The effective Chebotarev density theorem and modular forms modulo $\mathfrak m$

Abstract

Abstract. Suppose that f (resp. g) is a modular form of integral (resp. halfintegral) weight with coefficients in the ring of integers O K of a number field K. For any ideal m ⊂ O K , we bound the first prime p for which f | T p (resp. g | T p 2 ) is zero (mod m). Applications include the solution to a question of Ono (2001) concerning partitions.

On Fourier coefficients of elliptic modular forms $\bmod \, \ell$ with applications to Siegel modular forms

“…It is however not clear how one can ensure that ν ℓ (• • • ) = 0 without using 'analytic' inputs as in our paper, it does not help that we do not have much control on the set {n j } from [17]. If analytic inputs are used, then the ensuing bound on ℓ would be similar to ours.…”

On dependence of rational points on elliptic curves

“…One can use these versions to redefine the sets S j,l , S j and S, see Remark 2.2. The following can be found as Theorem 2.2 and Proposition 2.3 in [5].…”

Effective Congruences for Mock Theta Functions

“…The cusp form g(z) is determined by a result of Treener [17]. Moreover, work of Lichtenstein [14] allows us to bound the first prime p such that T (p 2 ) annihilates g(z). The details of the construction of g(z) follow.…”

“…We end this section by recalling the action of Hecke operators T (p 2 ) on half integral weight cusp forms and stating a result of Lichtenstein [14], which will allows us to bound the smallest prime p such that g(z) | T (p 2 ) ≡ 0 (mod ℓ j ). If χ is a quadratic character, g(z) = a(n)q n ∈ M λ+1/2 (4N, χ), and (p, 4N) = 1 then g(z) | T (p 2 ) := a(p 2 n) + (−1) λ n p χ(p) p λ−1 a(n) + p 2λ−1 a(n/p 2 ) q n .…”

“…We establish the existence of infinitely many linear congruences of the formwhere A is a multiple of ℓ and an auxiliary prime p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on prior works of Lichtenstein [14] and Treneer [17].…”