On the sharpness of the bound for the Local Converse Theorem of 𝑝-adic 𝐺𝐿_{𝑝𝑟𝑖𝑚𝑒}

Adrian, Moshe; Liu, Baiying; Stevens, Shaun; Tam, Geo Kam-Fai

doi:10.1090/bproc/32

Cited by 9 publications

(12 citation statements)

References 19 publications

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“…A natural question following Theorems 1.1, 1.2, and Corollary 1.3, is whether the bound ⌊ n 2 ⌋ is sharp. The sharpness of ⌊ n 2 ⌋ for the local converse theorem when A = C is proved in [ALST16] for the case of n being prime and p ≥ ⌊ n 2 ⌋. Thus, for ℓ-adic families of co-Whittaker representations, we also expect the bound ⌊ n 2 ⌋ is sharp.…”

Section: Introductionmentioning

confidence: 89%

On the local converse theorem and the descent theorem in families

Liu

Moss

2019

Math. Z.

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In this paper, we prove an analogue of Jacquet's conjecture on the local converse theorem for ℓ-adic families of co-Whittaker representations of GL n (F ), where F is a finite extension of Q p and ℓ = p. We also prove an analogue of Jacquet's conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an ℓadic family. These two theorems are closely related to the local Langlands correspondence in ℓ-adic families.

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Section: Introductionmentioning

confidence: 89%

On the local converse theorem and the descent theorem in families

Liu

Moss

2019

Math. Z.

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“…When n = 3, it is shown in [JPSS83] that t = 1 suffices, and Jacquet conjectured in 1999 that 1 t n 2 should suffice for arbitrary n. Jacquet's conjecture was proven in [Cha19] and [JL16], independently, for all n. We thus present Theorem 1.2 as a modanalogue of Jacquet's conjecture. It is shown in [ALST18] that the bound t n 2 cannot be improved in the K-setting.…”

Section: On the Mod-converse Theoremmentioning

confidence: 99%

Characterizing the Mod- Local Langlands Correspondence by Nilpotent Gamma Factors

Moss¹

2020

Nagoya Math. J.

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Let $F$ be a $p$ -adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell }$ , with $\ell$ different from $p$ . We define “nilpotent lifts” of irreducible generic $k$ -representations of $GL_{n}(F)$ , which take coefficients in Artin local $k$ -algebras. We show that an irreducible generic $\ell$ -modular representation $\unicode[STIX]{x1D70B}$ of $GL_{n}(F)$ is uniquely determined by its collection of Rankin–Selberg gamma factors $\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70B}\times \widetilde{\unicode[STIX]{x1D70F}},X,\unicode[STIX]{x1D713})$ as $\widetilde{\unicode[STIX]{x1D70F}}$ varies over nilpotent lifts of irreducible generic $k$ -representations $\unicode[STIX]{x1D70F}$ of $GL_{t}(F)$ for $t=1,\ldots ,\lfloor \frac{n}{2}\rfloor$ . This gives a characterization of the mod- $\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

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“…However, the sharpness of n 2 is not that obvious if we replace "generic" by "unitarizable supercuspidal" in the theorem. In the tame case, it is shown in [ALST18] that n 2 is indeed sharp for unitarizable supercuspidal representations of GL n (F ) when n is prime. For some certain families of supercuspidal representations, n 2 is no longer sharp and the GL 1 (F ) twisted Rankin-Selberg gamma factors might be enough to determine the structures of representations within these families.…”

Section: Introductionmentioning

confidence: 99%