\chapter{Visibility}
\section{Visible Subgroups of $H^1(K, A)$ and $\Sha(A / K)$.}
Suppose that $A$ is an abelian variety, defined over a number field $K$. In *** chapter ***
we introduced the notion of principal homogeneous spaces for $A$ and interpreted this notion
rather geometrically, in terms of $K$-rational points. We established a bijection between the
equivalence classes of principal homogeneous spaces, or elements of the Weil-Chatelet group
$WC(A/K)$ and the elements of the first Galois cohomology group $H^1(K, A)$. More precisely,
for each cohomology class $c \in H^1(K, A)$, there is a corresponding principal homogeneous
space $X$ over $K$, together with an action of $A$ on $X$, i.e. a map
$A \times X \ra X$, satisfying axioms, similar to those for a simply transitive group action
(can be explained better).
Our main motivation will be the following observation: in most cases, it is much more
difficult to understand elements of the first Galois cohomology group $H^1(K, A)$, than rational
points on some abelian variety. Thus, our goal will be to describe certain elements of the
$H^1(K, A)$ in terms of $K$-rational points on certain curves.
To formalize this, we use the notion of the visible subgroup of the first cohomology
$H^1(K, A)$, which was first introduced by Mazur *** CITE ***. Suppose that $i : A \ra J$ is
a morphism of abelian varieties over $K$ *** FOOTNOTE EXPLAINING WHY J ***.
\begin{defn}
We define the \emph{visible subgroup} of $H^1(K, A)$ with respect to $i$ and
$J$ as
$$
\textrm{Vis}_J^{(i)}(H^1(K, A)) := \textrm{ker }\{H^1(K, A) \ra H^1(K, J)\},
$$
where $H^1(K, A) \ra H^1(K, J)$ is the map on cohomology, induced from $i$.
\end{defn}
The notion is useful, because it relates to the geometric interpretation of the elements of
$H^1(K, A)$ as elements of the Weil-Chatelet group. Indeed, consider the short exact sequence on
abelian varieties
$$
0 \ra A \ra J \ra C \ra 0,
$$
where $C$ is the quotient $J / A$. Write the long exact sequence on Galois cohomology
$$
0 \ra A(K) \ra J(K) \ra C(K) \ra H^1(K, A) \ra H^1(K, J) \ra \dots.
$$
Using the definition of the visible subgroup, we can extract the following short exact
sequence
$$
0 \ra J(K) / A(K) \ra C(K) \ra \textrm{Vis}_J^{(i)}(H^1(K, A)) \ra 0.
$$
Let $c$ be a visible cohomology class. Then there exists a $K$-rational point $P$ on $C$,
which maps to $c$. The fiber over $P$ for the map $J \ra C$ is a subvariety of $J$, which
is equipped with a natural action of $A$. This variety belongs to the equivalence class
of principal homogeneous spaces, corresponding to the cohomology class $c$
(*** MORE DETAIL ***). This is the reason why one says that the visible element $c$
\emph{arises} from a rational point on the variety $C$.
Next, we define the visible subgroup of $\Sha(A)$ with respect to the same morphism
$i : A \ra J$.
\begin{defn}
The visible subgroup of $\Sha(A)$ with respect to the map $i$ is defined as
$$
\textrm{Vis}_J^{(i)}(A) := \textrm{Vis}_J^{(i)}(H^1(K, A)) \cap \Sha(A/K).
$$
\end{defn}
We can use the long exact sequence on Galois cohomology to understand better the
visible subgroup of $\Sha(A)$.
\section{Visibility Dimension}
The first interesting property, related to visibility is that each element of $H^1(K, A)$
becomes visible for suitable $J$ and an embedding $i : A \hookrightarrow J$.
\begin{thm}
Let $c \in H^1(K, A)$ be any cohomology class. Then there exists an abelian variety $J$ and
an embedding $i : A \hookrightarrow J$, such that $c \in \textrm{Vis}_J^{(i)}(H^1(K, A))$.
\end{thm}
\verb Proof: Recall that a cohomology class $c \in H^1(K, A)$ is trivial if and only if
the corresponding principal homogeneous space $C$ to $c$ has a $K$-rational point. Intuitively,
to trivialize $c$, it is enough to consider an extension $L / K$, so that $C$ has an
$L$-rational point. This always happens for some unramified extension $L / K$.
Choose such $L / K$, for which res$_L(c) = 0$. To produce the abelian variety $J$, one can
use Weil restriction of scalars. Indeed, for a scheme $X$ over $K$, consider the functor
the contravariant functor
$$
\textrm{Res}_{L / K}(X) : (\textrm{Sch} / K)^0 \ra \textrm{\{Sets\}},\
S \mapsto \textrm{Hom}_L(S \times_K L, X)
$$
By taking $X = A_L$, we obtained from [BLR, \S 7.6] that the functor is representable, and therefore
$J := \textrm{Res}_{L / K}(A_L)$ is an abelian variety over $K$ whose dimension is at most
$[L : K] \cdot \textrm{dim }A$. Moreover, for any group scheme $S$ over $K$, there is an
isomorphism $A_L(S \times_K L) \simeq J(S)$. Thus, we have an inclusion
$$
A(K) \hookrightarrow A_L(L) \simeq J(K)
$$
Using Yoneda's lemma, this corresponds to a proper morphism of groups schemes over $K$,
$\iota : A \ra J$. Therefore, *** EXPLAIN *** $\iota$ is a closed immersion. Finally,
one needs to check that $\iota : A \ra J$ makes $c \in H^1(K, A)$ visible. We know that
res$_{L/K}(c) = 0$. But by using Shapiro's lemma and computations, one concludes that
$$
H^1(K, J) \simeq H^1(L, A_L),
$$
and via this isomorphism $i^*(c) \mapsto \textrm{res}_{L/K}(c)$, so
$c \in \textrm{Vis}^{(i)}_J(H^1(K, A))$. $\hfill \Box$
The theorem gives rise to an interesting question, relating the order of
a cohomology class $c$ and the minimal dimension of an abelian variety $J$, for which $c$
is visible, under an embedding $i : A \hookrightarrow J$.
\begin{defn}
Let $c \in H^1(K, A)$. The \emph{visibility dimension} of $c$ is the minimal dimension of
an abelian variety $J$, such that $c$ is visible in $J$.
\end{defn}
First of all, we produce an upper bound for the visibility dimension of a cohomology class
$c \in H^1(K, A)$, in terms of the order $n$ of that element and the dimension of $A$.
\begin{lem}
Suppose that $G$ is a group and $M$ is a finite $G$-module. Let $c \in H^1(G, M)$ be any
cohomology class. Then there is a subgroup $H \subseteq G$, such that the restriction of
$c$ to $H^1(H, M)$ is trivial and $[G : H] \leq |M|$.
\end{lem}
\verb Proof: Consider $H = \textrm{ker}(f)$, where $f : G \ra M$ is any representative of
the cohomology class $c$. The map $f$ satisfies the cocycle condition
$$
f(\sigma \tau) = \sigma f(\tau) + f(\sigma).
$$
Clearly, the restriction of $c$ to $H^1(H, M)$ is trivial. To bound
the dimension, we construct an injection $G / H \hookrightarrow M$, by sending $\tau H \mapsto
f(\tau)$. By the definition of $H$, this map is well defined. Suppose $f(\sigma) = f(\tau)$.
