On the structure of prime-detecting quasimodular forms in higher levels
Yeong-Wook Kwon, Youngmin Lee
TL;DR
This work analyzes prime-detecting quasimodular forms on $\Gamma_{0}(N)$ and shows that any such form lies in the direct sum of quasimodular Eisenstein series and oldforms, providing a higher-level structural refinement of the prime-detection phenomenon. The authors deploy an $\,\ell$-adic Galois-representation framework and the linear independence of $G_{\mathbb{Q}}$-characters to show that the cuspidal component must vanish, yielding $\widetilde{\Omega}_{N}=\widetilde{\Omega}_{N}\cap(\widetilde{E}(N)\oplus\widetilde{S}^{\mathrm{old}}(N))$. They also establish density-type results for primes $p$ with $a_f(p)=0$: under algebraicity and independence hypotheses, the zero-coefficient primes have density bounded by $O\big(X/\log X \epsilon(X)^{\delta}\big)$, and if some $a_f(p_0)\neq 0$ for $p_0\nmid N$, the nonvanishing primes have positive density. The methods provide a conceptual alternative to analytic proofs at higher level by tying vanishing patterns to the arithmetic of Galois representations and their independence, illuminating the structure of quasimodular forms beyond the level 1 Eisenstein case.
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 $Γ_{0}(N)$ 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.