The $p$-adic constant for mock modular forms associated to CM forms

TL;DR

This paper investigates the $p$-adic constant associated with mock modular forms linked to CM forms, establishing its non-vanishing as a $p$-adic unit for inert primes under certain conditions.

Contribution

It proves that the $p$-adic constant $eta_{g}(f)$ is a $p$-adic unit when $p$ is inert in the CM field and the space of cusp forms is one-dimensional.

Findings

$eta_{g}(f)$ is a $p$-adic unit for inert primes $p$.

The result applies when the space of cusp forms $S_{k}( olinebreak ext{Gamma}_0(N))$ has dimension 1.

The paper extends understanding of $p$-adic constants for mock modular forms in CM cases.

Abstract

Let $g \in S_{k}(\Gamma_{0}(N))$ be a normalized newform and $f$ be a harmonic Maass form that is good for $g$. The holomorphic part of $f$ is called a mock modular form and denoted by $f^{+}$. For odd prime $p$, K. Bringmann, P. Guerzhoy, and B. Kane obtained a $p$-adic modular form of level $pN$ from $f^{+}$ and a certain $p$-adic constant $\alpha_{g}(f)$. When $g$ has complex multiplication by an imaginary quadratic field $K$ and $p$ is split in $\mathcal{O}_{K}$, it is known that $\alpha_{g}(f)$ is zero. On the other hand, we do not know much about $\alpha_{g}(f)$ for an inert prime $p$. In this paper, we prove that $\alpha_{g}(f)$ is a $p$-adic unit when $p$ is inert in $\mathcal{O}_{K}$ and $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N))=1$.