$ to $G\to E$ in the top row of
($None$) is the image of $[\alpha_{1},\beta_{1}]\cdots[\alpha_{g},\beta_{g}]$
in $\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})$. This image is a generator of
the image of $\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z})$ under $j_{b}$. So the
pushout in ($None$) factors through the pushout by the quotient of
$\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})$ by
$\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z})$:

$None$
${\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})/\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z})}$${E^{\prime}}$${G^{\mathrm{ab}}}$${{}[G,G]/[G,[G,G]]}$${G/[G,[G,G]]}$${G^{\mathrm{ab}}\,.}$

Using again the presentation of $G$ we can split this morphism of extensions,
and, using that
$\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})/\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z})$
is generated by commutators of lifts of elements of $G^{\mathrm{ab}}$,
conclude that all vertical arrows in ($None$) are isomorphisms.

In particular, we have that $[G,G]/[G,[G,G]]$ is the same as
$\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z})/\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z})$.
From ($None$) we see that the sub-$\mathbb{Z}$-module of
$\mathrm{H}^{2}(J(\mathbb{C}),\mathbb{Z}(1))$ (note the Tate twist, now we
take the Hodge structures into account) spanned by the $mf_{i}$ is obtained in
4 steps: take the kernel of
$\mathrm{H}^{2}(J(\mathbb{C}),\mathbb{Z}(1))\to\mathrm{H}^{2}(C(\mathbb{C}),\mathbb{Z}(1))$,
take the $(0,0)$-part (then we are dealing with
$\mathrm{Hom}(J(\mathbb{C}),J^{\vee}(\mathbb{C}))^{+}$), then
$\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ acts (because
$\mathrm{Hom}(J(\mathbb{C}),J^{\vee}(\mathbb{C}))^{+}$ and
$\mathrm{Hom}(J_{\overline{\mathbb{Q}}},J^{\vee}_{\overline{\mathbb{Q}}})^{+}$
are equal), through the Galois group of a finite extension of $\mathbb{Q}$,
take the invariants (because we only use morphisms $f_{i}$ defined over
$\mathbb{Q}$), then take the image of the multiplication by $m$ on that.

Dually, this means that $\pi_{1}(T(\mathbb{C}),1)$ arises as the pushout

$None$
${\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z}(-1))/\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z}(-1))}$${G/[G,[G,G]]}$${G^{\mathrm{ab}}}$${\left((\mathrm{H}_{2}(J(\mathbb{C}),\mathbb{Z}(-1))/\mathrm{H}_{2}(C(\mathbb{C}),\mathbb{Z}(-1)))_{(0,0)}\right)_{\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}}$${\pi_{1}(T(\mathbb{C}),1)}$${G^{\mathrm{ab}}\,,}$

where the subscript $(0,0)$ means the largest quotient of type $(0,0)$, where
the subscript $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ means co-
invariants modulo torsion, and where the left vertical map is $m$ times the
quotient map. We repeat that the morphism from $\pi_{1}(C(\mathbb{C}))=G$ to
$\pi_{1}(T(\mathbb{C}),1)$ given by the middle vertical map is induced by
$\widetilde{j_{b}}\colon C(\mathbb{C})\to T(\mathbb{C})$.

###  9.2 Finiteness of rational points

In this section we reprove Faltings’s finiteness result [16] in the special
case where $r<g+\rho-1$. This was already done in [4], Lemma 3.2 (where the
base field is either $\mathbb{Q}$ or imaginary quadratic). We begin by
collecting some ingredients on good formal coordinates of the
$\mathbb{G}_{\mathrm{m}}$-biextension $P^{\times,\rho-1}\to J\times
J^{\vee,\rho-1}$ over $\mathbb{Q}$, and on what $C$ looks like in such
coordinates.

