Distribution of zeta zeroes of Artin–Schreier covers

Bucur, Alina; David, Chantal; Feigon, Brooke; Laĺın, Matilde; Sinha, Kaneenika

doi:10.4310/mrl.2012.v19.n6.a12

Cited by 11 publications

(16 citation statements)

References 17 publications

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“…Xiong [Xi] extended their work to families of ℓ-fold covers of P 1 (F q ), for prime ℓ such that q ≡ 1 (mod ℓ), again obtaining Gaussian behavior. For other works in this direction see [BDFL1,BDFL2,BDFLS].…”

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confidence: 99%

Zeros of Dirichlet $L$-functions over function fields

Andrade

Miller

Pratt

et al. 2014

Communications in Number Theory and Physics

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ABSTRACT. Random matrix theory has successfully modeled many systems in physics and mathematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed 1-level density statistics for low-lying zeros of the family of primitive Dirichlet L-functions of fixed prime conductor Q, as Q → ∞, and verified the unitary symmetry predicted by random matrix theory. We compute 1-and 2-level statistics of the analogous family of Dirichlet L-functions over F q (T ). Whereas the Hughes-Rudnick results were restricted by the support of the Fourier transform of their test function, our test function is periodic and our results are only restricted by a decay condition on its Fourier coefficients. We show the main terms agree with unitary symmetry, and also isolate error terms. In concluding, we discuss an F q (T )-analogue of Montgomery's Hypothesis on the distribution of primes in arithmetic progressions, which Fiorilli and Miller show would remove the restriction on the Hughes-Rudnick results.

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mentioning

confidence: 99%

Zeros of Dirichlet $L$-functions over function fields

Andrade

Miller

Pratt

et al. 2014

Communications in Number Theory and Physics

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“…This result is consistent with the random matrix model predicted by the theory of Katz and Sarnak when both g and q tend to infinity. This beautiful work of Faifman and Rudnick was extended to the family of l-fold covers of the projective line ( [21]), and more recently to the family of Artin-Schreier covers of the projective line ( [4]), on which similar distribution results were obtained.…”

Section: Introductionmentioning

confidence: 78%

Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field

Xiong¹

2015

Journal of Number Theory

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For a function field k over a finite field with F q as the field of constants, and a finite abelian group G whose exponent divides q − 1, we study the distribution of zeta zeroes for a random G-extension of k, ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution.

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“…a j (n)K itj (2π|n|y)e(nx), 4 where a j (n) ∈ R, ̺ j (1) = 0 and K ν is the K-Bessel function of order ν. We order the f j 's so that 0 < t 1 ≤ t 2 ≤ t 3 ≤ .…”

Section: Introductionmentioning

confidence: 99%

“…This refined technique owes its origin to the work of Faifman and Rudnick [8], who used it to prove a central limit theorem for the number of zeros of the zeta functions of a family of hyperelliptic curves defined over a fixed finite field as the genus of the curves varies. The ideas of Faifman and Rudnick have since been fruitfully adapted by various authors (for example, [3], [4], [23]) to study similar statistics for different families of smooth projective curves over finite fields. Nagoshi [16] proved another remarkable theorem.…”

Section: Introductionmentioning

confidence: 99%

Fluctuations in the Distribution of Hecke Eigenvalues about the Sato–Tate Measure

Prabhu

Sinha

2017

International Mathematics Research Notices

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We study fluctuations in the distribution of families of p-th Fourier coefficients a f (p) of normalised holomorphic Hecke eigenforms f of weight k with respect to SL 2 (Z) as k → ∞ and primes p → ∞. These families are known to be equidistributed with respect to the Sato-Tate measure. We consider a fixed interval I ⊂ [−2, 2] and derive the variance of the number of a f (p)'s lying in I as p → ∞ and k → ∞ (at a suitably fast rate). The number of a f (p)'s lying in I is shown to asymptotically follow a Gaussian distribution when appropriately normalised. A similar theorem is obtained for primitive Maass cusp forms.

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Distribution of zeta zeroes of Artin–Schreier covers

Cited by 11 publications

References 17 publications

Zeros of Dirichlet $L$-functions over function fields

Zeros of Dirichlet $L$-functions over function fields

Distribution of zeta zeroes for abelian covers of algebraic curves over a finite field

Fluctuations in the Distribution of Hecke Eigenvalues about the Sato–Tate Measure

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