Quasimodular forms that detect primes are Eisenstein

TL;DR

The paper identifies prime-detecting quasimodular forms as lying entirely in the Eisenstein subspace and provides an explicit description: every such form is a linear combination of derivatives $D^nH_k$ with $k\\ge 6$, where $H_k$ are distinguished Eisenstein-based quasimodular forms. The authors achieve this by decomposing mixed-weight quasimodular forms into Eisenstein and cuspidal parts and proving a fundamental lemma on the $\\ell$-adic Galois representations of cusp forms that yields maximal freedom for Hecke eigenvalues modulo large primes. This $\\\

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.

Theorems & Definitions (12)

• Remark
• Theorem 1.1
• Remark
• Corollary 1.2
• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof