Traces of Hecke Operators

Knightly, Andrew; Li, Charles

doi:10.1090/surv/133

Cited by 40 publications

(31 citation statements)

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“…For the fixed-central-character trace formula, it is worth noting that the versions of the trace formula for GL 2 stated, for instance, in [GJ79, (7.14)-(7.19)], [Shi63], [KL06,Theorem 22.1] and [Pal12] are all fixed-central-character versions. However, we believe it is easiest to take the version from [Art02] as a starting point since it fits most nicely into the framework of [Art88] and [Art89], and the geometric terms of trace formulae in these papers are easiest to manage.…”

Section: Representation-theoretic Results For Fixed Central Charactermentioning

confidence: 99%

“…We also need a fixed-central-character version of the trace formula. For GL 2 , the fixed-central-character trace formula is classical and has been stated in [Shi63], [GJ79], [KL06], [Pal12] and elsewhere; a more general invariant fixedcentral-character version has been stated in [Art02]. Versions of Arthur's (non-fixed central character) trace formula in [Art88] and [Art89], however, are significantly more 'user friendly' in that it is easier to bound the noncentral terms.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Representation-theoretic Results For Fixed Central Charactermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fields of rationality of cusp forms

Binder

2017

Isr. J. Math.

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“…Traditionally, the non-elliptic contribution is written in a different form than the above (cf. [GJ79], [Gel96], [KL06]; in the case at hand we sum over all characters of Z.Q/nZ.A/ 1 ). Namely, the regular hyperbolic contribution is written as the sum over t 2 T .Q/ n Z.Q/ of…”

Section: The Main Resultsmentioning

confidence: 99%

“…The purpose of this paper is to resolve the problem for the group G D GL.2/ over Q, where a completely explicit form of the trace formula is available (see [GJ79], [Gel96], [KL06] for detailed accounts). The restriction to Q is merely for the sake of exposition.…”

mentioning

confidence: 99%

On the Arthur–Selberg trace formula for $\mathrm{GL}(2)$

Finis¹,

Lapid²

2011

Groups Geom. Dyn.

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“…where we group the curves into isomorphism classes, as discussed in Section 2.1. On the other hand, using the Eichler-Selberg trace formula (see [12] or [11]), for k ≥ 4,…”

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confidence: 99%

The Error Term in the Sato–tate Theorem Of birch

Murty¹,

Prabhu²

2019

Bull. Aust. Math. Soc.

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We establish an error term in the Sato–Tate theorem of Birch. That is, for$p$prime,$q=p^{r}$and an elliptic curve$E:y^{2}=x^{3}+ax+b$, we show that$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$for any interval$I\subseteq [0,\unicode[STIX]{x1D70B}]$, where the quantity$\unicode[STIX]{x1D703}_{a,b}$is defined by$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$and$\unicode[STIX]{x1D707}_{ST}(I)$denotes the Sato–Tate measure of the interval$I$.

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Traces of Hecke Operators

Cited by 40 publications

References 0 publications

Fields of rationality of cusp forms

Fields of rationality of cusp forms

On the Arthur–Selberg trace formula for $\mathrm{GL}(2)$

The Error Term in the Sato–tate Theorem Of birch

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