A quadratic large sieve inequality over number fields

Goldmakher, Leo; Louvel, Benoit

doi:10.1017/s0305004112000370

Cited by 9 publications

(10 citation statements)

References 8 publications

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“…). This is [GL,Lemmas 2.4 and 2.5] which in turn is a slight generalization of [HB2,Lemmas 4 and 5]. From the reciprocity formula in Lemma 2.1 one concludes easily…”

Section: A Large Sieve Inequality For N-th Order Charactersmentioning

confidence: 70%

L-Functions with n-th-Order Twists

Blomer

Goldmakher

Louvel

2012

International Mathematics Research Notices

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Let K be a number field containing the n-th roots of unity for some n 3. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s, χ) with χ an n-th order Hecke character: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.

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“…). This is [GL,Lemmas 2.4 and 2.5] which in turn is a slight generalization of [HB2,Lemmas 4 and 5]. From the reciprocity formula in Lemma 2.1 one concludes easily…”

Section: A Large Sieve Inequality For N-th Order Charactersmentioning

confidence: 70%

L-Functions with n-th-Order Twists

Blomer

Goldmakher

Louvel

2012

International Mathematics Research Notices

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“…However, the lack of good enough bounds of type Lindelöf-on-average for the mean square of central values was a critical obstacle in order to go beyond the case of the number field Q. Building on a quadratic large sieve inequality on number fields due to Goldmakher and Louvel [Goldmakher and Louvel, 2013], generalizing a result of Heath-Brown [Heath-Brown, 1995], we are able to overcome these difficulties. This is also the opportunity to fill in some gaps usually left under cover of the standardness of the method and that we believe to be written nowhere.…”

Section: Statement Of the Resultsmentioning

confidence: 99%

“…is known [Goldmakher and Louvel, 2013] to have meromorphic continuation to the whole complex plane, more precisely a continuation to an analytic function on C\{1} and a simple pole at 1 if and only if χ D has conductor one. Moreover the completed L-function satisfies the functional equation…”

Section: Quadratic Symbols On Number Fieldsmentioning

confidence: 99%

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Quadratic twists of central values for GL(3)

Kuan¹,

Lesesvre²

2020

Preprint

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We prove that a cuspidal automorphic representation of GL(3) over any number field is determined by the quadratic twists of its central value. This generalizes a result of Chinta and Diaconu that is valid only over the field of rational numbers and is restricted to Gelbart-Jacquet lifts. A consequence is a result of non-vanishing for infinitely many quadratic twists of the L-functions at the central point.

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“…A large sieve over Hecke families was established in [18] and [4]. From the analysis in these papers, we take the following bounds.…”

Section: Proposition 32mentioning

confidence: 99%

Average Bateman--Horn for Kummer polynomials

Balestrieri¹,

Rome²

2020

Preprint

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For any prime r ∈ N and almost all k ∈ N smaller than x r , we show that the polynomial f (n) = n r + k takes the expected number of prime values, as n ranges from 1 to x. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form N K/Q (z) = t r + k = 0, where K/Q is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order r.

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A quadratic large sieve inequality over number fields

Abstract: Abstract. We formulate and prove a large sieve inequality for quadratic characters over a number field. To do this, we introduce the notion of an n-th order Hecke family. We develop the basic theory of these Hecke families, including versions of the Poisson summation formula.

Cited by 9 publications

References 8 publications

L-Functions with n-th-Order Twists

L-Functions with n-th-Order Twists

Quadratic twists of central values for GL(3)

Average Bateman--Horn for Kummer polynomials

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