On the structure of prime-detecting quasimodular forms in higher levels

TL;DR

This paper investigates the structure of prime-detecting quasimodular forms at higher levels, proving they are composed of Eisenstein series and oldforms, and provides bounds on primes related to Fourier coefficient vanishing.

Contribution

It proves that prime-detecting quasimodular forms at higher levels are sums of Eisenstein series and oldforms, using an approach based on Galois representations, extending previous conjectures and results.

Findings

Prime-detecting quasimodular forms at higher levels are sums of Eisenstein series and oldforms.

Established an upper bound on the number of primes with vanishing Fourier coefficients for non-prime-detecting forms.

Extended the structure understanding of quasimodular forms beyond level 1.

Abstract

Craig, van Ittersum, and Ono conjectured that every prime-detecting quasimodular form of level $1$ is a quasimodular Eisenstein series. This conjecture was proved by Kane--Krishnamoorthy--Lau and by van Ittersum--Mauth--Ono--Singh independently. However, in higher levels, prime-detecting quasimodular forms need not be Eisenstein. Recently, Kane, Krishnamoorthy, and Lau formulated a natural higher level analogue of the above conjecture and proved it by analytic methods. In a similar direction, but via an alternative approach based on the independence of characters of $\ell$-adic Galois representations, we prove that any prime-detecting quasimodular form on $\Gamma_{0}(N)$ belongs to the direct sum of the spaces of quasimodular Eisenstein series and quasimodular oldforms. Moreover, for a quasimodular form $f$ that is not prime-detecting, we give an upper bound for the number of primes $p$ less than $X$ for which the $p$-th Fourier coefficient of a quasimodular form vanishes.