On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$. We prove that the degree of the Shimura parametrization of $E$ arising from the optimal quotient $J^D(M)\to E$ is equal to the $D$-new modular degree of $E$, under mild assumptions on the local Galois representations of $E$. As an application, we prove a conjecture of Deines asserting the equality of the $D$-new modular degree, the $D$-new congruence number, and the Shimura degree of $E$. We introduce the Shimura congruence number into this framework and prove, under the same assumptions, that all four quantities coincide; moreover, the two congruence numbers are always equal. We establish new cases of multiplicity one for Shimura Jacobians and use them to prove the main theorems, following a strategy introduced by Agashe, Ribet, and Stein. Building on these results, we obtain new examples of the failure of multiplicity one. Finally, we give a negative answer to a question of Papikian and Rabinoff asking whether the functorial map between the component groups of the N\'eron models of $J^D(M)$ and $E$ is surjective when $D>1$, and we relate this phenomenon to the failure of multiplicity one.