On the size of 𝑝-adic Whittaker functions

“…In view of this we define t ν = (t ν,1 − t ν,2 )/2. Then the correct spectral parameter for the Eisenstein series E v (s, ·) is (λ ν (s)) ν , where λ ν (s) = 1 4 + (t ν + s) 2 if ν is real, 1 + 4(t ν + s) 2 if ν is complex.…”

Section: Set-up and Basic Definitionsmentioning

confidence: 99%

“…The strategy is to start from (A.1) and insert the expressions from[2, Lemma 2.2]. Let us first deal with some easy cases.If 0 ≤ l < a 2 , we have t = −n p + r andS t,l = µ∈X ′ l , t=−np ζ Fp (1)q −l p χ 1 (̟ a2−a1 p ) a 1 < l ≤ n, we have t = −2l + r and S t,l = µ∈X ′ l , t=−2l ζ Fp (1)q −l p χ 1 (̟ 0 p ) 2 ≤ ζ Fp (1) if r = 0, 0 else.…”

mentioning

confidence: 99%

On sup-norm bounds part II: GL(2) Eisenstein series

Assing¹

2019

Forum Mathematicum

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In this paper we consider the sup-norm problem in the context of analytic Eisenstein series for GL(2) over number fields. We prove a hybrid bound which is sharper than the corresponding bound for Maaß forms. Our results generalise those of Huang and Xu where the case of Eisenstein series of square-free levels over the base field Q had been considered. theory of Eisenstein series is used to give lower bounds for ζ(1 + it). This last method has vast generalizations. See for example [8]. Our case is reverse. We exploit that the theory of Hecke L-functions over number fields is well developed and use this to produce an improved amplifier. This leads to the remarkable fact that (as in [23]) we get sup-norm bounds which are in some aspects better than the state of the art results for cusp forms. It was already observed in [13] that one can improve the sup-norm bound in the spectral aspect if one has a better understanding of the Hecke eigenvalues. This is exactly the ingredient we gain from the classically well developed GL 1 theory. More precisely, we use highly non-trivial zero free regions for Hecke L-functions to derive an asymptotic expansion for generalized divisor sums. This will serve as a lower bound for the amplifier.Note that recently in [16] it was shown that the quality of the sup-norm bound that we achieve for Eisenstein series holds for SL 2 (Z)-Maaß forms on average.Before we can give a precise statement of our theorems we have to introduce some notation. We assume that the reader is familiar with the results and the notation in [1] and only focus on the aspects where Eisenstein series differ from cusp forms.

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“…It should be possible to extend the argument to prove a non-trivial hybrid bound (simultaneously in the depth and eigenvalue aspects) for the sup-norm; however, we do not attempt to do so here. The method of this paper can be combined with the Fourier/Whittaker expansion at various cusps in the adelic context (the necessary machinery for which is now available thanks to recent work of Assing [Ass19] building on earlier work of the second author [Sah16, Sah17]) to give a depth-aspect sub-local bound in the case (possibly with a different exponent than in Theorem A due to some differences in the counting argument). Finally, this paper provides a general strategy of how one should go about improving the local bound in the level aspect in cases where the local vectors are not sufficiently localized.…”

Section: Introductionmentioning

confidence: 99%

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

Hu¹,

Saha²

2020

Compositio Math.

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We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

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On the size of 𝑝-adic Whittaker functions

Cited by 10 publications

References 14 publications

On sup-norm bounds part I: ramified Maaß newforms over number fields

On sup-norm bounds part I: ramified Maaß newforms over number fields

On sup-norm bounds part II: GL(2) Eisenstein series

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

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