Abstract
Let p be a prime not dividing the integer n. By an Ihara result, we mean existence of a cokernel torsion-free injection from a full lattice in the space of p-old modular forms into a full lattice in the space of all modular forms of level pn. In this paper, we will prove an Ihara result in the number field case, for Siegel modular forms. The case of elliptic modular forms is discussed in Ihara (Discrete subgroups of Lie groups and applications to moduli, Oxford University Press, Bombay, 1975). We will use a geometric formulation for the notion of p-old Siegel modular forms (Rastegar in BIMS 43(7):1–23, 2017). Then, we apply an argument by Pappas, and prove the Ihara result using density of Hecke orbits (Chai in Invent Math 121(3):439–479, 1995). This result is meant to pave the way for modularity results in higher genera.
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Acknowledgements
We have benefited from conversations with G. Pappas, A. Rajaei, C. Skinner, R. Takloo-Bighash, R. Taylor, and A. Wiles. We shall thank Princeton University, IPM and Institute for Advanced Study. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1128155.
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Communicated by Amir Akbary.
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Rastegar, A. Ihara-Type Results for Siegel Modular Forms. Bull. Iran. Math. Soc. 46, 693–716 (2020). https://doi.org/10.1007/s41980-019-00285-5
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DOI: https://doi.org/10.1007/s41980-019-00285-5