Then, the cocycle condition
$$
f(\tau) = f(\sigma (\sigma^{-1}\tau)) = \sigma f(\sigma^{-1}\tau) + f(\sigma).
$$
Thus, $\sigma f(\sigma^{-1}\tau) = 0$, i.e. $f(\sigma^{-1}\tau) = 0$, which means that
$\sigma^{-1}\tau \in H$. This proves injectivity of the map $G / H \hookrightarrow M$, and so
the bound follows. $\hfill \Box$
\begin{prop}
The visibility dimension of any $c \in H^1(K, A)$ is at most $d \cdot n^{2d}$, where $n$ is the
order of $c$ in $H^1(K, A)$ and $d$ is the dimension of $A$.
\end{prop}
\verb Proof: It follows from the proof of Theorem 2.0.1 that the dimension of $J$, which was
constructed using Weil restriction of scalars is at most $[L : K] \cdot \textrm{dim }A$.
Thus, we need an upper bound for the degree of the unramified extension $[L : K]$. To get this,
consider the surjective map $H^1(K, A[n]) \ra H^1(K, A)[n]$, which is induced from the long
exact sequence on Galois cohomology. Since $c \in H^1(K, A)[n]$, it suffices to trivialize one
of its preimages. By the lemma, there exists a finite index subgroup of
Gal$(\overline{\Q} / K)$ (which by Galois theory corresponds to some unramified field
extension $L / K$), such that $c$ is trivialized in $H^1(L, A[n])$ and the index of the subgroup
$[L : K]$ is at most $|A[n]| = n^{2d}$. Thus, one can choose $L$, so that
$[L : K] \cdot \textrm{dim }A \leq d \cdot n^{2d}$, so we get an upper bound for the
dimension. $\hfill \Box$
\section{Visualizing Elements for Modular Abelian Varieties Attached to Newforms via
Level Raising}
\subsection{Computing the Modular Degree}
Let $A_f$ be an abelian variety attached to a newform $f$. Then $A_f$ is a quotient of $J_0(N)$,
which means that there is a surjective morphism $J_0(N) \ra A_f$. Consider the dual morphism
$A_f^{\vee} \ra J_0(N)$, which is an injection. By taking the composition of these two maps,
we obtain a finite degree morphism $\theta_f : A_f^{\vee} \ra A_f$. This finite morphism is
in fact a polarization. It turns out that the degree of $\theta_f$ is a perfect square, which
is a consequence of the following
\begin{prop}
Suppose $A$ is an abelian variety over a field $k$ and let $\lambda : A \ra \Adual$ be
a polarization. Suppose that char$(k)$ is either zero, or prime to the degree of $\lambda$.
There exists a finite abelian group $H$, such that
$$
\textrm{ker}(\lambda) \cong H \times H,
$$
where the above identification is a group isomorphism.
\end{prop}
Before proving the proposition, we will need the following lemma:
\begin{lem}
Suppose that $G$ is a finite abelian group for which there exists a nondegenerate,
alternating, bilinear pairing $\Gamma : G \times G \ra \Q/\Z$. There exists a group $H$,
such that $G \cong H \times H$.
\end{lem}
\begin{proof}
Using the structure theorem for abelian groups, one can reduce the statement to the case
when $G$ is a $p$-group for some prime number $p$. Let $x$ be an element of $G$ of maximal
order $p^h$ for some integer $h$. First, we show that there exists $y \in G$, such that
$\Gamma(x, y) = 1/p^h$. Indeed, if no such $y$ exists, then $\Gamma(p^{h-1}x, y) = 0$
for each $y \in G$, so $\Gamma$ is degenerate, which is a contradiction. Notice that
every such $y$ still has maximal order $p^h$, since $0 \ne p^{h-1}\Gamma(x, y) =
\Gamma(x, p^{h-1}y)$. Moreover, we show that
$\langle x \rangle \cap \langle y \rangle = \varnothing$. Indeed, if $mx = ny$ for some
$0 < m, n < p^h$, then
$$
0 = m\Gamma(x, x) = \Gamma(x, mx) = n\Gamma(x, y) \ne 0,
$$
which is a contradiction.
After choosing such $y$ one can define
$$
H = \{z : \Gamma(x, z) = \Gamma(y, z) = 0\}.
$$
We claim that $G \simeq (\langle x \rangle + \langle y \rangle) \oplus H$. Indeed,
for any $g \in G$, the alternating pairing $\Gamma$ gives us
$$
g - (p^h\Gamma(g, y))x - (p^h\Gamma(g, x))y \in H,
$$
It is easy to check that this produces a group isomorphism
$$
G \simeq (\langle x \rangle + \langle y \rangle) \oplus H.
$$
But $\Gamma$ restricts to an alternating, nondegenerate, bilinear pairing to $H \times H$.
This means that we can use induction on the size of the group $G$ to prove the statement.
If $G$ is trivial, there is noting to prove. If not, we construct $H$ and apply the
hypothesis for $H$, i.e. there exists a subgroup $H'$ of $H$, such that
$H \simeq H' \oplus H'$. This means that
$$
G \simeq (\langle x \rangle \oplus H') \oplus (\langle y \rangle \oplus H'),
$$
because $\langle x \rangle \cap \langle y \rangle = \varnothing$.
\end{proof}
\begin{proof}[Proof of Proposition ***]
The idea is to prove the existence of a nondegenerate,
alternating, bilinear pairing $\beta : \textrm{Ker}(\lambda) \times \textrm{Ker}(\lambda) \ra
\Q / \Z$ and then to use lemma ***
Let $m$ be an integer that kills $\textrm{Ker}(\lambda)$.
Define
$$
e^\lambda : \textrm{Ker}(\lambda) \times \textrm{Ker}(\lambda) \ra \mathbb{\mu}_m \subset
\overline{k}^*
$$
in the following way: suppose that $P, P' \in \textrm{Ker}(\lambda)$. Choose a point
$Q \in A(\overline{k})$, such that $mQ = P'$ and let
$$
e^\lambda(P, P') := \overline{e}_m(P, \lambda Q),
$$
where $\overline{e}_m : A[m] \times \Adual[m] \ra \mathbb{\mu}_m$ is the Weil pairing.
The pairing is well defined, since $m(\lambda Q) = \lambda(mQ) = \lambda(P') = 0$.
Moreover, it is nondegenerate, alternating and bilinear, because of the properties of
the Weil pairing. Thus, we can apply lemma *** to get
$\textrm{Ker}(\lambda) \cong H \times H$.
\end{proof}
Using the above proposition, we can define the \emph{modular degree}.
\begin{defn}
The \emph{modular degree} of $A_f$ is defined as
$$
\textrm{moddeg}(A_f) = \sqrt{\textrm{deg}(\theta_f)},
$$
where $\theta_f : A_f^{\vee} \ra A_f$ is the dual isogeny.
