A p-adic completion of Zagier's Eisenstein series

TL;DR

This paper demonstrates that Zagier's weight 3/2 mock Eisenstein series can be extended to a p-adic modular form for any odd prime p, revealing a new p-adic completion analogous to its harmonic Maass form completion.

Contribution

It introduces a novel p-adic completion of Zagier's Eisenstein series, bridging mock modular forms and p-adic modular forms in a new way.

Findings

Zagier's series admits a p-adic modular form completion for odd primes

The p-adic completion resembles its harmonic Maass form counterpart

Establishes a new connection between mock Eisenstein series and p-adic modular forms

Abstract

This note points out that for any odd prime $p$, Zagier's weight $3/2$ mock Eisenstein series can be completed to a $p$-adic modular form in a way that bears some resemblance to its completion to a harmonic Maass form.