####  9.2.1 Formal trivialisations

Let $A$, $B$ and $G$ be connected smooth commutative group schemes over a
field $k\supset\mathbb{Q}$, and let $E\to A\times B$ be a commutative
$G$-biextension. Let $a$ be in $A(k)$, $b\in B(k)$ and $e\in E(k)$. For
$n\in\mathbb{N}$, let $A^{a,n}$ be the $n$th infinitesimal neighborhood of $a$
in $A$, hence its coordinate ring is $\mathcal{O}_{A,a}/m_{a}^{n+1}$. We use
similar notation for $B$ with $b$, and $E$ with $e$, and also for the points
$0$ of $A$, $B$ and $E$, and, similarly, the formal completion of $A$ at $a$
is denoted by $A^{a,\infty}$, etc. We also use such notation in a relative
context, for example, for the group schemes $E\to B$ and $E\to A$. We view
completions as $A^{a,\infty}$ as set-valued functors on the category of local
$k$-algebras with residue field $k$ such that every element of the maximal
ideal is nilpotent. For such a $k$-algebra $R$, $A^{a,\infty}(R)$ is the
inverse image of $a$ under $A(R)\to A(k)$. Then $A^{0,\infty}$ is the formal
group of $A$.

We now want to show that the formal $G^{0,\infty}$-biextension
$E^{0,\infty}\to A^{0,\infty}\times B^{0,\infty}$ is isomorphic to the trivial
biextension (the object $G^{0,\infty}\times A^{0,\infty}\times B^{0,\infty}$
with $+_{1}$ given by addition on the 1st and 2nd coordinate, and $+_{2}$ by
addition on the 1st and 3rd coordinate). As exp for $A^{0,\infty}$ gives a
functorial isomorphism
$T_{A/k}(0)\otimes_{k}{\mathbb{G}_{\mathrm{a}}}_{k}^{0,\infty}\to
A^{0,\infty}$, and similarly for $B$ and $G$, it suffices to prove this
triviality for $\mathbb{G}_{\mathrm{a}}^{0,\infty}$-biextensions of
$\mathbb{G}_{\mathrm{a}}^{0,\infty}\times\mathbb{G}_{\mathrm{a}}^{0,\infty}$
over $k$. One easily checks that the group of automorphisms of the trivial
$\mathbb{G}_{\mathrm{a}}^{0,\infty}$-biextension of
$\mathbb{G}_{\mathrm{a}}^{0,\infty}\times\mathbb{G}_{\mathrm{a}}^{0,\infty}$
over $k$ that induce the identity on all three
$\mathbb{G}_{\mathrm{a}}^{0,\infty}$’s is $(k,+)$, with $c\in k$ acting as
$(g,a,b)\mapsto(g+cab,a,b)$. As this group is commutative, it then follows
that the group of automorphisms of the $G^{0,\infty}$-biextension
$E^{0,\infty}\to A^{0,\infty}\times B^{0,\infty}$ that induce identity on
$G^{0,\infty}$, $A^{0,\infty},$and $B^{0,\infty}$, is equal to the $k$-vector
space of $k$-bilinear maps $T_{A/k}(0)\times T_{B/k}(0)\to T_{G/k}(0)$. This
indicates how to trivialise $E^{0,\infty}$. We choose a section $\tilde{e}$ of
the $G$-torsor $E\to A\times B$ over the closed subscheme $A^{0,1}\times
B^{0,1}$ of $A\times B$:

${E}$${A^{0,1}\times B^{0,1}}$${A\times B\,,}$$\scriptstyle{\tilde{e}}$ with
$\tilde{e}(0,0)=e$ in $E(k)$.

Note that

$\mathcal{O}(A^{0,1}\times B^{0,1})=(k\oplus m_{A^{0,1}})\otimes(k\oplus
m_{B^{0,1}})=k\oplus m_{A^{0,1}}\oplus m_{B^{0,1}}\oplus(m_{A^{0,1}}\otimes
m_{B^{0,1}})\,.$

