Year: 2026
Type: preprint
Abstract: We relate the systems of Hecke eigenvalues arising from the (mod $p$) modular forms on the Shimura curve attached to the indefinite quaternion algebra $B$ of discriminant $δ$ over $\mathbf{Q}$ to the systems of Hecke eigenvalues arising from the (mod $p$) algebraic modular forms attached to the definite quaternion algebra $D$ of discriminant $pδ$. Moreover, we discuss details of the Hecke structure on the latter spaces, following ideas of Serre. The entire setup can be seen as a Shimura curve analogue of Serre's letter to Tate on quaternions and modular forms. The bijection between the sets of systems of Hecke eigenvalues is a special case of recent work of Terakado and Yu; the novelty of this paper is the explicit nature of the construction, allowing for finer control of its behaviour with respect to weights, as well as the results on the Hecke module structure.
Source: arXiv (Cornell University)
Authors: Fam, Yiannis, Ghitza, Alexandru
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213287364
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