2017/06/08 by Popa, Alexandru A. · 1 citation
#11F11 #11F25 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1706.02691
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 Γ0(N) and Γ1(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 Γ1(N): we determine explicit trace forms for Γ0(4) with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level N tends to infinity.