Hence two such $\tilde{e}$ differ by a $k$-algebra morphism from $k\oplus
m_{G^{0,2}}=k\oplus m_{G^{0,1}}\oplus\mathrm{Sym}^{2}m_{G^{0,1}}$ (use the
exponential map) to $k\oplus m_{A^{0,1}}\oplus
m_{B^{0,1}}\oplus(m_{A^{0,1}}\otimes m_{B^{0,1}})$, hence by a triple of
$k$-linear maps from $m_{G^{0,1}}$ to $m_{A^{0,1}}$, $m_{B^{0,1}}$, and
$m_{A^{0,1}}\otimes m_{B^{0,1}}$. The linear maps $m_{G^{0,1}}\to m_{A^{0,1}}$
and $m_{G^{0,1}}\to m_{B^{0,1}}$ correspond to the differences on
$A^{0,1}\times B^{0,0}$ and on $A^{0,0}\times B^{0,1}$, respectively. There
are unique such linear maps such that the adjusted $\tilde{e}$ is compatible
with the given trivialisations of $E\to A\times B$ over $A^{0,1}\times
B^{0,0}$ and over $A^{0,0}\times B^{0,1}$. In geometric terms, $\tilde{e}$,
assumed to be adjusted, is then a splitting of $T_{G}(0)_{B}\hookrightarrow
T_{E/B}(0)\twoheadrightarrow T_{A}(0)_{B}$ over $B^{0,1}$ that is compatible
with the already given splitting over $0\in B(k)$, and it is also a splitting
of $T_{G}(0)_{A}\hookrightarrow T_{E/A}(0)\twoheadrightarrow T_{B}(0)_{A}$
over $A^{0,1}$ that is compatible with the already given splitting over $0\in
A(k)$. The splitting over $B^{0,1}$ gives an isomorphism from $(T_{G}(0)\oplus
T_{A}(0))_{B^{0,1}}$ to $(T_{E/B})_{B^{0,1}}$. So the exponential map, for
$+_{1}$, for the pullback to $B^{0,1}$ of $E\to B$, gives an isomorphism of
formal groups over $B^{0,1}$:

${\left((T_{G}(0)\oplus
T_{A}(0))\otimes_{k}\mathbb{G}_{\mathrm{a}}^{0,\infty}\right)_{B^{0,1}}}$${E_{B^{0,1}}^{0,\infty}\,.}$

Viewing $E_{B^{0,1}}^{0,\infty}$ as the tangent space at the zero section of
the pullback to $A^{0,\infty}$ of $E\to A$, this isomorphism gives a splitting
of $T_{G}(0)_{A}\hookrightarrow T_{E/A}(0)\twoheadrightarrow T_{B}(0)_{A}$
over $A^{0,\infty}$. The exponential map for $+_{2}$ for the pulback to
$A^{0,\infty}$ of $E\to A$ then gives an isomorphism of formal groups over
$A^{0,\infty}$:

${G^{0,\infty}\times B^{0,\infty}\times A^{0,\infty}}$${(G^{0,\infty}\times
B^{0,\infty})_{A^{0,\infty}}}$${E_{A^{0,\infty}/A^{0,\infty}}^{0,\infty}}$${E^{0,\infty}\,,}$

where $E_{A^{0,\infty}/A^{0,\infty}}^{0,\infty}$ denotes the completion along
the zero section of the pullback via $A^{0,\infty}\to A$ of $E\to A$. The
compatibility between $+_{1}$ and $+_{2}$ on $E$ ensures that this isomorphism
is an isomorphism of biextensions, with the trivial biextension structure on
the left.

Now that we know what good formal coordinates at $0$ in $E(k)$ are, we look at
the point $e$ in $E(k)$, over $(a,b)$ in $(A\times B)(k)$. We produce an
isomorphism $E^{0,\infty}\to E^{e,\infty}$, using the partial group laws. Let
$E_{b}$ be the fibre over $b$ of $E\to B$. We choose a section

${E_{b}}$${A^{a,1}\times\\{b\\}}$${A\times\\{b\\}}$$\scriptstyle{\tilde{e}_{1}}$
with $\tilde{e}_{1}(a,b)=e$ in $E(k)$.

The exponentials for the group laws of $E_{b}$ and $A$ then give a section

${E_{b}}$${A^{a,\infty}\times\\{b\\}}$${A\times\\{b\\}\,,}$$\scriptstyle{\tilde{e}^{\infty}_{1}}$

