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

On the trace formula for Hecke operators on congruence subgroups, II 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 $$\Gamma _0(N)$$ Γ 0 ( N ) and $$\Gamma _1(N)$$ Γ 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 $$\Gamma _1(N)$$ Γ 1 ( N ) : we determine explicit trace cusp forms for $$\Gamma _0(4)$$ Γ 0 ( 4 ) with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level N tends to infinity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by SpringerNature
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
D.O.I.
10.1007/s40687-018-0125-5
Publisher site
See Article on Publisher Site

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 $$\Gamma _0(N)$$ Γ 0 ( N ) and $$\Gamma _1(N)$$ Γ 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 $$\Gamma _1(N)$$ Γ 1 ( N ) : we determine explicit trace cusp forms for $$\Gamma _0(4)$$ Γ 0 ( 4 ) with Nebentypus, and we compute the limit of the trace of a fixed Hecke operator as the level N tends to infinity.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jan 22, 2018

References

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