On $p$-adic quaternionic Eisenstein series

TL;DR

This paper investigates the nature of $p$-adic Eisenstein series for quaternionic modular groups, revealing their transformation into classical modular forms of level $p$ and identifying cases with transcendental coefficients.

Contribution

It introduces a $U(p)$ operator to connect $p$-adic Eisenstein series with classical forms and demonstrates the existence of transcendental coefficient examples.

Findings

Certain $p$-adic Eisenstein series become classical modular forms of level $p$

Existence of $p$-adic Eisenstein series with transcendental coefficients

Previous examples were algebraic, this extends understanding of their nature

Abstract

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).