that we view as an $A^{a,\infty}$-valued point of $E_{b}$, and as a section of
the group scheme $E_{A^{a,\infty}}\to A^{a,\infty}$, with group law $+_{2}$.
The translation by $\tilde{e}^{\infty}_{1}$ on this group scheme induces
translation by $b$ on $B_{A^{a,\infty}}$, and maps $(a,0)$, the $0$ element of
$E_{a}$, to $e$. Hence it induces an isomorphism of formal schemes
$E^{(a,0),\infty}\to E^{e,\infty}$. In order to get an isomorphism
$E^{0,\infty}\to E^{(a,0),\infty}$, we repeat the process above, but with the
roles of $A$ and $B$ exchanged. We choose a section
$\tilde{0}_{2}\colon\\{a\\}\times B^{0,1}\to E_{a}$ of $E_{a}\to\\{a\\}\times
B$. Then the exponential for $+_{2}$ gives us a section
$\tilde{0}^{\infty}_{2}\colon\\{a\\}\times B^{0,\infty}\to E_{a}$ of
$E_{a}\to\\{a\\}\times B$. This $\tilde{0}^{\infty}_{2}$ is a section of the
group scheme $E_{B^{0,\infty}}\to B^{0,\infty}$, and the translation on it by
$\tilde{0}^{\infty}_{2}$ sends $0$ in $E(k)$ to $(a,0)$, hence gives an
isomorphism of formal schemes $E^{0,\infty}\to E^{(a,0),\infty}$. Composition
then gives us an isomorphism $E^{0,\infty}\to E^{e,\infty}$, and the good
formal coordinates on $E$ at $0\in E(k)$ give what we call good formal
coordinates at $e$. Similarly, we get a section $\widetilde{0}_{1}^{\infty}$
of $E_{A^{0,\infty}}\to A^{0,\infty}$ and a section
$\widetilde{e}_{2}^{\infty}$ of $E_{B^{b,\infty}}\to B^{b,\infty}$ giving
isomorphisms $E^{0,\infty}\to E^{(0,b),\infty}$ and $E^{(0,b),\infty}\to
E^{e,\infty}$, hence by composition a 2nd isomorphism $E^{0,\infty}\to
E^{e,\infty}$. These isomorphisms are equal for a unique choice of
$\tilde{0}_{1}$ and $\tilde{e}_{2}$ (given the choices of $\tilde{0}_{2}$ and
$\tilde{e}_{1}$).

In Section 9.2.3 we will use that these isomorphisms transport all additions
that occur in ($None$) to additions in $E^{0,\infty}$ and therefore to
additions in the trivial formal biextension.

####  9.2.2 Zariski density of the curve in formally trivial coordinates

Let $C$ be as in the beginning of Section 2. Let $\widetilde{C(\mathbb{C})}$
be the inverse image of $C(\mathbb{C})$ under the universal cover
$\widetilde{T(\mathbb{C})}\to T(\mathbb{C})$. Then $\widetilde{C(\mathbb{C})}$
is connected since $\widetilde{j_{b}}\colon C\to T$ gives a surjection on
complex fundamental groups. Now we consider the complex analytic variety
$\widetilde{T(\mathbb{C})}$ as a complex algebraic variety via the bijection
$\widetilde{T(\mathbb{C})}=\mathbb{C}^{g+\rho-1}$ as given in ($None$). The
analytic subset $\widetilde{C(\mathbb{C})}$ contains the orbit of $0$ under
$\pi_{1}(T(\mathbb{C}),1)$. This orbit surjects to the lattice of
$J(\mathbb{C})$ in $\mathrm{M}_{g,1}(\mathbb{C})$, and over each lattice
point, its fibre in $\mathrm{M}_{\rho-1,1}(\mathbb{C})$ contains a translate
of $2\pi i\mathrm{M}_{\rho-1,1}(\mathbb{Z})$. Hence this orbit is Zariski
dense in $\mathbb{C}^{g+\rho-1}$. It follows that the formal completion of
$\widetilde{C(\mathbb{C})}$ at any of its points is Zariski dense in
$\mathbb{C}^{g+\rho-1}$: if a polynomial function on $\mathbb{C}^{g+\rho-1}$
is zero on such a completion, then it vanishes on the connected component of
$\widetilde{C(\mathbb{C})}$ of that point, hence on
$\widetilde{C(\mathbb{C})}$ and consequently on $\widetilde{T(\mathbb{C})}$.

