2026/07/25 by Mohammad Masih Hamidi
#math.NT
Let E be an elliptic curve over ℚ. We prove that the degree of the Shimura parametrization of E arising from the optimal quotient JD(M)→ 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 Néron models of JD(M) and E is surjective when D>1, and we relate this phenomenon to the failure of multiplicity one.