Traces of Hecke operators

Knightly, Andrew; Li, Charles

doi:10.1090/surv/133/01

Cited by 11 publications

(30 citation statements)

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“…Comparing [8, Propositions 5.2.2 and 5.3.1] and [20,Proposition 3.46] shows that the Hecke operators defined above coincide with those of [8]. Thus if (dq, N M ) = 1 we have that…”

Section: Trace Formulasmentioning

confidence: 95%

“…One can check that the diamond operators and Hecke operators commute with each other and that

T_{p}

respects the decomposition

S_{k} false(normalΓ (N, M) false) = ⨁_{χ (mod N)} S_{k} false(Γ_{0} (N M), χ false) .

Comparing [, Propositions 5.2.2 and 5.3.1] and [, Proposition 3.46] shows that the Hecke operators defined above coincide with those of . Thus if

(d q, N M) = 1

we have that

0true Tr (false⟨ d false⟩ T_{q} | S_{k} false(normalΓ (N, M) false)) = \sum_{χ false(prefixmod N false)} χ (d) Tr (T_{q} | S_{k} false(Γ_{0} (N M), χ false)) .

We use to prove a trace formula for Hecke operators on the groups

Γ (N, M)

.…”

Section: Trace Formulasmentioning

confidence: 96%

“…We will prove Theorem using the trace formula for

{normalΓ}_{0} false(N false)

with nebentypus. We refer to for the proof and also to for the statement. Theorem Let

k ⩾ 2

and

χ false(- 1 false) = {(- 1)}^{k}

.…”

Section: Trace Formulasmentioning

confidence: 99%

“…We will prove Theorem 9 using the trace formula for Γ 0 (N ) with nebentypus. We refer to [20] for the proof and also to [21] for the statement.…”

Section: Trace Formulasmentioning

confidence: 99%

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Elliptic curves over a finite field and the trace formula

Kaplan

Petrow

2017

Proc. London Math. Soc.

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Abstract. We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2 (Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2 (Z) which include as special cases the groups Γ 1 (N ) and Γ(N ). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup, and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z) 2 and congruence subgroups Γ 0 (2), Γ 0 (4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vlǎduţ, Gekeler, Howe, and others.

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“…Comparing [8, Propositions 5.2.2 and 5.3.1] and [20,Proposition 3.46] shows that the Hecke operators defined above coincide with those of [8]. Thus if (dq, N M ) = 1 we have that…”

Section: Trace Formulasmentioning

confidence: 95%

“…One can check that the diamond operators and Hecke operators commute with each other and that

T_{p}

respects the decomposition

S_{k} false(normalΓ (N, M) false) = ⨁_{χ (mod N)} S_{k} false(Γ_{0} (N M), χ false) .

Comparing [, Propositions 5.2.2 and 5.3.1] and [, Proposition 3.46] shows that the Hecke operators defined above coincide with those of . Thus if

(d q, N M) = 1

we have that

0true Tr (false⟨ d false⟩ T_{q} | S_{k} false(normalΓ (N, M) false)) = \sum_{χ false(prefixmod N false)} χ (d) Tr (T_{q} | S_{k} false(Γ_{0} (N M), χ false)) .

We use to prove a trace formula for Hecke operators on the groups

Γ (N, M)

.…”

Section: Trace Formulasmentioning

confidence: 96%

“…We will prove Theorem using the trace formula for

{normalΓ}_{0} false(N false)

with nebentypus. We refer to for the proof and also to for the statement. Theorem Let

k ⩾ 2

and

χ false(- 1 false) = {(- 1)}^{k}

.…”

Section: Trace Formulasmentioning

confidence: 99%

“…We will prove Theorem 9 using the trace formula for Γ 0 (N ) with nebentypus. We refer to [20] for the proof and also to [21] for the statement.…”

Section: Trace Formulasmentioning

confidence: 99%

See 2 more Smart Citations

Elliptic curves over a finite field and the trace formula

Kaplan

Petrow

2017

Proc. London Math. Soc.

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“…where X is the Chebyshev polynomial of degree defined by X (2 cos θ) = sin(( +1)θ) sin θ (see, e.g., [KL1,Prop. 29.8], where ω corresponds to ψ −1 ).…”

Section: Preliminaries On Modular Formsmentioning

confidence: 99%

Weighted Distribution of Low-lying Zeros of GL(2) -functions

Knightly¹,

Reno²

2019

Can. J. Math.-J. Can. Math.

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We show that if the zeros of an automorphic L-function are weighted by the central value of the L-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the L-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted L-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

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Fields of rationality of cusp forms

Binder

2017

Isr. J. Math.

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Abstract. In this paper, we prove that for any totally real field F , weight k, and nebentypus character χ, the proportion of Hilbert cusp forms over F of weight k and character χ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible GL 2 representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for GL 2 ; and third, a Plancherel equidsitribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin-Templier and mirrors work of Finis-Lapid-Mueller by introducing an explicit bound for certain families of orbital integrals.

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Traces of Hecke operators

Cited by 11 publications

References 0 publications

Elliptic curves over a finite field and the trace formula

Elliptic curves over a finite field and the trace formula

Weighted Distribution of Low-lying Zeros of GL(2) -functions

Fields of rationality of cusp forms

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