We express our conclusion in more algebraic terms: for $c\in C(\mathbb{C})$,
with images $t\in T(\mathbb{C})$ and in $P^{\times,\rho-1}(\mathbb{C})$, each
polynomial in good formal coordinates at $t$ of the biextension
$P^{\times,\rho-1}\to J\times J^{\vee}$ over $\mathbb{C}$ that vanishes on
$\widetilde{j_{b}}(C_{\mathbb{C}}^{c,\infty})$, vanishes on
$T_{\mathbb{C}}^{t,\infty}$. This statement then also holds with $\mathbb{C}$
replaced by any subfield, or even any subring of the form $\mathbb{Z}_{(p)}$
with $p$ a prime number, or the localisation of $\overline{\mathbb{Z}}$ (the
integral closure of $\mathbb{Z}$ in $\mathbb{C}$) at a maximal ideal.

####  9.2.3 The $p$-adic closure in good formal coordinates

We stay in the situation of Section 2, but we denote
$G:=\mathbb{G}_{\mathrm{m}}^{\rho-1}$, $A:=J$ and $B:=J^{\vee,0\rho-1}$, and
$E:=P^{\times,\rho-1}$. Let $d_{G}$, $d_{A}$, and $d_{B}$ be their dimensions:
$d_{G}=\rho-1$, $d_{A}=g$ and $d_{B}=(\rho-1)g$.

Let $p>2$ be a prime number. From Section 9.2.1 and Lemma 5.1.1 we conclude
that we can choose _formal_ parameters for $E$ at $0$, over
$\mathbb{Z}_{(p)}$, such that they converge on the residue polydisk
$E(\mathbb{Z}_{p})_{\overline{0}}$, and such that they induce the trivial
biextension structure on
$\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$.
We keep the notation of Section 9.2.1, for $e$ in $E(\mathbb{Z}_{p})$, lying
over $(a,b)$ in $(A\times B)(\mathbb{Z}_{p})$. This $e$ plays the role that
$\widetilde{t}$ has at the beginning of Section 4. As explained at the end of
Section 9.2.1, we may and do assume that $e$ is in
$E(\mathbb{Z}_{p})_{\overline{0}}$, and hence $a\in
A(\mathbb{Z}_{p})_{\overline{0}}$ and $b\in B(\mathbb{Z}_{p})_{\overline{0}}$.

Assume now that, as in Section 4, for $i,j\in\\{1,\ldots,r\\}$, we have
$x_{i}$ in $A(\mathbb{Z}_{p})_{\overline{0}}$ and $y_{j}$ in
$B(\mathbb{Z}_{p})_{\overline{0}}$, and $e_{i,j}$ in
$E(\mathbb{Z}_{p})_{\overline{0}}$ over $(x_{i},y_{j})$, and $r_{i}$ in
$E(\mathbb{Z}_{p})_{\overline{0}}$ over $(x_{i},b)$ and $s_{j}$ in
$E(\mathbb{Z}_{p})_{\overline{0}}$ over $(a,y_{j})$. We denote the images of
all these elements under the bijection

$E(\mathbb{Z}_{p})_{\overline{0}}\longrightarrow\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$

as follows:

$\displaystyle x_{i}$ $\displaystyle\mapsto(0,x_{i},0)\,,$ $\displaystyle
y_{j}$ $\displaystyle\mapsto(0,0,y_{j})\,,$ $\displaystyle e_{i,j}$
$\displaystyle\mapsto(g_{i,j},x_{i},y_{j})$ $\displaystyle r_{i}$
$\displaystyle\mapsto(r_{i}^{\prime},x_{i},b)\,,$ $\displaystyle s_{j}$
$\displaystyle\mapsto(s_{j}^{\prime},a,y_{j})\,,$ $\displaystyle e$
$\displaystyle\mapsto(e^{\prime},a,b)\,.$

Then a straightforward computation shows that the image of $D(n)$ as defined
in ($None$) is

$\left(e^{\prime}+\sum_{i}n_{i}r^{\prime}_{i}+\sum_{j}n_{j}s^{\prime}_{j}+\sum_{i,j}n_{i}n_{j}g_{i,j},\,\,a+\sum_{i}n_{i}x_{i},\,\,b+\sum_{j}n_{j}y_{j}\right)\quad\text{in
$\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$.}$

The conclusion is that in these coordinates, the map

$\kappa\colon\mathbb{Z}_{p}^{r}\longrightarrow\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$

is a polynomial map, hence the Zariski closure of its image is an algebraic
variety of dimension at most $r$.

