Weight 2 CM newforms as p-adic limits

TL;DR

This paper demonstrates that all weight 2 CM newforms that are eta quotients can be expressed as p-adic limits of derivatives of Weierstrass mock modular forms, extending previous results using new techniques.

Contribution

It introduces a method to represent weight 2 CM newforms as p-adic limits of derivatives of mock modular forms, broadening the scope of prior results.

Findings

All weight 2 CM newforms as eta quotients are p-adic limits.

These forms are limits of derivatives of Weierstrass mock modular forms.

Extends previous results using holomorphic modular forms techniques.

Abstract

Previous works have shown that certain weight $2$ newforms are $p$-adic limits of weakly holomorphic modular forms under repeated application of the $U$-operator. The proofs of these theorems originally relied on the theory of harmonic Maass forms. Ahlgren and Samart obtained strengthened versions of these results using the theory of holomorphic modular forms. Here, we use such techniques to express all weight $2$ CM newforms which are eta quotients as $p$-adic limits. In particular, we show that these forms are $p$-adic limits of the derivatives of the Weierstrass mock modular forms associated to their elliptic curves.