Periods and nonvanishing of central L-values for GL(2n)

Feigon, Brooke; Martin, Kimball; Whitehouse, D. R.

doi:10.1007/s11856-018-1657-5

Cited by 25 publications

(18 citation statements)

References 33 publications

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“…Remark 4.1. In [FMW17], Conjecture 1.4 (a reformulation of Conjecture 1 of Prasad and Takloo-Bighash in [PTB11]) is stated for general representations but only in the quaternionic case. It is checked for supercuspidal representations with extra conditions.…”

Section: The Epsilon Factor Satisfiesmentioning

confidence: 99%

Distinction of the Steinberg representation and a conjecture of Prasad and Takloo-Bighash

Chommaux¹

2019

Journal of Number Theory

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Section: The Epsilon Factor Satisfiesmentioning

confidence: 99%

Distinction of the Steinberg representation and a conjecture of Prasad and Takloo-Bighash

Chommaux¹

2019

Journal of Number Theory

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“…[Laf97], [Laf02]). • GL n × GL n -periods of cusp forms for GL n+n (for n = n or n + 1) are intensely studied (e.g., see [FMW18]). • Zagier's computation of T E -periods of Eisenstein series is generalized by…”

Section: Introductionmentioning

confidence: 99%

Automorphic forms for PGL(3) over elliptic function fields. Part 1: Graphs of Hecke operators

Alvarenga,

Lorscheid,

Júnior

2021

Preprint

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This is a first part of a series of papers in which we develop explicit computational methods for automorphic forms for GL 3 and PGL 3 over elliptic function fields. In this first part, we determine explicit formulas for the action of the Hecke operators on automorphic forms on GL 2 and GL 3 in terms of their graphs. Our primary result consists in a complete description of the graphs of degree 1 Hecke operators for GL 3 . As complementary results, we describe the 'even component' of the graphs of degree 2 Hecke operators for GL 2 and the 'neighborhood of the identity' of the graphs of degree 2 Hecke operators for GL 3 . In addition, we establish two dualities for Hecke operators for GL n and PGL n , which hold for all n, all degrees and all function fields. Dedicated to Gunther Cornelissen on the occasion of his 50th birthday. Contents Introduction 1 1. Graphs of Hecke operators 10 2. Duality 13 3. Hecke operators for elliptic curves 17 4. Graphs of rank 3 and degree 1 23 5. Proof of the main theorem 33 6. Degree 2 graphs 57 References 71

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“…It is inspired by Jacquet's new proof [19] of Waldspurger's theorem. Although such a formula has not been established in full generality, its simple form was used by Feigon-Martin-Whitehouse [14] to obtain some evidence for the conjecture of Guo-Jacquet. For applications, one needs to compare geometric sides of Guo-Jacquet trace formulae for different symmetric pairs.…”

Section: Introductionmentioning

confidence: 99%

A local trace formula for $p$-adic infinitesimal symmetric spaces: the case of Guo-Jacquet

Hua-jie¹

2021

Preprint

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We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo-Jacquet trace formulae and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe's finiteness for weighted orbital integrals and the representability of the Fourier transform of weighted orbital integrals. These local results are prepared for the comparison of regular semi-simple terms, which are weighted orbital integrals, of an infinitesimal variant of Guo-Jacquet trace formulae. Contents 1. Introduction 1 2. Notation and preliminaries 4 3. Symmetric pairs 7 4. Weighted orbital integrals 17 5. The noninvariant trace formula 21 6. Howe's finiteness for weighted orbital integrals 27 7. Representability of the Fourier transform of weighted orbital integrals 33 8. Invariant weighted orbital integrals 41 9. The invariant trace formula 47 10. A vanishing property at infinity 52 References 55 Date: May 4, 2021.Theorem 1.1 (see Theorems 5.3 and 5.12). For all f, f ′ ∈ C ∞ c (s(F )), we have the equalityThis formula results from the Plancherel formula and an analogue of Arthur's truncation process in [5]. We cannot deduce it via the exponential map as in [33, §V] for lack of a local trace formula for symmetric spaces. One needs to return to the proof of [5] instead.In Section 6, we show Howe's finiteness for weighted orbital integrals on s(F ). For r ⊆ s(F ) an open compact subgroup, denote by C ∞ c (s(F )/r) the subspace of C ∞ c (s(F )) consisting of the functions invariant by translation of r.Proposition 1.2 (see Corollaries 6.6 and 6.9). Let r be an open compact subgroup of s(F ), M ∈ L G,ω (M 0 ) and σ ⊆ (m ∩ s rs )(F ). Suppose that there exists a compact subset σ 0 ⊆ (m ∩ s)(F ) such that σ ⊆ Ad((M H )(F ))(σ 0 ). Then there exists a finite subset {X i : i ∈ I} ⊆ σ and a finite subset) such that for all X ∈ σ and all f ∈ C ∞ c (s(F )/r), we have the equalityThe proof originates from Howe's seminal work [18] which is extended to weighted orbital integrals on Lie algebras by Waldspurger. We modify the argument in [33, §IV] to make it apply to our case.In Section 7, we show that the Fourier transform of weighted orbital integrals on s(F ) is represented by locally integrable functions on s(F ).Proposition 1.3 (see Propositions 7.2 and 7.10). Let M ∈ L G,ω (M 0 ) and X ∈ (m ∩ s rs )(F ). Then there exists a locally constant function ĵG M (η, X, •) on s rs (F ) such that for all f ∈ C ∞ c (s(F )), we have

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Periods and nonvanishing of central L-values for GL(2n)

Cited by 25 publications

References 33 publications

Distinction of the Steinberg representation and a conjecture of Prasad and Takloo-Bighash

Distinction of the Steinberg representation and a conjecture of Prasad and Takloo-Bighash

Automorphic forms for PGL(3) over elliptic function fields. Part 1: Graphs of Hecke operators

A local trace formula for $p$-adic infinitesimal symmetric spaces: the case of Guo-Jacquet

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