####  9.2.4 Proof of finiteness

The proof is by contradiction. So assume that $r<g+\rho-1$, and that
$C(\mathbb{Q})$ is infinite. Let $p>2$ be a prime number. Then there is a
$u\in C(\mathbb{F}_{p})$ such that the residue disk $C(\mathbb{Z}_{p})_{u}$
contains infinitely many elements of $C(\mathbb{Q})$, hence infinitely many
elements in the image of $\kappa$ of Section 4.10. By construction,
$\kappa(\mathbb{Z}_{p}^{r})$ is contained in $T(\mathbb{Z}_{p})_{t}$. The
image of $T(\mathbb{Z}_{p})_{t}$ in
$\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$
is $\mathbb{Z}_{p}^{\rho-1}\times\mathbb{Z}_{p}^{g}$, with
$\mathbb{Z}_{p}^{g}$ embedded in
$\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$ as a
sub-$\mathbb{Z}_{p}$-module. By the previous section, the Zariski closure of
$\kappa(\mathbb{Z}_{p}^{r})$ in
$\mathbb{Z}_{p}^{d_{G}}\times\mathbb{Z}_{p}^{d_{A}}\times\mathbb{Z}_{p}^{d_{B}}$
is of dimension at most $r$. Hence there are non-zero polynomial functions on
$\mathbb{Z}_{p}^{\rho-1}\times\mathbb{Z}_{p}^{g}$ that are zero on infinitely
many points of $C(\mathbb{Z}_{p})_{u}$, and hence are zero on a non-empty open
smaller disk. This contradicts, via a ring morphism
$\mathbb{Z}_{p}\to\mathbb{C}$, the conclusion of Section 9.2.2.

###  9.3 The relation with $p$-adic heights

We want to compare the approach to quadratic Chabauty in this article to the
one in [4], by answering the question: which local analytic coordinates on
$T(\mathbb{Z}_{p})$ and $C(\mathbb{Q}_{p})$ lead to the equations, in terms of
$p$-adic heights, for the quadratic Chabauty set $C(\mathbb{Q}_{p})_{2}$ in
[4]? Before we do this, we note that the Poincaré biextension has played a
role in Arakelov theory, and in the theory of $p$-adic heights, since a long
time: see [32], [23] and [25]. Moreover, [8] gives a detailed description how
Kim’s cohomological approach relates to $p$-adic heights in the context of
$\mathbb{G}_{\mathrm{m}}$-torsors on abelian varieties.

Let $p>2$ be a prime number of good reduction for $C$. We consider the
Poincaré torsor as $\mathcal{M}^{\times}$ on $(J\times J)_{\mathbb{Q}_{p}}$
via ($None$), and we use the description of $\mathcal{M}^{\times}$ given in
($None$).

Let $\mathcal{D}$ be the subset
$\mathrm{Div}^{0}(C_{\mathbb{Q}_{p}})\times\mathrm{Div}^{0}(C_{\mathbb{Q}_{p}})$
made of pairs of divisors $(D_{1},D_{2})$ having disjoint support. Let $W$ be
an isotropic complement of
$\Omega^{1}_{C_{\mathbb{Q}_{p}}/\mathbb{Q}_{p}}(C_{\mathbb{Q}_{p}})$ in
$\mathrm{H}^{1}_{\mathrm{dR}}(C_{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ and let
$\log\colon\mathbb{Q}_{p}^{\times}\to\mathbb{Q}_{p}$ be a group morphism
extending the formal logarithm on $1+p\mathbb{Z}_{p}$. With these choices
made, Coleman and Gross ([11, (5.1)]) define the function (there denoted
$\langle{\cdot},{\cdot}\rangle$)

$h_{p}\colon\mathcal{D}\to\mathbb{Q}_{p}\,,$

the $p$-part of the $p$-adic height pairing. We define the function

$\psi\colon\mathcal{M}^{\times}(\mathbb{Q}_{p})\longrightarrow\mathbb{Q}_{p}$

by demanding that for every effective $D_{1}$ and $D_{2}$ in
$\mathrm{Div}(C_{\mathbb{Q}_{p}})$ of the same degree and every $E$ in
$\mathrm{Div}^{0}(C_{\mathbb{Q}_{p}})$, and every $\lambda$ in
$\mathbb{Q}_{p}^{\times}$, the element

