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Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms

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Abstract

We show that the systems of prime-to-p Hecke eigenvalues arising from automorphic forms\(\pmod p\) for a good prime p associated to an algebraic group \(G/{\mathbb {Q}}\) of Hodge type are the same as those arising from algebraic modular forms\(\pmod p\) associated to an inner form of G.

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Notes

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Acknowledgements

Part of this work was done during our visits in Wuhan University, the Morningside Center of Mathematics, CAS and POSTECH. We thank Jiangwei Xue and Xu Shen for their invitations and the institutions for hospitality and good research conditions. This paper relies on results of Professor Mark Kisin on Shimura varieties of Hodge type and the construction of automorphic forms by Professor Kai-Wen Lan. We also thank Professor Benedict Gross for helpful comments and sharing his preprint [10] to us. Further we thank the referee for a thorough reading of the manuscript and valuable suggestions. Yasuhiro Terakado thanks Chao Zhang for answering questions on stratifications of Hodge type Shimura varieties. Chia-Fu Yu is partially supported by the MoST grants 107-2115-M-001-001-MY2 and 109-2115-M-001-002-MY3.

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Authors and Affiliations

  1. Institute of Mathematics, Academia Sinica, Astronomy Mathematics Building, No. 1, Roosevelt Rd., Sec. 4, Taipei, 10617, Taiwan

    Yasuhiro Terakado & Chia-Fu Yu

  2. National Center for Theoretical Sciences, National Taiwan University, No. 1, Roosevelt Rd., Sec. 4, Taipei, 10617, Taiwan

    Chia-Fu Yu

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  1. Yasuhiro Terakado
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  2. Chia-Fu Yu
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Correspondence to Chia-Fu Yu.

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Communicated by Wei Zhang.

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Terakado, Y., Yu, CF. Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms. Math. Ann. 382, 69–102 (2022). https://doi.org/10.1007/s00208-021-02200-y

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