\end{defn}
There is an explicit algorithm for computing the modular degree of a modular
abelian variety, attached to a newform. For an abelian group $G$, we denote by
$G^*$ the group Hom$_\Z(G, \Z)$. The Hecke algebra acts on the space of modular
symbols $H_1(X_0(N), \Z)$ and on its dual. There is a natural restriction map
$$
r_f : H_1(X_0(N), \Z)^*[I_f] \ra (H_1(X_0(N), \Z)[I_f])^*.
$$
It turns out that the modular degree is related to the cokernel of this map.
\begin{prop}
There exists an isomorphism
$$
\textrm{coker}(r_f) \simeq \textrm{ker}(\theta_f).
$$
\end{prop}
\begin{proof}
\end{proof}
\subsection{Intersecting Complex Tori}
In this subsection, we discuss how to compute intersections of abelian varieties. The whole
idea is pretty straightforward if one thinks of the varieties as complex tori.
Suppose that $V$ is a finite dimension vector space over $\C$ and $\Lambda$ be a lattice
in $V$. One can contruct the complex tori $T = V / \Lambda$. Suppose that $V_A$ and $V_B$
are vector subspaces of $V$. The lattices $\Lambda_A = V_A \cap \Lambda$ and
$\Lambda_B = V_B \cap \Lambda$ give us complex subtori $A = V_A / \Lambda_A$ and
$B = V_B \cap \Lambda_B$ of $T$. The following proposition gives us an explicit way to
compute the intersection group of $A$ and $B$, using only the lattices $\Lambda$,
$\Lambda_A$ and $\Lambda_B$.
\begin{prop}
Suppose that $A \cap B$ is finite. Then
$$
A \cap B \cong \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\textrm{tors}}.
$$
\end{prop}
\begin{proof}
Since $A \cap B$ is finite, then $V_A \cap V_B = \varnothing$.
There is a map $A \oplus B \ra T$ given by
$(v_A + \Lambda_A) + (v_B + \Lambda_B) \mapsto (v_A-v_B) + \Lambda$. The kernel of
consists precisely of pairs $(x, x)$, where $x \in A \cap B$. Indeed,
$(v_A + \Lambda_A) + (v_B + \Lambda_B)$ is in the kernel if and only if
$v_A - v_B \in \Lambda$, which means precisely that the points $x = v_A + \Lambda_A$ and
$y = v_B + \Lambda_B$ viewed as points in $T$. Therefore, we have an exact sequence
$$
0 \ra A \cap B \ra A \oplus B \xrightarrow{(x,y) \mapsto x - y} T
$$
One can construct the following commutative diagram with exact rows and columns:
\begin{equation*}
\xymatrix{
& & 0 \ar[d] & A\cap B \ar[d] & \\
0 \ar[r] & \Lambda_A \oplus \Lambda_B \ar[r]\ar[d] & V_A \oplus V_B \ar[r]\ar[d] & A \oplus B \ar[r]\ar[d] & 0\\
0 \ar[r] & \Lambda \ar[r]\ar[d] & V \ar[r]\ar[d] & T \ar[r]\ar[d] & 0 \\
& \Lambda / \left(\Lambda_A + \Lambda_B\right) & V / \left(V_A + V_B\right) & T / (A+B) &
}
\end{equation*}
Using the snake lemma, we obtain an exact sequence
$$
0 \ra A \cap B \ra \Lambda / (\Lambda_A + \Lambda_B) \ra V / (V_A + V_B)
$$
Finally, observe that $V/ (V_A + V_B)$ is a $\C$-vector space and therefore has no torsion.
Therefore, the kernel of $\Lambda / (\Lambda_A + \Lambda_B) \ra V / (V_A + V_B)$ contains
$\ds \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\textrm{tors}}$. Conversely, it is
easy to check that any element which is not torsion is mapped to a nonzero element of
$V / (V_A + V_B)$. Therefore
$\ds A \cap B \cong \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\textrm{tors}}$.
\end{proof}
The proposition can be applied to compute intersections of modular abelian varieties.
Indeed, consider the modular Jacobian $J_0(N)$ as an abelian variety over $C$.
The tangent space at the identity is precisely $V=$ Hom$(S_2(\Gamma_0(N)), \C)$. By considering
$\Lambda = H_1(X_0(N), \Z)$ and the integration pairing, we get $J_0(N)(\C) = V / \Lambda$.
Let $f$ and $g$ be newforms, which are not Galois conjugates and let $I_f$ and $I_g$ be the
annihilators of $f$ and $g$ in the Hecke algebra $\mathbb{T}$. Let $A = A_f^{\vee}$ and
$B = B_f^{\vee}$. Then $V_A = V[I_f]$ and $V_B = V[I_g]$ are the tangent spaces at the
identity to $A$ and $B$. According to the above proposition, we have
$$
A \cap B \cong \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right),
$$
where $\Lambda_A = \Lambda[I_f]$ and $\Lambda_B = \Lambda[I_g]$.
\subsection{Producing a Multiple of the Order of the Torsion Subgroup}
We first consider some methods for providing upper bounds on the size of the torsion
subgroup of a modular abelian variety $A := A_f$, attached to a newform $f$, which is a
normalized eigenform, i.e. $\ds f = q + \sum_{n = 1}^\infty a_nq^n$.
The basic idea for bounding the size of the torsion group $A_{tors}(\Q)$ is to inject
the torsion subgroup into the group of $\mathbb{F}_p$-rational points of the reduction of $A$
for various primes $p$. We start with the following
\begin{prop}
Suppose that $A$ is a modular abelian variety, which is a quotient of $J_0(N)$ and
$p \nmid 2N$ is a prime. Then there exists an injective map
$$
A(\Q)_{\textrm{tors}} \ra A_{\mathbb{F}_p}(\mathbb{F}_p).
$$
\end{prop}
\begin{rem}
The above statement is not true for arbitrary number fields. For example, *** COMPLETE ***
\end{rem}
Using the above proposition, one can get an upper bound on the torsion, by taking the greatest
common divisor of all $|A_{\mathbb{F}_p}(\mathbb{F}_p)|$, where $p$ runs over all primes, for
which $p \nmid N$. In short,
$$
|A(\Q)_{\textrm{tors}}| \leq \textrm{gcd}\{|A_{\mathbb{F}_p}(\mathbb{F}_p)|\ :\ \forall p
\nmid 2N \}.
$$
To complete the computation of the upper bound, we need an algorithm for computing the
order of the group $A_{\mathbb{F}_p}(\mathbb{F}_p)$. The observation is that the
$\mathbb{F}_p$-rational points can be recovered as the fixed points of the Frobenius
automorphism, acting on $A_{\mathbb{F}_p}(\mathbb{\overline{F}_p})$. Indeed,
the automorphism $\textrm{Frob}_p : \mathbb{\overline{F}}_p \ra \mathbb{\overline{F}}_p$ that
sends $x \mapsto x^p$ induces an automorphism $\textrm{Frob}_p : A_{\mathbb{F}_p}
(\mathbb{\overline{F}}_p) \ra A_{\mathbb{F}_p}(\mathbb{\overline{F}}_p)$. Thus,
$$
|A_{\mathbb{F}_p}(\mathbb{F}_p)| = |\textrm{ker}(1 - \textrm{Frob}_p)|.
$$
A useful tool for computing degrees, such as the above one are characteristic polynomials.
Since we cannot define characteristic polynomials of an endomorphism of an abelian variety,
we should somehow relate this automorphism to an automorphism of vector spaces, or
modules of finite rank.
