Invisible Elements that Becomes Visible at Higher Level

Consider the elliptic curve $E$ of conductor $5389=17\cdot 317$ defined by the equation

$$y^2+xy+y =x^3 - 35590x-2587197.$$

In [CM00], Cremona and Mazur observe that the BSD conjecture implies that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)=9$, but they find that $\Vis_{J_0(5389)}({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)[3])=\{0\}$. We will now verify, without assuming any conjectures, that $9\mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ and that these $9$ elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$ are visible in $J_0(5389\cdot 7)$.

First note that the mod $3$ representation $\rho_{E,3}$ attached to $E$ is irreducible because $E$ is semistable and admits no $3$-isogeny (according to [Cre]). The newform attached to $E$ is

$$f_E = q + q^2 - 2q^3 - q^4 + 2q^5 - 2q^6 - 2q^7 + \cdots,$$

and $a_7^2 = (-2)^2 \equiv (7+1)^2 \pmod{3}$, so Ribet's level-raising theorem [Rib90] implies that there is a newform $g$ of level $7\cdot 5389$ that is congruent modulo $3$ to $f_E$. This observation led us to the following proposition.

Proposition 4.2   Map $E$ to $J_0(7\cdot 5389)$ by the sum of the two maps on Jacobians induced by the two degeneracy maps $X_0(7\cdot 5389) \rightarrow X_0(5389)$. The image $E'$ of $E$ in $J_0(7\cdot 5389)$ is $2$-isogenous to $E$ and

$$(\mathbf{Z}/3\mathbf{Z})^2 \subset \Vis_{J_0(7\cdot 5389)}({\mbox... ...coding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E')).$$

Proof. It is easy to see from the discussion in [Rib90] that the kernel of the sum of the two degeneracy maps $J_0(5389) \rightarrow J_0(7\cdot 5389)$ is a group of $2$-power order, so $E'$ is isogenous to $E$ via an isogeny of degree a power of $2$.

Consider the elliptic curve $F$ defined by $y^2 - y = x^3 + x^2 + 34x - 248$. Using Cremona's programs tate and mwrank we find that $F$ has conductor $7\cdot 5389$, and that $F(\mathbf{Q}) \cong \mathbf{Z}\times \mathbf{Z}$. The Tamagawa numbers of $F$ at $7$, $17$, and $317$ are $1$, $2$, and $1$, respectively. The newform attached to $F$ is

$$f_F = q - 2q^2 + q^3 + 2q^4 - q^5 - 2q^6 - q^7 + \cdots$$

and, by [Stu87], we prove that $f_E(q) + f_E(q^7) \equiv f_F\pmod{3}$ by checking this congruence for the first $7632=[\SL_2(\mathbf{Z}):\Gamma_0(7\cdot 5389)]/6$ terms. Since $2\leq k < 3$ and $3\nmid 7\cdot 5389$, the first part of the multiplicity one theorem of [Edi92, §9] implies that $F[3] = E'[3]$.

Finally, we apply Theorem 3.1 with $A=E'$, $B=F$, $J=A+B\subset J_0(7\cdot 5389)$, $N=7\cdot 5389$, and $n=3$. It is routine to check the hypothesis. For example, the hypothesis that $J/B$ has no $\mathbf{Q}$-rational $3$-torsion can be checked as follows. Cremona's online tables imply that $E$ admits no $3$-isogeny, so $E[3]$ is irreducible. Since $J/B$ is isogenous to $E$, the representation $(J/B)[3]$ is also irreducible, so $(J/B)(\mathbf{Q})[3]=\{0\}$. Thus, by Theorem 3.1, we have $(\mathbf{Z}/3\mathbf{Z})^2 \subset \Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E')).$ To finish the proof, note that $\Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}... ...oding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E')).$ $\qedsymbol$

Since $E'$ is $2$-isogenous to $E$ and $9\mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E')$, it follows that $9\mid \char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$, as predicted by the BSD conjecture.