On The Structure Of Prime-Detecting Quasimodular Forms In Higher Levels

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 ℓ-adic Galois representations, we prove that any prime-detecting quasimodular form on Γ 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.