On the trace formula for Hecke operators on congruence subgroups

Popa, Alexandru

doi:10.1090/proc/13896

Cited by 3 publications

(19 citation statements)

References 13 publications

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“…We perform this computation for an arbitrary Fuchsian group of the first kind with cusps, both because the formula may be of independent interest, and because we expect the resulting formula for the cuspidal trace to hold for such a Fuchsian group as well. The formula we obtain for the Eisenstein trace matches the hyperbolic split terms in the formula we proved in [9] for (1.1), which allows us to obtain a very general formula for the cuspidal trace as well.…”

Section: Introductionsupporting

confidence: 71%

“…To state the main result of [9], let p w (t, n) be the Gegenbauer polynomial, defined by the power series expansion (1 − tx + nx 2 ) −1 = w 0 p w (t, n)x w . Theorem 1 [9] Let be a finite index subgroup of 1 , k 2 an integer, χ a character of with kernel of finite index in , and a double coset of such that …”

Section: A General Trace Formula On the Cuspidal Subspacementioning

confidence: 99%

“…In [9], we gave a short proof of a trace formula for Hecke operators on modular forms of weight k 2 for finite index subgroups of SL 2 (Z). The formula expresses the following combination of traces…”

Section: Introductionmentioning

confidence: 99%

“…The approach he initiated in [19] for proving the trace formula for SL 2 (Z), based on computing the action of Hecke operators on period polynomials, was brought to completion in the joint paper [10], and generalized to congruence subgroups in [9]. Here we show, based on the formula we proved in [9], that simple formulas in terms of the extended class numbers hold for any congruence subgroup satisfying a mild assumption (Assumption 2.1), which is verified by the congruence subgroups considered in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In Sect. 2 we state the main results: after recalling the main result of [9] in Theorem 1, the general trace formula on the cuspidal space is stated in Theorem 2. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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

Popa

2018

Res Math Sci

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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 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.

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Section: Introductionsupporting

confidence: 71%

Section: A General Trace Formula On the Cuspidal Subspacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Sect. 2 we state the main results: after recalling the main result of [9] in Theorem 1, the general trace formula on the cuspidal space is stated in Theorem 2. In Sect.…”

Section: Introductionmentioning

confidence: 99%

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

Popa

2018

Res Math Sci

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A generalization of Ramanujan's congruence to modular forms of prime level

Gaba

Popa

2018

Journal of Number Theory

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We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup Γ 0 (N ), and on a version of Ihara's lemma.

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An elementary proof of the Eichler–Selberg trace formula

Popa¹,

Zagier²

2018

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We give a purely algebraic proof of the trace formula for Hecke operators on modular forms for the full modular group SL 2 (Z), using the action of Hecke operators on the space of period polynomials. This approach, which can also be applied for congruence subgroups, is more elementary than the classical ones using kernel functions, and avoids the analytic difficulties inherent in the latter (especially in weight two). Our main result is an algebraic property of a special Hecke element that involves neither period polynomials nor modular forms, yet immediately implies both the trace formula and the classical Kronecker-Hurwitz class number relation. This key property can be seen as providing a bridge between the conjugacy classes and the right cosets contained in a given double coset of the modular group.

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On the trace formula for Hecke operators on congruence subgroups

Cited by 3 publications

References 13 publications

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

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

A generalization of Ramanujan's congruence to modular forms of prime level

An elementary proof of the Eichler–Selberg trace formula

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