Examples of nontrivial Manin constants

Earlier this was a section; I made it into a subsection, since it is short. -AmodWe present two sets of examples in which the Manin constant is not $1$ .

I rewrote this subsection, and kept a copy of William's original version after it. Feel free to pick the one you like. -AmodUsing results of [Kil02], Adam Joyce [Joy05] proves that there is a new optimal quotient of $J_0(431)$ with Manin constant $2$ . Joyce's methods also produce examples with Manin constant $2$ at levels $503$ and $2089$ . For the convenience of the reader, we breifly discuss his example at level $431$ . There are exactly two elliptic curves $E_1$ and $E_2$ of prime conductor $431$ , and $E_1\cap E_2 = 0$ as subvarieties of $J_0(431)$ , so $A=E_1 \times E_2$ is an optimal quotient of $J_0(431)$ attached to a saturated ideal $I$ . If $f_i$ is the newform corresponding to $E_i$ , then one finds that $f_1\equiv f_2\pmod{2}$ , and so $g = (f_1 - f_2)/2 \in S_2({\bf {Z}})[I]$ . However $g$ is not in the image of $\H ^0(A_{{\bf {Z}}},\Omega^1_{A/{\bf {Z}}})$ . Thus the Manin constant of $A$ is divisible by $2$ .

As another class of examples, one finds by computation for each prime $\ell\leq 100$ that $W_\ell$ does not leave $S_2(\Gamma_0(11\ell);{{\bf {Z}}_{(\ell)}})$ stable. Theorem 3.5 (with $I=0$ ) then implies that the Manin constant of $J_0(11\ell)$ is divisible by $\ell$ for these values of $\ell$ .