Integrality of nearly (holomorphic) Siegel modular forms

TL;DR

This paper investigates the integrality properties of nearly holomorphic Siegel modular forms, establishing links between their Fourier coefficients, CM values, and p-adic counterparts, thus advancing understanding in modular form theory.

Contribution

It introduces nearly Siegel modular forms, proves their Fourier expansion integrality implies CM value integrality, and establishes a correspondence with nearly holomorphic forms.

Findings

Integrality of Fourier expansion implies integrality of CM values.

Existence of a one-to-one correspondence between integral nearly Siegel and nearly holomorphic forms.

CM values integrality extends to nearly overconvergent p-adic Siegel modular forms.

Abstract

In order to considering the integrality of nearly holomorphic (vector-valued) Siegel modular forms, we introduce nearly Siegel modular forms and study their integrality. We show that the integrality of nearly Siegel modular forms in terms of their Fourier expansion implies the integrality of their CM values. Furthermore, we show that there exists a one-to-one correspondence between integral nearly Siegel modular forms and integral nearly holomorphic ones. By these results, the integrality of CM values holds for nearly holomorphic Siegel modular forms and for nearly overconvergent p-adic Siegel modular forms.