AI 中文总结
研究有理数域上椭圆曲线E，在局部伽罗瓦表示假设下，证明其志村参数化次数等于D-新模次数，应用此结果证明相关猜想，建立志村雅可比簇一重性新情况，得到一重性不成立例子，否定特定问题并关联相关现象。
AI 中文摘要
设E是有理数域上的椭圆曲线。在对E的局部伽罗瓦表示的温和假设下，我们证明了由最优商J^D(M)→E产生的E的志村参数化的次数等于E的D-新模次数。作为应用，我们证明了Deines的一个猜想，断言E的D-新模次数、D-新同余数和志村次数相等。我们将志村同余数引入此框架并证明，在相同假设下，所有四个量一致；此外，两个同余数总是相等。我们建立了志村雅可比簇的一重性新情况并用它们证明主要定理。基于这些结果，我们得到了一重性不成立的新例子。最后，我们对Papikian和Rabinoff的一个问题给出否定答案，即当D>1时，J^D(M)和E的奈龙模型的分支群之间的函子映射是否满射，并将此现象与一重性不成立联系起来。
英文摘要
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 Néron models of $J^D(M)$ and $E$ is surjective when $D>1$, and we relate this phenomenon to the failure of multiplicity one.
CommentsRevised Section 5 to clarify Helm's tensor-product convention and correct an exact multiplicity statement to the corresponding upper bound. The main theorems are unchanged
