Quasimodular forms that detect primes are Eisenstein

TL;DR

This paper proves that prime-detecting quasimodular forms are essentially Eisenstein series by employing $ ext{l}$-adic Galois representations, providing an alternative to previous analytic proofs.

Contribution

It offers a new proof that prime-detecting quasimodular forms are Eisenstein, using Galois representations instead of analytic methods.

Findings

Prime-detecting quasimodular forms are Eisenstein series.

Alternative proof via $ ext{l}$-adic Galois representations.

Supports the conjecture on the structure of prime-detecting forms.

Abstract

MacMahon's partition functions and their extensions provide equations that identify prime numbers as solutions. These results depend on the theory of (mixed weight) quasimodular forms on $SL_2(\mathbb{Z})$. Two of the authors, along with Craig, conjectured an explicit description of the set of prime-detecting quasimodular forms in terms of Eisenstein series and their derivatives. Kane et al.\ recently verified this conjecture using analytic methods. We offer an alternative proof using the theory of $\ell$-adic Galois representations associated to modular forms.