Large Values of Newforms on GL(2) with Highly Ramified Central Character

Saha, Abhishek

doi:10.1093/imrn/rnv259

Cited by 22 publications

(63 citation statements)

References 9 publications

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“…If π is non-spherical then the situation looks completely different. However, recent lower bounds for W π were used to justify large sup-norms of modular forms in the level aspect, see [12,15]. In [13], the author used second moments of W π to prove the, at present, best hybrid bound for the sup-norm of Maaß forms.…”

Section: Introductionmentioning

confidence: 99%

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On the size of 𝑝-adic Whittaker functions

Assing¹

2018

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

“…In [15], the author raised a question about the exact size of the functions W π , and in [12] the author put forward a conjecture for the case of a weakly ramified central character. In this paper we will answer [12,Question 1]. It turns out that [12,Conjecture 2] is wrong in general.…”

Section: Introductionmentioning

confidence: 99%

On the size of 𝑝-adic Whittaker functions

Assing¹

2018

Trans. Amer. Math. Soc.

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“…• This theorem generalizes the methods from [23] and [12] to number fields. We also implement some ideas from [17] in order to deal with arbitrary level and central character.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Here we used the definition of the Gauß sum given in [17]. We evaluate this explicitly in terms of ǫ-factors using [17, (6)].…”

Section: The Constants Term Of E(s G)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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“…Thus, Theorem 1.3 hinges on the p-adic analysis of the values of W f, p , which is a purely local question about π f, p . To access these values, we use the local Fourier expansion of W f, p and analyze the resulting local Fourier coefficients c t, ℓ (χ) with the help of the recent "basic identity" (reviewed in §3.5) that was derived by the third-named author in [Sah16] from the GL 2 local functional equation of Jacquet-Langlands [JL70].…”

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confidence: 99%

The Manin constant in the semistable case

Česnavičius¹

2018

Compositio Math.

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The Manin constant c of an elliptic curve E over Q is the nonzero integer that scales the pullback of a Néron differential under a minimal parametrization φ : X0(N ) Q ։ E into the differential ω f determined by the normalized newform f associated to E. Manin conjectured that c = ±1 for optimal parametrizations, and we prove that in general c | deg(φ) under a minor assumption at 2 and 3 that is not needed for cube-free N or for parametrizations by X1(N ) Q . Since c is supported at the additive reduction primes, which need not divide deg(φ), this improves the status of the Manin conjecture for many E. For the proof, we settle a part of the Manin conjecture by establishing an integrality property of ω f necessary for it to hold. We reduce the latter to p-adic bounds on denominators of the Fourier expansions of f at all the cusps of X0(N ) C and then use the recent basic identity for the p-adic Whittaker newform to establish stronger bounds in the more general setup of newforms of weight k on X0(N ). To overcome obstacles at 2 and 3, we analyze nondihedral supercuspidal representations of GL2(Q2) and exhibit new cases in which X0(N ) Z has rational singularities. E may be more canonical within the same isogeny class: for instance, X 1 (11) Q and X 0 (11) Q are distinct isogenous elliptic curves. The multiplicity one theorem ensures that the φ-pullback of a Néron differential ω E is a nonzero multiple of the differential ω f ∈ H 0 ((X Γ ) Q , Ω 1 ) associated to the normalized newform f whose Hecke eigenvalues agree with the Frobenius traces of E:

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Large Values of Newforms on GL(2) with Highly Ramified Central Character

Cited by 22 publications

References 9 publications

On the size of 𝑝-adic Whittaker functions

On the size of 𝑝-adic Whittaker functions

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

The Manin constant in the semistable case

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