Level $767$

Consider the modular Jacobian $J_0(767)$ . Using the modular symbols package in Magma, one decomposes $J_0(767)$ (up to isogeny) into a product of six optimal quotients of dimensions 2, 3, 4, 10, 17 and 23. The duals of these quotients are subvarieties ${A_2}, {A_3}, {A_4}, A_{10}, {A_{17}}$ and ${A_{23}}$ defined over $\mathbb{Q}$ , where $A_d$ has dimension $d$ . Consider the subvariety $A_{23}$ .

We first show that the Birch and Swinnerton-Dyer conjectural formula predicts that the orders of the groups ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ and ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23}^\vee)$ are both divisible by 9.

Proposition 6.1.1   Assume [AS05, Conj. 2.2]. Then

$$3^2 \mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fon... ...wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A^{\vee}_{23}).$$

Proof. Let $A = A^\vee_{23}$ . We use [AS05, §3.5 and §3.6] (see also [Ka81]) to compute a multiple of the order of the torsion subgroup $A(\mathbb{Q})_{{\mathrm{tor}}}$ . This multiple is obtained by injecting the torsion subgroup into the group of $\mathbb{F}_p$ -rational points on the reduction of $A$ for odd primes $p$ of good reduction and then computing the order of that group. Hence, the multiple is an isogeny invariant, so one gets the same multiple for $A^\vee(\mathbb{Q})_{{\mathrm{tor}}}$ . For producing a divisor of $\char93 A(\mathbb{Q})_{{\mathrm{tor}}}$ , we use the injection of the subgroup of rational cuspidal divisor classes of degree 0 into $A(\mathbb{Q})_{{\mathrm{tor}}}$ . Using the implementation in Magma we obtain $120 \mid \char93 A(\mathbb{Q})_{{\mathrm{tor}}} \mid 240$ . To compute a divisor of $A^\vee(\mathbb{Q})_{{\mathrm{tor}}}$ , we use the algorithm described in [AS05, §3.3] to find that the modular degree $m_A = 2^{34}$ , which is not divisible by any odd primes, hence $15 \mid \char93 A^\vee(\mathbb{Q})_{{\mathrm{tor}}} \mid 240$ .

Next, we use [AS05, §4] to compute the ratio of the special value of the $L$ -function of $A_{/\mathbb{Q}}$ at 1 over the real Néron period $\Omega_{A}$ . We obtain
$$\frac{L(A_{/\mathbb{Q}}, 1)}{\Omega_{A}} = c_{A} \cdot \frac{2^9 \cdot 3}{5}$$ , where $c_{A}\in\mathbb{Z}$ is the Manin constant. Since $c_{A} \mid 2^{\textrm{dim}(A)}$ by [ARS06] then

$$\frac{L(A_{/\mathbb{Q}}, 1)}{\Omega_{A}} = \frac{2^{n+2} \cdot 3}{5},$$

for some $0 \leq n \leq 23$ . In particular, the modular abelian variety $A_{/\mathbb{Q}}$ has rank zero over $\mathbb{Q}$ .

Next, using the algorithms from [CS01,KS00] we compute the Tamagawa number $c_{A,13} = 1920 = 2^3 \cdot 3 \cdot 5$ . We also find that $2 \mid c_{A,59}$ is a power of $2$ because $W_{59}$ acts as $1$ on $A$ , and on the component group ${\mathrm{Frob}}_{59}=-W_{59}$ , so the fixed subgroup $\Phi_{A,59}(\mathbb{F}_{59})$ of Frobenius is a $2$ -group (for more details, see [Rib90a, Prop. 3.7-8]).

Finally, the Birch and Swinnerton-Dyer conjectural formula for abelian varieties of Mordell-Weil rank zero (see [AS05, Conj. 2.2]) asserts that

$$\frac{L(A_{/\mathbb{Q}},1)}{\Omega_{A}} = \frac{\char93 {\mbox{{\... ...hbb{Q})_{{\mathrm{tor}}} \cdot \char93 A^\vee(\mathbb{Q})_{{\mathrm{tor}}}}.$$

By substituting what we computed above, we obtain $3^2 \mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ . Since $L(A_{/\mathbb{Q}}, 1) \ne 0$ , [KL89] implies that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A)$ is finite. By the nondegeneracy of the Cassels-Tate pairing, $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{... ...ontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A^\vee/\mathbb{Q})$ . Thus, if the BSD conjectural formula is true then $3^2 \mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\f... ...ntfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A^\vee)$ . $\qedsymbol$

We next observe that there are no visible elements of odd order for the embedding ${A_{23}}_{/\mathbb{Q}} \hookrightarrow {J_0(767)}_{/\mathbb{Q}}$ .

Lemma 6.1.2   Any element of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ which is visible in $J_0(767)$ has order a power of $2$ .

Proof. Since $m_{A_{23}} = 2^{34}$ , [AS05, Prop. 3.15] implies that any element of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ that is visible in $J_0(767)$ has order a power of $2$ . $\qedsymbol$

Finally, we use Theorem 5.4.2 to prove the existence of non-trivial elements of order 3 in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ which are invisible at level $767$ , but become visible at higher level. In particular, we prove unconditionally that $3 \mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ which provides evidence for the Birch and Swinnerton-Dyer conjectural formula.

Proposition 6.1.3   There is an element of order 3 in ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(\mathbb{Q}, A_{23})$ which is not visible in $J_0(767)$ but is strongly visible in $J_0(2 \cdot 767)$ .

Proof. Let $A=A_{23}$ , and note that $A$ has rank 0 , since $L(A_{/\mathbb{Q}},1)\neq 0$ . Using Cremona's database [Cre] we find that the elliptic curve

$$E:\qquad y^2 + xy = x^3 - x^2 + 5x + 37$$

has conductor $2\cdot 767$ and Mordell-Weil group $E(\mathbb{Q}) = \mathbb{Z}\oplus \mathbb{Z}$ . Also

$$c_2 = 2, c_{13} = 2, c_{59} = 1, \overline{c}_2 = 6, \overline{c}_{13} = 2, \overline{c}_{59}=1.$$

We apply Theorem 5.4.2 with $\ell=3$ and $p=2$ . Since $E$ does not admit any rational $3$ -isogeny (by [Cre]), Hypothesis 3 is satisfied. The level is square free and the modular degree of $A$ is a power of $2$ , so Hypothesis 2 is satisfied.

We have $a_3(E) = -3$ . Using Magma we find

$$\det(T_3\vert _{\overline{W}} - (-3)) \equiv 1\pmod{3},$$

which verifies the noncongruence hypothesis and completes the proof.

$\qedsymbol$