Indeed, it is helpful to introduce the $\ell$-adic Tate module. Indeed, it is defined
as the inverse limit
$$
T_{\ell} A := \varprojlim_n A[\ell^n],
$$
taken with respect to the natural map
$$
A[\ell^{n+1}] \xrightarrow{\cdot l} A[\ell^{n}].
$$
Let $\varphi : A \ra A$ be any element of End$(A)$. There is an induced homomorphism
$\varphi_l : T_\ell A \ra T_\ell A$.
\begin{lem}
For any $\varphi \in \textrm{End}(A)$,
$$
\textrm{deg}(\varphi) = |\textrm{det}(\varphi_l)|.
$$
\end{lem}
\begin{proof}
This is proved in [Mil, \S12, Prop. 12.9].
\end{proof}
Thus, all we need to do is compute the characteristic polynomial of the Frobenius
homomorphism, acting on the $\ell$-adic Tate module $T_\ell A$. This is achieved in the
following
\begin{prop}
Let $F_p$ be the characteristic polynomial of the homomorphism
$\textrm{Frob}_p : T_\ell A \ra T_\ell A$. Then
$$
F_p(x) = \prod_{\sigma : K_f \hookrightarrow \overline{\Q}} \left(x^2 - \sigma(a_p)x + p\right).
$$
\end{prop}
\begin{proof}
The statement is proved in [Shi, Ch. 7].
\end{proof}
Finally, let $G_p(x)$ be the characteristic polynomial of multiplication by $a_p$ on
the Tate module $T_\ell A$. Then
$$
F_p(x) = x^{[K_f : \Q]}G_p\left(x + \frac{p}{x}\right).
$$
But the polynomial $G_p(x)$ is easily computable from the coefficients of $f$. Therefore,
$F_p(x)$ is computable.
Finally, we obtain
$$
|A_{\mathbb{F}_p}(\mathbb{F}_p)| = |\textrm{det}(1 - \textrm{Frob}_p)| = |F_p(1)| = |G_p(1+p)|.
$$
This gives the explicit way of computing the number of $\mathbb{F}_p$-rational points on the
reduced variety $A_{\mathbb{F}_p}$.
\subsection{Producing a Divisor of the Order of the Torsion Subgroup}
Producing a lower bound for the order of the torsion subgroup is more subtle than
the upper bound. We start by the following
\begin{defn}
The \emph{rational cuspidal subgroup} $C$ is defined as the subgroup
of $J_0(N)(\Q)_{tors}$, generated by the divisors of the form
$(\alpha) - (\infty)$, where $\alpha$ is a rational cusp for $\Gamma_0(N)$
(recall that a divisor is called $\Q$-rational if it is fixed by the action of
the absolute Galois group Gal$(\overline{\Q} / \Q)$).
\end{defn}
Introducing the group $C$ is very useful for our purposes, since the size of its image
in $A(\Q)_{tors}$ produces a divisor of the order of the torsion subgroup.
We compute a list of cusp representatives for all the cusps for $\Gamma_0(N)$, using the
following
\begin{prop}
\end{prop}
Next, we compute a sublist of all $\Q$-rational cusps. This can be done, provided we know the
action of Gal$(\overline{\Q} / \Q)$ on the cusps for $\Gamma_0(N)$. This is done by Stevens
*** CITE ***, and the essential result is contained in the following proposition:
\begin{prop}
(i) The cusps of $X_0(N)$ are rational over $\Q(\zeta_N)$ (i.e. they are fixed by the
elements of Gal$(\overline{\Q} / \Q(\zeta_N))$). \\
(ii) The absolute Galois group Gal$(\overline{\Q} / \Q)$ acts on the cusps for $\Gamma_0(N)$
through the subgroup Gal$(\Q(\zeta_N) / \Q) = (\Z / N\Z)^{\times}$. The element
$d \in (\Z / N\Z)^{\times}$ acts on the cusp representative $x / y$ by
$x/y \mapsto x / (d'y)$, where $d'$ is the
multiplicative inverse of $d$ in $(\Z / N\Z)^{\times}$, i.e. $dd' \equiv 1$ (mod $N$).
\end{prop}
\begin{proof}
\end{proof}
Next, we compute the subgroup $\mathcal C$ of the space of modular symbols of weight 2 for
$\Gamma_0(N)$, generated by the rational cusps. In other words, $\mathcal C$ is the
space of all symbols $\{\alpha, \infty\}$, where $\alpha$ is a $\Q$-rational
cusp. The image of $C$ in $A_f(\Q)_{tors}$ is isomorphic to the image of
$\mathcal C$ in the quotient group
$$
P := \Phi_f(\mathfrak M_2 (\Gamma_0(N))) / \Phi_f(\mathfrak S_2(\Gamma_0(N))),
$$
where $\Phi_f$ is the integration pairing, defined by
$$
\Phi_f : \mathfrak M_2(\Gamma_0(N)) \ra \textrm{Hom}(S_2(\Gamma_0(N)), \C),\
\{\alpha, \beta\} \mapsto \left\{f \mapsto \int_\gamma f\right\},
$$
where $\gamma$ is a path, representing the homology class $\{\alpha, \beta\}$.
\subsection{Computation of the Tamagawa Numbers}
Let $A$ be an abelian variety over a number field $K$ and let $\mathcal A$ be its
N\'eron model. The closed fiber of the N\'eron model at a place $\nu$ is a commutative group
scheme, which we
denote by $\mathcal A_{\mathbb{F}_\nu}$ (by $\mathbb{F}_\nu$ we mean the residue field at
$\nu$). This scheme is not necessarily connected, so we will denote by
$\mathcal A_{\mathbb{F}_\nu}^0$ the connected component of the identity.
\begin{prop}
The identity component of the fiber over $\nu$ satisfies
$$
\mathcal A_{\mathbb{F}_\nu}^0 \cong A_{\mathbb{F}_\nu}^{\textrm{ns}}(\mathbb{F}_\nu),
$$
where $A_{\mathbb{F}_\nu}^{\textrm{ns}}$ is the subvariety of the reduced variety
$A_{\mathbb{F}_\nu}$, consisting of the points of nonsingular reduction. If $A_{\textrm{ns}}$
is the subvariety of $A$, consisting of all points with nonsingular reduction at $\nu$, then
$$
\mathcal A_{\mathbb{F}_\nu} / \mathcal A_{\mathbb{F}_\nu}^0 \cong A(K_\nu) / A_{\textrm{ns}}(K_\nu).
$$
In particular, if $A$ has a nonsingular reduction at $\nu$, then the there is only one component
of $A_{\mathbb{F}_\nu}$, which is the identity component.
\end{prop}
\begin{defn}
The \emph{component group} $\Phi_{A, \nu}$ of the N\'eron model is defined to be the quotient
$\mathcal A_{\mathbb{F}_\nu} / \mathcal A_{\mathbb{F}_\nu}^0$. The order of the component group
$c_{A, \nu}$ is called the \emph{Tamagawa number} of $A$ at $\nu$.
\end{defn}
For example, if $A$ has a nonsingular reduction at $\nu$ then $c_{A, \nu} = 1$.
There is no general algorithm to compute the Tamagawa numbers. There exists, however, an
algorithm for the case of elliptic curves and it is known as Tate's algorithm. A detailed
exposition is given in *** CITE *** and *** CITE ***.
