On the trace formula for Hecke operators on congruence subgroups, II
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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 \(\Gamma _0(N)\) and \(\Gamma _1(N)\), 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 \(\Gamma _1(N)\): we determine explicit trace cusp forms for \(\Gamma _0(4)\) with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level N tends to infinity.
KeywordsTrace formula Hecke operators Holomorphic modular forms
Mathematics Subject Classification11F11 11F25
This work was partly supported by the European Community Grant PIRG05-GA-2009-248569 and by the CNCS-UEFISCDI Grant TE-2014-4-2077. Part of this work was completed during several visits at MPIM in Bonn, whose support I gratefully acknowledge. It is a great pleasure to dedicate this paper to Don Zagier on the occasion of his 65th birthday.
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