On the trace formula for Hecke operators on congruence subgroups, II

Abstract

In a previous paper, we obtained a general trace formula for double coset operators acting on modular forms for congruence subgroups, expressed as a sum over conjugacy classes. Here we specialize it to the congruence subgroups 0 (N) and 1 (N), obtaining explicit formulas in terms of class numbers for the trace of a composition of Hecke and Atkin-Lehner operators. The formulas are among the simplest in the literature and hold without any restriction on the index of the operators. We give two applications of the trace formula for 1 (N): we determine explicit trace cusp forms for 0 (4) with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level N tends to infinity.

Computing classical modular forms

“…A comprehensive treatment with references is the book of Knightly-Li [56], and a tidy presentation is given by Schoof-van der Vlugt [83, Theorem 2.2]; proofs of the trace formula with different developments continue, see e.g. Popa [72]. This method has been implemented in Pari/GP [70] by Belabas-Cohen [4] and in a standalone implementation by Bober, described in §7.2.…”

An elementary proof of the Eichler–Selberg trace formula

“…Remark. The formula in Corollary 1 is generalized to modular forms on congruence subgroup Γ with Nebentypus in [9], using the same operator T n as in Theorem 1 acting on the space of period polynomials for Γ. The trace on the Eisenstein subspace is also explicitly computed there, yielding a simple formula for the trace of a composition of arbitrary Hecke and Atkin-Lehner operators on cusp forms for Γ 0 (N ).…”

“…The proof given here depends on purely algebraic properties of a special Hecke element, independent of its action on period polynomials. The same Hecke element has been used by the first author in two sequels to this paper to obtain simple trace formulae on modular forms for congruence subgroups as well [8,9]. Our approach is more elementary than the classical automorphic kernel method, and applies uniformly in all weights, whereas the classical approach requires additional technicalities in weight two.…”