For the case of modular abelian varieties over $\Q$, there is a known algorithm
to compute $|\Phi_{A, \nu}(\mathbb{\overline{F}}_p)|$ in the case when
$p \parallel N$. Furthermore, it is possible to compute
$|\Phi_{A, \nu}(\mathbb{F}_p)|$ up to power of 2 *** CITE *** and *** CITE ***.
\subsection{Computing the $L$-Ratio}
To motivate the definition and the interpretation of the $L$-Ratio, we start with the
simpler case of an elliptic curves.
First, suppose that $f \in S_2(\Gamma_0(N))$ is a newform. The $L$-function $L(f, s)$, associated to $f$
is defined via the Mellin transform
$$
L(f, s) := (2\pi)^s \Gamma(s) \int_0^{i \infty} (-iz)^s f(z) \frac{dz}{z}.
$$
It is not hard to check (by using the Fourier expansion $f = \sum_{n=1}^\infty a_nq^n$) that
$$
L(f, s) = \sum_{n = 0}^\infty \frac{a_n}{n^s}.
$$
Next, we look at the special value of the $L$-function at $s = 1$.
$$
L(f, 1) = -2\pi i \int_{0}^{i \infty} f(z)dz = -\langle \{0, \infty\}, f \rangle,
$$
where
$$
\langle, \rangle : \mathfrak M_2(\Gamma_0(N)) \times S_2(\Gamma_0(N)) \ra \C
$$
is the integration pairing, defined by
$\ds \langle [\gamma], f \rangle = 2\pi i\int_\gamma f(z)dz$.
Since the modular symbols $\{0, \infty\}$ is in the rational homology of the curve
$X_0(N)$, then $\langle \{0, \infty\}, f\rangle$ is a rational multiple of a period
of $f$. To compute that rational multiple, we use the Hecke operator $T_p$ on modular
symbols. Indeed, we have
$$
T_p \{0, \infty\} = \{0, \infty\} + \sum_{k = 0}^{p-1}\{k/p, \infty\} =
(1+p)\{0, \infty\} + \sum_{k = 0}^{p-1}\{0, k/p\}.
$$
But it is a general result about modular symbols ** CITE ** that whenever $p \nmid N$,
the symbol $\{0, k/p\}$ is integral, i.e. $\{0, k/p\} \in H_1(X_0(N), \Z)$. Thus,
$\ds \sum_{k = 0}^{p-1}\langle \{k/p, 0\}, f \rangle$ is a period of the modular form $f$.
Another observation is that it is a real period. Indeed,
$$
\overline{\langle \{0, k / p\}, f\rangle} = -\langle \{0, k / p\}, f\rangle =
\langle \{0, (p-k)/p\}\rangle,
$$
so after summing all the contribution, we see that the period
$\ds \left \langle \sum_{k = 0}^{p-1}\{k/p, \infty\}, f\right \rangle$ is real.
Now, let $\Omega(f)$ be twice the minimal real part of a period of the lattice.
Then
$$
\frac{L(f, 1)}{\Omega(f)} = \frac{n(p, f)}{2(1 + p - a_p)},
$$
where $n(p, f)$ is an integer.
Suppose that $A$ is an abelian variety, attached to a newform $f$. Consider the
$\R$-rational points on $A$.
One can make sense of the \emph{volume} of $A(\R)$, which we call
\emph{the real volume} and denote it by $\Omega_A$. Let $\mathcal A$ be the
N\'eron model for $A$ and $\mathcal A_{\R}$ be the generic fiber of $\mathcal A$.
\begin{defn}
Suppose that $\Lambda$ and $\Lambda'$ are lattices in the vector space $V$. The
\emph{lattice index} $[\Lambda : \Lambda']$ is defined to be the determinant of the
linear transformation of $V$, which takes $\Lambda$ to $\Lambda'$.
\end{defn}
Let $T_e \mathcal A_\R$ be the tangent space at the identity of $\mathcal A_\R$.
Fix a lattice $\Lambda$
for the tangent space and declare that the complex torus $T_e \mathcal A_\R / \Lambda$ has
a unit measure. Let $A(\R)^0$ be the connected
component at the identity of $A(\R)$. Then
$A(\R)^0 = T_e \mathcal A_\R / H_1(\mathcal A_R, \Z)$.
We can define a measure on $A(\R)^0$ by using the lattice index
$[\Lambda : H_1(A(\R), \Z)]$. Indeed, set
$$
\mu_\Lambda(A(\R)^0) := [\Lambda : H_1(\mathcal A_\R, \Z)].
$$
This is, of course, a relative measure, which depends on the choice of $\Lambda$.
Finally, we can define the volume of $A(R)$ as
$$
\mu_\Lambda(A(\R)) = \mu_\Lambda(A(\R)^0) \cdot c_{\infty},
$$
where $c_{\infty} = |\mathcal A_\R / \mathcal A_\R^0|$ is the number of connected
components of the generic fiber of the N\'eron model $\mathcal A$.
\begin{defn}
The \emph{real volume} of the abelian variety $A$ is defined as
$$
\Omega_A := \mu_\Lambda(A(\R)),
$$
where $\Lambda$ is the dual lattice to the lattice defined by the N\'eron
differentials $H^0(\mathcal A, \Omega_{A / \Z})$ in the cotangent space
$T_e^*\mathcal A$.
\end{defn}
\begin{thm}
The following formula allows us to compute the $L$-ratio
$$
\frac{L(A, 1)}{\Omega_A} = \frac{1}{c_\infty \cdot c_A} \cdot
[\Phi(H_1(X_0(N), \Z))^+ : \Phi(\mathbb{T}\{0, \infty\})].
$$
\end{thm}
Before proving the theorem, we need the following
\begin{lem}
\end{lem}
\begin{proof}[Proof of theorem ***]
\end{proof}
\subsection{Producing Visible Elements of the Shafarevich-Tate Group}
Here we describe a technique which produces visible elements of $\Sha$. The basic
idea is that under certain conditions, $\Sha(A)$ will contain a weak Mordell-Weil group
of some abelian variety, which in turn will produce element of certain order of $\Sha$.
The precise statement is the following
\begin{thm}
Suppose that $A$ and $B$ are abelian subvarieties of $J$, which have finite intersection
over $\overline{\Q}$. Let $N$ be the product of all primes of bad reduction for $A$ and $B$
(in the case of modular abelian varieties, this is simply the level $N$). Let $p$ be a prime,
which satisfies the following conditions:
\begin{itemize}
\item $p \nmid N \cdot |(J/B)(\Q)_{tors}| \cdot |B(\Q)_{tors}|
\cdot \prod_{q}c_{A, q} \cdot c_{B, q}$, where $c_{A, q}$ and $c_{B, q}$ are the Tamagawa
numbers.
\item $B(\overline{\Q})[p] \subset A(\overline{\Q})$, where both $A(\overline{\Q})$ and
$B(\overline{\Q})$ are viewed as subgroups of $J(\overline{\Q})$.
\end{itemize}
Under these hypothesis, there is a natural map
$$
\varphi : B(\Q) / pB(\Q) \ra \Sha(A),
$$
such that the dimension of the kernel of $\phi$ over $\F_p$ does not exceed the Mordell-Weil
rank of $A$. In particular, the map is injective if the Mordell-Weil rank of $A$ is 0.
