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

  • Mohammad Masih Hamidi
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{hamidi2026comparison,
  title = {{On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties}},
  author = {Mohammad Masih Hamidi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.23244},
  eprint = {2607.23244},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.23244},
}
RIS
TY  - UNPB
TI  - On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties
AU  - Mohammad Masih Hamidi
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2607.23244
UR  - https://arxiv.org/abs/2607.23244
AB  - Let E be an elliptic curve over ℚ. We prove that the degree of the Shimura parametrization of E arising from the optimal quotient J^D(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 J^D(M) and E is surjective when D>1, and we relate this phenomenon to the failure of multiplicity one.
ER  -