$\lambda{\cdot}\mathrm{Norm}_{D_{1}/\mathbb{Q}_{p}}(1)\otimes\mathrm{Norm}_{D_{2}/\mathbb{Q}_{p}}(1)^{-1}$

in

$\mathcal{M}^{\times}(\mathcal{O}_{C_{\mathbb{Q}_{p}}}(E),\Sigma(D_{1})-\Sigma(D_{2}))=\left(\mathrm{Norm}_{D_{1}/\mathbb{Q}_{p}}\mathcal{O}_{C_{\mathbb{Q}_{p}}}(E)\otimes\mathrm{Norm}_{D_{2}/\mathbb{Q}_{p}}\mathcal{O}_{C_{\mathbb{Q}_{p}}}(-E)\right)^{\times}$

is sent to

$\psi(\lambda{\cdot}\mathrm{Norm}_{D_{1}/\mathbb{Q}_{p}}(1)\otimes\mathrm{Norm}_{D_{2}/\mathbb{Q}_{p}}(1)^{-1}):=h_{p}(D_{1}-D_{2},E)+\log\lambda\,.$

That this depends only on the linear equivalence classes of $D_{1}{-}D_{2}$
and $E$ follows from ($None$), plus (see [11, Proposition 5.2]) the fact that
$h_{p}$ is biadditive, symmetric and, for any non-zero rational function $f$
on $C_{\mathbb{Q}_{p}}$ and any $D$ in $\mathrm{Div}^{0}(C_{\mathbb{Q}_{p}})$
with support disjoint from that of $\mathrm{div}(f)$, we have
$h_{p}(D,\mathrm{div}(f))=\log(f(D))$. Moreover, expressing $h_{p}$ in terms
of a Green function $G$ as in [7, Theorem 7.3], we deduce that, in each
residue disk of $\mathcal{M}^{\times}(\mathbb{Z}_{p})$, $\psi$ is given by a
power series. Let $\omega_{1},\ldots,\omega_{g}$ be a basis of
$\Omega^{1}_{C_{\mathbb{Q}_{p}}/\mathbb{Q}_{p}}(C_{\mathbb{Q}_{p}})$. This
basis gives a unique morphism of groups $\log_{J}\colon
J(\mathbb{Q}_{p})\to\mathbb{Q}_{p}^{g}$ that extends the logarithm of Lemma
5.1.1. We define

$\Psi:=(\log_{J}\circ\mathrm{pr}_{J,1},\log_{J}\circ\mathrm{pr}_{J,2},\psi)\colon\mathcal{M}^{\times}(\mathbb{Q}_{p})\longrightarrow\mathbb{Q}_{p}^{g}\times\mathbb{Q}_{p}^{g}\times\mathbb{Q}_{p}\,.$

By the biadditivity of $h_{p}$, $\Psi$ is a morphism of biextensions, with the
trivial biextension structure on
$\mathbb{Q}_{p}^{g}\times\mathbb{Q}_{p}^{g}\times\mathbb{Q}_{p}$ as in
($None$). As $p>2$, $\Psi$ induces, from each residue polydisk to its image, a
homeomorphism given by power series. Pulling back the coordinate functions on
$\mathbb{Q}_{p}^{2g+1}$ gives, for every
$x\in\mathcal{M}^{\times}(\mathbb{F}_{p})$, coordinates on
$\mathcal{M}^{\times}(\mathbb{Z}_{p})_{x}$.