\end{thm}
\begin{proof} There are two major steps for the proof of the theorem. First, we construct
a map from the weak Mordell-Weil group $B(\Q) / pB(\Q)$ to the
visible part of $H^1(K, A)$, using the hypothesis that $B(\overline{\Q})[p] \subset
A(\overline{\Q})$. The second step is proving that the image of $B(\Q) / pB(\Q)$
in $H^1(K, A)$ consists of locally trivial cohomology classes, which immediately implies
that this image is contained in Vis$_J^{(i)}(\Sha(A))$. \\
\noindent \textit{\bf Step 1:} \textit{Constructing a map $B(\Q) / pB(\Q) \ra H^1(K, A)$.} \\
The argument we will use is purely algebraic and is based on diagram chasing. Start with the
short exact sequence
$$
0 \ra A \ra J \ra C \ra 0,
$$
where $C$ is simply the quotient $J / A$, considered over $K$. The associated long exact
sequence on Galois cohomology is
\begin{equation}\label{eqn:les}
0\ra A(K) \ra J(K) \ra C(K) \xrightarrow{\,\delta\,}
H^1(K,A) \ra \cdots.
\end{equation}
One can construct a map $\psi : B \ra C$ by composing the inclusion map $B \hookrightarrow J$
and the map $J \ra C$. Since $B[p] \subset A$ and $A$ is the kernel of the map $J \ra C$, then
the map $\psi : B \ra C$ factors through the multiplication by $p$ map
$B \xrightarrow{\, \cdot p\,} B$. This gives us the following commutative diagram:
\begin{equation}\label{eqn:bigcd}
\xymatrix{
& & B \ar[d] \ar[r]^{.p} \ar[dr]^{\psi} & B\ar[d] \\
0 \ar[r] & A \ar[r] & J \ar[r] & C
}
\end{equation}
We still have not used the fact that $B(\Q)[p]$ is empty. We take
$\Q$-rational points and use this fact to get the following diagram, with exact rows
and columns:
\begin{equation}\label{eqn:bigcd}
\xymatrix{
& K_0\ar[d] & K_1\ar[d]& K_2\ar[d]\\
0 \ar[r] & B(\Q) \ar[r]^{.p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d]
& B(\Q)/pB(\Q)\ar[r]\ar[d]^{\varphi} & 0\\
0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\
& K_3
}
\end{equation}
By definition, the visible part of $H^1(K, A)$ is the kernel of the
map $H^1(\Q, A) \ra H^1(\Q, J)$, which by the long exact sequence on Galois cohomology is
exactly the image of the map $H^0(\Q, C) \xrightarrow{\,\delta\,} H^1(\Q, A)$, which is
$\delta(C(K))$. Now, we apply the snake lemma to get an exact sequence
$$
K_0 \ra K_1 \ra K_2 \ra K_3.
$$
We can analyze further the sequence, by observing that $K_1$ is finite. Indeed, $B \ra C$
has finite kernel by the definition of $C$ as the quotient $J/A$ and also $A \cap B$ has finite
intersection. Therefore, $K_1 \subset B(\Q)_{tors}$. But $B(\Q)$ does not contain $p$-torsion
elements. Since $K_2 \subset B(\Q) / pB(\Q)$ is $p$-torsion, then $K_1 \ra K_2$ must
necessarily be the zero map. Therefore, $K_2 \ra K_3$ is an injective map. \\
\noindent {\it {\bf Step 2:} The Local Analysis.} \\
In the previous step we constructed a map $\varphi : B(K)/pB(K) \ra H^1(K, A)$. Consider the
composition $\pi : B(K) \ra H^1(K, A)$ of the quotient map $B(K) \ra B(K)/pB(K)$ and the map
$\varphi$. Let $x \in B(K)$ be a $K$-rational point. To show that the image of $B(K)/pB(K)$
under $\varphi$ lies in $\Sha(A)$, it suffices to show that for each place $\nu$, the
restriction res$_\nu(\pi(x)) = 0$.
We prove this by considering the different possibilities for $\nu$: \\
\noindent {\it {\bf Case 1:} $\nu$ is real archimedian.} Don't quite understand this case -
need to do some more work. \\
\noindent {\it {\bf Case 2:} $p \ne char(\nu)$.}
Let $m$ be the order of the component group $\Phi_{B, \nu}(\mathbb{F}_\nu)$ of the closed
fiber $\mathcal B_{\mathbb{F}_\nu}$ at $\nu$ of the Neron model $\mathcal B$
(i.e. the Tamagawa number $c_{B, \nu}$). Then $mx$ is in the identity component
$\mathcal B_{\mathbb{F}_\nu}^0$. Hence, we can apply *** LEMMA *** to get that there
exists $z \in B(K_\nu^{ur})$, such that $pz = mx$. Now, look at
res$_\nu(\pi(mx)) \in H^1(K_\nu, A(K_\nu))$. It follows (cf. cohomological interpretation of
Kummer pairing) that this cohomology class is represented by the cocycle
$$
f : \textrm{Gal}(\overline{K_\nu}/K_\nu) \ra A(\overline{K_\nu}),\ \sigma \mapsto \sigma(z) - z.
$$
Since $z \in B(K_\nu^{ur})$, it follows that $f$ is unramified cocycle, i.e.
res$_\nu(\pi(mx))$ is an unramified cohomology class. Thus, res$_\nu(\pi(mx)) \in
H^1(K_\nu^{ur} / K_\nu, A(K_\nu^{ur}))$.
Next, we need to prove the following duality lemma, which gives us a relationship between the
unramified cohomology and the the cohomology of a component group.
\begin{lem}[Duality Lemma]
Let $A$ be an abelian variety over a local field $K$, which is the fraction field of a
discrete valuation ring $R$ with residue field $k$.
Let $\mathcal A$ be the N\'eron model of $A$ and $\Phi(\overline{k})$ be the component
group of the closed fiber $\mathcal A_k$ of $\mathcal{A}$, i.e.
$\Phi(\overline{k}) := \mathcal{A}_{k}(\overline{k}) / \mathcal{A}_{k}^0(\overline{k})$.
Then
$$
H^1(K^{ur}/K, A(K^{ur})) = H^1(K^{ur}/K, \Phi(\overline{k})),
$$
where $K^{ur}$ denotes the maximal unramified extension of $K$.
\end{lem}
It follows from the duality lemma that
$$
H^1(K_\nu^{ur}/K_\nu, A(K_\nu^{ur})) = H^1(K_\nu^{ur}/K_\nu,
\Phi_{A, \nu}(\overline{\mathbb{F}_\nu})).
$$
The cohomology of the component group is easier to work with, because the component
group is a finite Gal$(K_\nu^{ur}/K_\nu)$-module.
\begin{lem}
Suppose that $G$ is a cyclic group and $A$ is a finite $G$-module. Let
$$
h(A) := |H^0(G, A)| / |H^1(G, A)|
$$
be the Herbrand quotient of the $G$-module $A$. Then $h(A) = 1$.
\end{lem}
\begin{proof}
Let $g$ be a generator for $G$ and $A^G$ denote the fixed submodule of $A$. Let
$I_G$ denote the kernel of the homomorphism $\Z[G] \ra G$, which maps $g \ra 1$.
