Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms

Knightly, Andrew; Li, C.

doi:10.1090/s0065-9266-2012-00673-3

Cited by 37 publications

(44 citation statements)

References 52 publications

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“…When

c

is prime, the Weil bound (cf. [, Theorem 9.3]) from algebraic geometry can be used. In the general case, one obtains Lemma For all

c \in C (a, a)

m, n \in Z

, we have

S_{a a} false(m, n; c false) ≪ {(m, n, c)}^{1 / 2} τ {(c)}^{O (1)} {(c q_{0})}^{1 / 2},

where

q_{0}

is the modulus of

χ

.…”

Section: Sums Of Kloosterman Sums In Arithmetic Progressionsmentioning

confidence: 99%

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Drappeau

2017

Proc. London Math. Soc.

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International audienceWe prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As a consequence, assuming the Riemann hypothesis for Dirichlet $L$-functions, we prove a power-saving error term in the Titchmarsh divisor problem of estimating $\sum_{p\leq x}\tau(p-1)$. Unconditionally, we isolate the possible contribution of Siegel zeroes, showing it is always negative. Extending work of Fouvry and Tenenbaum, we obtain power-saving in the asymptotic formula for $\sum_{n\leq x}\tau_k(n)\tau(n+1)$, reproving a result announced by Bykovski\u{i} and Vinogradov by a different method. The gain in the exponent is shown to be independent of $k$ if a generalized Lindel\"of hypothesis is assumed

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“…When

c

is prime, the Weil bound (cf. [, Theorem 9.3]) from algebraic geometry can be used. In the general case, one obtains Lemma For all

c \in C (a, a)

m, n \in Z

, we have

S_{a a} false(m, n; c false) ≪ {(m, n, c)}^{1 / 2} τ {(c)}^{O (1)} {(c q_{0})}^{1 / 2},

where

q_{0}

is the modulus of

χ

.…”

Section: Sums Of Kloosterman Sums In Arithmetic Progressionsmentioning

confidence: 99%

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Drappeau

2017

Proc. London Math. Soc.

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“…However, one can still derive estimates for the sum. See for example [10]. Over Q p a general sharp bound for S χ (A, B, m) is given in [3, Section 5].…”

Section: Evaluation Of Oscillatory Integralsmentioning

confidence: 99%

On the size of 𝑝-adic Whittaker functions

Assing¹

2018

Trans. Amer. Math. Soc.

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“…The notation is a little easier here, because there are no cuspidal oldforms and also the Eisenstein spectrum contains in some sense only newforms, in particular the classical parametrization in terms of the cusps ∞ and 0 and the adelic parametrization in terms of the two pairs (χ, triv), (triv, χ) is the same. Following [KL,Section 5] (or [DFI2, Section 7]), the Fourier coefficients ρ a,t (n) are…”

Section: Versions Of the Kuznetsov Formulamentioning

confidence: 99%

Uniform subconvexity and symmetry breaking reciprocity

Blomer

Khan

2019

Journal of Functional Analysis

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A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the ℓ-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level ℓ and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.2010 Mathematics Subject Classification. Primary: 11M41, 11F72.

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Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms

Cited by 37 publications

References 52 publications

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

On the size of 𝑝-adic Whittaker functions

Uniform subconvexity and symmetry breaking reciprocity

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