We describe $\widetilde{j_{b}}$ and $\kappa$ in these coordinates. It suffices
to describe, for each $i$ in $\\{1,\ldots,\rho{-}1\\}$,
$\widetilde{j_{b,i}}\colon C\to T_{i}$, and from now on we omit the dependence
on $i$. For each $c\in C(\mathbb{F}_{p})$, on
$T(\mathbb{Z}_{p})_{\widetilde{j_{b}}(x)}$ we use the coordinates
$x_{1}:=f^{*}t_{1},\ldots,x_{g}:=f^{*}t_{g}$, $z:=f^{*}t_{2g+1}$ where $f$ is
the map $T\to\mathcal{M}^{\times}$ and $t_{1},\ldots,t_{2g+1}$ are the
coordinates on $\mathcal{M}^{\times}(\mathbb{Z}_{p})_{\widetilde{j_{b}}(c)}$
we just defined. Since the map $\Psi$ is a morphism of biextensions, for $j$
in $\\{1,\ldots,g\\}$, $x_{j}\circ\kappa$ is a polynomial of degree at most
$1$, and $z\circ\kappa$ is a polynomial of degree at most $2$. As explained in
Section 7, over $\mathbb{Z}_{p}$, $\widetilde{j_{b}}$ is given by a line
bundle $\mathcal{L}$ over $(C\times C)_{\mathbb{Z}_{p}}$ rigidified along
$(C\times\\{b\\})_{\mathbb{Z}_{p}}$ and along the diagonal with two sections
$l_{b}$ and $l$. Choosing a section that trivializes $\mathcal{L}$ on an open
subset of $(C\times C)_{\mathbb{Z}_{p}}$ containing $(b,b)$, $(c,b)$, and
$(c,c)$ in $(C\times C)(\mathbb{F}_{p})$ we get a divisor $D$ on $(C\times
C)_{\mathbb{Z}_{p}}$ whose support is disjoint from $(c,b)$ and $(c,c)$, and
an isomorphism between $\mathcal{L}$ and $\mathcal{O}(D)$ on $(C\times
C)_{\mathbb{Z}_{p}}$. After modifying $D$ with a principal horizontal divisor
and a principal vertical divisor $D|_{C{\times}\\{b\\}}$ and
$\mathrm{diag}^{*}D$ are both equal to the the zero divisor on
$C_{\mathbb{Z}_{p}}$, hence $l_{b}$ and $l$ are the extensions of elements of
$\mathbb{Q}_{p}$, interpreted as rational sections of $\mathcal{O}(D)$ on
$(C\times C)_{\mathbb{Z}_{p}}$. By Propositions 7.5 and 7.8, there exists a
unique $\lambda\in\mathbb{Q}_{p}^{\times}$ such that, for each $d\in
C(\mathbb{Z}_{p})_{c}$,

$\widetilde{j_{b}}(d)=\lambda\cdot\mathrm{Norm}_{d/\mathbb{Z}_{p}}(1)\otimes\mathrm{Norm}_{b/\mathbb{Z}_{p}}(1)^{-1}\in\mathcal{M}^{\times}(j_{b}(d),D|_{\\{d\\}{\times}C})\,.$

Since $x_{j}$ is the $j$-th coordinate of $\log_{J}$ and since $z$ is the
pullback of $\psi$, we deduce that

$x_{1}(\widetilde{j_{b}}(d))=\int_{b}^{d}\omega_{1},\,\ldots\,,\quad
x_{g}(\widetilde{j_{b}}(d))=\int_{b}^{d}\omega_{g},\quad
z(\widetilde{j_{b}}(d))=h_{p}(d-b,D|_{\\{d\\}{\times}C})+\log\lambda\,,$

and, by [4, Proof of Theorem 1.2] and [5, Lemma 5.5], the function $d\mapsto
h_{p}(d{-}b,D|_{\\{d\\}{\times}C})$ is a sum of double Coleman integrals.

It should now be easy to exactly interpret geometrically the cohomological
approach, showing that in the coordinates used here, the equations for
$C(\mathbb{Q}_{p})_{2}$ are precisely equations for the intersection of
$C(\mathbb{Q}_{p})$ and the $p$-adic closure of $T(\mathbb{Z})$. For doing
computations, one can do them in the geometric context of this article, or, as
in [5], in terms of the étale fundamental group of $C$. The connection between
these is then given by $p$-adic local systems on $T$.

##### Author contributions

This project started with an idea of Edixhoven in December 2017. From then on
Edixhoven and Lido worked together on the project. Section 8 is due entirely
to Lido. Section 9 was written in July and August 2020.

##### Acknowledgements

We thank Steffen Müller, Netan Dogra, Jennifer Balakrishnan, Kamal Khuri-
Makdisi, Jan Vonk, Barry Mazur and Gerd Faltings for discussions and
correspondence we had with them, Michael Stoll for suggesting the title of
this article, Pim Spelier for his MSc thesis [30], and Sachi Hashimoto for the
cartoon guide [19]. Finally, we thank the referee for their very thorough work
and for making us increase the number of propositions and lemmas so that,
hopefully, this article has become easier to parse and to digest.

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