There is an exact sequence
$$
0 \ra A^G \ra A \xrightarrow{\cdot(g-1)} A \ra A / I_GA \ra 0.
$$
Since $A$ is a finite module, it follows that $|A/I_GA| = |A^G|$. Next, let
$N : A \ra A$ be the homomorphism, obtained by multiplication by
$\ds N = \sum_{h \in G}h$. This homomorphism induces a homomorphism
$N^* : A / I_GA \ra A^G$. Moreover, we have an exact sequence
$$
0 \ra H^1(G, A) \ra A/I_GA \xrightarrow{N^*} A^G \ra H^0(G, A) \ra 0,
$$
which immediately implies that $|H^1(G, A)| = |H^0(G, A)|$, so $h(A) = 1$.
\end{proof}
Next, using [Serre, VIII.4.8] we get that the order of $H^1(K_\nu^{ur}/K_\nu,
\Phi_{A, \nu}(\overline{\mathbb{F}_\nu}))$ is equal to the order of the component group
$\Phi_{A, \nu}(\overline{\mathbb{F}_\nu})$. But
$p \nmid c_{A, \nu} = |\Phi_{A, \nu}(\overline{\mathbb{F}_\nu})|$ by assumption. But the
order of res$_\nu(\pi(mx))$ divides $p$. Hence, the order of res$_\nu(\pi(mx))$ is 1, which
means that $m$res$_{\nu}(\pi(x)) = 0$. Since the order of $p\pi(x)$ and
$p \nmid m = c_{B, \nu}$, then it follows that res$_{\nu}(\pi(x)) = 0$, i.e. $\pi(x)$ is a
locally trivial element. \\
\noindent {\it {\bf Case 3:} $char(\nu) = p$.}
Consider the maximal unramified extension $K_\nu^{ur}$ of $K_\nu$.
Let $\mathcal A$, $\mathcal J$ and $\mathcal C$ be the
Neron models of $A, J$ and $C$ respectively.
The first observation is that the induced sequence on the N\'eron models
$$
0 \ra \mathcal A \ra \mathcal J \xrightarrow{\psi} \mathcal C \ra 0
$$
is exact. Indeed, one can derive this using the following lemma:
\begin{lem}
Suppose that $0 \ra A' \ra A \ra A'' \ra 0$ is an exact sequence of abelian
varieties over a field $K$, which is the fraction field of a discrete valuation ring $R$.
Assume that the ramification index $e = \nu(p)$ satisfies $e < p-1$, where $p$ is the residue
characteristic and $\nu$ is the normalized valuation on $R$. Let $\mathcal A'$, $\mathcal A$
and $\mathcal A''$ be the N\'eron models of $A'$, $A$ and $A''$ respectively. If $\mathcal A$
has abelian reduction, then the induced sequence
$$
0 \ra \mathcal A' \ra \mathcal A \ra \mathcal A'' \ra 0
$$
is exact and consists of abelian $R$-schemes.
\end{lem}
\begin{proof}
This is proved in [BLR, \S 7.5, Thm 4.].
\end{proof}
Therefore, the sequence of the N\'eron models is exact. Hence,
$\psi : \mathcal J \ra \mathcal C$ is a flat morphism, which is surjective and with
a smooth kernel $\mathcal A$. This is enough to claim that $\psi$ is smooth
[BLR, \S 2.4, Prop. 8]. Next, using *** lemma ***, it follows that $\mathcal J(R) \ra
\mathcal C(R)$ is surjective, and therefore
$\mathcal J(K^{ur}_\nu) \ra \mathcal C(K^{ur}_\nu)$ is surjective. Therefore,
res$_\nu(\pi(x))$ is a unramified cohomology class. Using the duality lemma, we have
$$
H^1(K^{ur}_\nu / K_\nu, A) \cong H^1(K^{ur}_\nu / K_\nu, \Phi_{A, \nu}(\overline{\mathbb{F}_\nu}))
$$
But $A$ has good reduction at $\nu$, since $p \nmid N$, so $\Phi_{A, \nu}(\overline{\mathbb{F}}_\nu)$ is trivial. Therefore, $H^1(K^{ur}_\nu / K_\nu, \Phi_{A, \nu}(\overline{\mathbb{F}_\nu}))$
is trivial, so res$_\nu(\pi(x)) = 0$. This completes the proof of the visibility theorem.
\end{proof}
\section{Examples of Visible Elements.}
\subsection{A 20-Dimensional Quotient of $J_0(389)$.}
Consider the cuspidal subspace $S_2(\Gamma_0(389))$. One can use the correspondence
between Galois conjugacy classes of newforms and modular abelian varieties, attached
to newforms (quotients of $J_0(389)$) to recover the quotients of $J_0(389)$.
Using modular symbols, it is possible *** CITE *** to decompose the new subspace of
the cuspidal subspace $S_2(\Gamma_0(389))$ into Galois conjugacy classes.
There are five abelian varieties in the decomposition of dimensions 1, 2, 3, 6, 20, which we
denote by $A_1, A_2, A_3, A_6, A_{20}$ respectively. First of all, we can apply the
algorithm for computing the $L$-ratio from *** SECTION ***
to verify that $L(A_1, 1) = L(A_2, 1) = L(A_3, 1) = L(A_6, 1) = 0$, but
$L(A_{20}, 1) \ne 0$. More precisely, the algorithm, applied to $A_{20}$ gives us
$$
c_A \cdot \frac{L(A_{20}, 1)}{\Omega_A} = \frac{2^{11} \cdot 5^2}{97},
$$
where $c_A$ is the Manin constant. But according to *** THEOREM ***, the Manin constant
is $c_A = 2^{?}$. Thus, we recover that
$$
L(A_{20}, 1) = \frac{2^{?} \cdot 5^2}{97}.
$$
Next, we can compute the bounds on the order of the torsion subgroup
$A_{20}(\Q)_{\textrm{tors}}$. For the upper bound, we try only the primes 3 and 5 to get
a bound 97. For the lower bound, we compute the rational cuspidal subgroup, which turns out
to be cyclic of order 97. Therefore $|A_{20}(\Q)_{\textrm{tors}}| = 97$.
The Birch and Swinnerton-Dyer conjecture states that
$$
\frac{L(A, 1)}{\Omega_A} = \frac{|\Sha(A)| \cdot \prod_{p \mid N} c_p}
{|A(\Q)_{\textrm{tors}}| \cdot |A^{\vee}(\Q)_{\textrm{tors}}|}
$$
\subsection{Visibility After Level Raising for an Elliptic Curve.}
\subsection{Evidence for the Birch and Swinnerton-Dyer Conjecture for an 18-Dimensional
Quotient of $J_0(551)$.}
Similarly to the case of $J_0(389)$, we compute the newform quotients of $J_0(551)$.
There are four elliptic curves $E_1$, $E_2$, $E_3$ and $E_4$, and abelian varieties
$A_2$, $A_3$, $A_{16}$ and $A_{18}$ of dimensions 2, 3, 16 and 18. We will be interested
in studying the 18-dimensional quotient $A_{18}$.
First, we look at the polarization $\theta : A_{18}^{\vee} \ra A_{18}$. Using the ideas
from *** CITE ***, the modular kernel has order $|\textrm{ker}(\theta)| = 2^{14} \cdot 13^4$.
