Tables

We computed all $19608$ abelian varieties $A=A_f$ attached to newforms of level $N\leq 2333$. Suppose that $A$ is one of the $10360$ of these for which $L(A,1)\neq 0$, so Conjecture 2.2 asserts that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ has order

$$\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m... ...\char93 A(\mathbf{Q})\cdot \char93 A^{\vee}(\mathbf{Q})}{\prod_{p\mid N} c_p}.$$

For any rational number $x$, let $x^{\op}$ be the odd part of $x$. Define integers $S_l$ and $S_u$ such that that

$$S_l\mid \numer(\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}_?^{\oddpart}) \mid S_u$$

as follows:

$\mathbf{S}_u$

The upper bound $S_u$ is the odd part of the numerator of

$$\frac{L(A,1)}{\tilde{\Omega}_A} \cdot \frac{T^2}{\prod_{p\mid\mid N} c_p},$$

where $T$ is the upper bound on $\char93 A(\mathbf{Q})$ and $\char93 A^{\vee}(\mathbf{Q})$ computed using Section 3.5 using $a_p$ for $p\leq 17$. Since the Manin constant and the Tamagawa numbers are integers, $S_u$ is an upper bound on the odd part of $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}_?$.

$\mathbf{S}_l$

The lower bound $S_l$ is defined as follows: Let $S_{l,1}$ be the odd part of the rational number

$$\frac{L(A,1)}{\tilde{\Omega}_A} \cdot \frac{\char93 C\cdot D}{\prod_{p\mid\mid N} c_p},$$

where $C\subset A(\mathbf{Q})_{\tor}$ and $D$ is the part of $C$ coprime to the modular degree of $A$. Usually $C$ is generated by the image of $(0)-(\infty)$, though we should have let $C$ be the subgroup generated by all rational cusps, but didn't realize this until it was too late. When $A$ is an elliptic curve, $C=A(\mathbf{Q})_{\tor}$.

If $N$ is square free, we let $S_l=S_{l,1}$ and are done. Otherwise, let $S_{l,2}$ be the largest part of $S_{l,1}$ coprime to all primes whose square divides $N$. This takes care of the Manin constant, which only involves primes whose square divides $N$. To take care of Tamagawa numbers, remove all primes $p\leq 2\dim(A)+1$ from $S_{l,2}$ to obtain $S_l$.

Remark 5.1   When $N$ is square free we have

$$S_l\mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}_?^{\oddpart} \mid S_u$$

since $c_A$ is a power of $2$ and no Tamagawa numbers have been omitted from the formulas for $S_l$ and $S_u$. For every $N\leq 2333$ we found that $S_l$ is an integer, so when $N\leq 2333$ is squarefree, $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}_?^{\oddpart}$ is an integer. Since Conjecture 2.2 asserts that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}_?$ is the order of a group, hence an integer, our data gives evidence for Conjecture 2.2.

Tables 1-4 list every $A$ of level $N\leq 2333$ such that $S_l>1$. The $A$ column contains the label of $A$ (see Section 3.2), and the next column (labeled dim) contains $\dim A$. A star next to the label for $A$ indicates that we have proved that the odd part of $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ is at least as large as conjectured by the Birch and Swinnerton-Dyer conjecture. This is the case for $33$ of the $168$ examples. The columns labeled $S_l$ contains the number $S_l$ defined above. If $S_l=S_u$ then the column labeled $S_u$ contains a dot, and otherwise, it contains $S_u$ (there are only $13$ cases in which $S_u\neq S_l$). The column labeled $\moddeg(A)^*$ contains the odd part $m$ of the modular degree of $A$, written as a product $\gcd(m,S_u)\cdot$    $m/\gcd(S_u,m)$, where $m/\gcd(S_u,m)$ is shrunk to save space. The only non-square-free levels of $A_f$ for which $S_l>1$ are $1058$, $1664$, $2224$, and $2264$.

The column labeled $B$ contains all $B$ such that $L(B,1)=0$ and $\gcd(S_l,\char93 (A^{\vee}\cap \tilde{B}^{\vee}))>1$, where $\tilde{B}^{\vee}$ is the abelian subvariety of $J_0(N)$ generated by all images under the degeneracy maps of  $B^{\vee}=A_g^{\vee}$, where $g$ is a newform of level dividing $N$. Thus, e.g., when $B^{\vee}$ is of level $N$, $\tilde{B}^{\vee}=B^{\vee}$. The next column, labeled $\dim$, contains the dimension of $B$.

The final two columns contain information about the relationship between $A$ and $B$. The one labeled $A^{\vee}\cap \tilde{B}^{\vee}$ contains the abelian group structure of the indicated abelian group, where e.g., $[a^bc^d]$ means the abelian group $(\mathbf{Z}/a\mathbf{Z})^b\times (\mathbf{Z}/c\mathbf{Z})^d$. The column labeled $\Vis$ contains a divisor of the order of $\Vis_{C}({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A^{\vee}))$, where $C=A^{\vee} + \tilde{B}^{\vee}$ (note that $\Vis_{C}({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{... ...g{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A^{\vee}))$).

The table is divided into three vertical regions, with the columns in the first region about $A$ only, the columns of the second region about about $B$ only, and the third column about the relationship between $A$ and $B$.