This shows that the only odd primes $p$, for which one might get visible elements
of $\Sha(A_{18})$ in $J_0(551)$ is $p = 13$.
We next compute the torsion bounds for the order of $A_{18}(\Q)_{\textrm{tors}}$. The
algorithm for the upper bound gives us $|A_{18}(\Q)_{\textrm{tors}}| \leq 80$ if we look at
all primes up to 31 (without the prime 31, the bound is 160). The order of the rational
cuspidal subgroup is 40. Therefore, $|A_{18}(\Q)_{\textrm{tors}}| = 40$ or 80.
Notice that the order of the modular kernel is not divisible by 5, i.e.
the degree of the isogeny $\theta : A_{18}^{\vee} \ra A_{18}$ is not divisible by 5.
Therefore, the order of the torsion subgroup $|A_{18}^{\vee}(\Q)_{\textrm{tors}}|$ must
necessarily be divisible by 5.
The most difficult part is to compute the Tamagawa numbers for the abelian variety $A_{18}$.
We can use the techniques from *** CITE ***. Since $551 = 19 \cdot 29$ then we need to
compute $c_{A_{18}, 19}$ and $c_{A_{18}, 29}$. The second number is $c_{A_{18}, 29} = 40$,
but the algorithm does not work for the first one. Instead, we compute the component group
order over $\overline{\mathbb{F}}_{19}$, which is $2^2 \cdot 13^2$ and
conclude that $c_{A_{18}, 19} = 2$ or 4 *** EXPLAIN ***.
We also use the algorithm to compute
$$
\frac{L(A_{18}, 1)}{\Omega_{A_{18}}} = c_A \cdot \frac{2^2 \cdot 3^2}{5}.
$$
Since $c_A \mid 2^{\textrm{dim}(A)}$ by *** CITE ***, then it follows that
$$
\frac{L(A_{18}, 1)}{\Omega_{A_{18}}} = \frac{2^{n+2} \cdot 3^2}{5},
$$
for some $0 \leq n \leq 18$. We have all the quantities for the strong Birch and
Swinnerton-Dyer conjecture, so we obtain
$$
\frac{2^{n+2} \cdot 3^2}{5} = \frac{|\Sha(A_{18})| \cdot 2^m \cdot (2^3 \cdot 5)}
{(2^k \cdot 5) \cdot (2^l \cdot 5)},
$$
where $1 \leq m \leq 2$, $3 \leq k \leq 4$ and $0 \leq l \leq 4$. Thus, it follows
that $|\Sha(A_{18})| = 2^s\cdot 3^2$ for some $2\leq s \leq 24$, so there is no chance to
get visible elements for $A_{18}$ inside $J_0(551)$.
We saw that the conjectural order of $\Sha(A_{18})$ is divisible by $3^2$. Although we
produced no visible elements, there is a way of proving that $\Sha(A_{18})$ has a
subgroup of order 9 by using the following algorithm:
\begin{enumerate}
\item Let $A$ be a quotient of $J_0(N)$. Choose a prime number $\ell \nmid N$ and
consider the degeneracy maps $\alpha^*, \beta^* : J_0(N) \ra J_0(\ell N)$.
\item By Shimura's construction, $A^{\vee}$ is a subvariety of $J_0(N)$, so we can consider
its image in $J_0(\ell N)$ under a linear combination of the degeneracy maps. For instance,
look at $C = \alpha^*(A^{\vee}) + \beta^*(A^{\vee})$.
\item Compute explicitely (using modular symbols) the image of the variety $A_{18}^{\vee}$
in $J_0(\ell N)$.
\item If possible, apply the visibility theorem $C$ (or some other technique) to prove that $C$
contains a subgroup of the form $B(\Q) / pB(\Q)$ for some abelian variety $B$.
\item To prove that $\Sha(A)$ has element, whose order is divisible by $p$, it suffices
to show that the degree of the isogeny, which is the composition of the maps
$$
A \ra A^{\vee} \ra A^{\vee} \times A^{\vee} \xrightarrow{\alpha^* + \beta^*} C
$$
is not divisible by $p$ (and then to perform analysis on the cochain level to show that
the kernel of $\Sha(A) \ra \Sha(C)$ does not have an element of order 3). This can be
done by noting that the kernel is annihilated by the operators $T_r|_A - (r+1)$ for
all primes $r \nmid N$, so it suffices to show that the order of the kernel of some of
these operators is not divisible by $p$.
\end{enumerate}
To perform the above algorithm in practice, we choose the prime $\ell = 2$. By decomposing
the modular Jacobian $J_0(2\cdot551)$ as a product of quotients, we notice that five of the
factors are elliptic curves, one of which is the rank 2 elliptic curve $E$ with
Weierstrass equation
$$
E : y^2 + xy = x^3 + x^2 - 29x + 61.
$$
We can use the techniques from section *** CITE *** to compute the intersection of $E$ with $C$
and verify that $C$ contains $E[3]$. We wish to apply the visibility theorem for
$A = C$ and $B = E$ and $p = 3$. Indeed, the only hypothesis that we have to check is that the
Tamagawa numbers for $B$ are not divisible by $3$. But Tate's algorithm *** CITE *** computes
these numbers for elliptic curves. Using this algorithm, one checks that
$c_{B, 2} = 2$, $c_{B, 19} = 2$ and $c_{B, 29} = 1$. Since the Mordell-Weil rank of $C$ is
zero, then there is an injection
$$
B(\Q) / 3B(\Q) \hookrightarrow \Sha(C).
$$
Finally, we need to check that the degree of the above isogeny $\varphi : A \ra C$ is not
divisible by 3. To do this, we compute the kernel of the operator $T_3|_A - (3 + 1)$. It is
possible to perform the last step in practice, because one knows precisely how
the Hecke operator acts on the space of modular symbols. The kernel is
$$
\textrm{ker}(T_3|_A - (3+1)) = 12625812402998886400 = 2^{14} \cdot 5^2 \cdot 5552003^2,
$$
which is not divisible by 3. Therefore, 3 does not divide the order of the kernel of the
isogeny $A_{18} \ra C$, so $\Sha(A_{18})$ contains an element of order 3.
\subsection{Existence of Visible Element of Order 3 for a Quotient of $J_0(767)$}
We provide a computational example of an invisible element of $\Sha$ becomes visible
after level-raising. Consider the Jacobian $J_0(767)$ of the modular curve $X_0(767)$.
We consider a modular abelian variety $A_f$, which is a quotient of $J_0(767)$ and is attached
to a newform $f$. In practice, one uses the fact that abelian varieties $A_f$, which are
quotients of $J_0(N)$ correspond precisely to the Galois conjugacy classes of newforms.
These in turn correspond to the decomposition subspaces of $H_1(X_0(N), \Q)$ (or the space
of modular symbols $\mathfrak M_2(\Gamma_0(N))$) under the action of the Hecke operator.
Using the modular symbols package in the computer algebra system MAGMA, we find these
abelian varieties. For the case $N = 767$, there are six varieties of dimensions
2, 3, 4, 10, 17 and 23. Consider the largest quotient $A$, which is the 23 dimensional
abelian variety. We will study in more detail the Shafarevich-Tate group for this particular
abelian variety.