scholarly journals Correlations of Sums of Two Squares and Other Arithmetic Functions in Function Fields

2017 ◽  
Vol 2019 (14) ◽  
pp. 4469-4515 ◽  
Author(s):  
Lior Bary-Soroker ◽  
Arno Fehm
Keyword(s):  
Finite Field ◽  
Function Field ◽  
Function Fields ◽  
Arithmetic Functions ◽  
Quadratic Character ◽  
Field Limit ◽  
Numerical Evidence ◽  
Divisor Functions ◽  
Shifted Primes ◽  
Sums Of Two Squares

Abstract We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg, and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

scholarly journals Shifted convolution and the Titchmarsh divisor problem over 𝔽 q [ t ]

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140308 ◽  
Author(s):  
J. C. Andrade ◽  
L. Bary-Soroker ◽  
Z. Rudnick
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Function Field ◽  
Analytic Number Theory ◽  
Divisor Problem ◽  
Divisor Functions ◽  
Analytic Number

In this paper, we solve a function field analogue of classical problems in analytic number theory, concerning the autocorrelations of divisor functions, in the limit of a large finite field.

scholarly journals The -parity conjecture for abelian varieties over function fields of characteristic

2014 ◽  
Vol 150 (4) ◽  
pp. 507-522 ◽  
Author(s):  
Fabien Trihan ◽  
Seidai Yasuda
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Prime Number ◽  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Selmer Group ◽  
Smooth Variety ◽  
Tate Conjecture ◽  
Parity Conjecture

AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

Large values of Dirichlet L-functions over function fields

2020 ◽  
Vol 16 (05) ◽  
pp. 1081-1109
Author(s):  
Dragan Đokić ◽  
Nikola Lelas ◽  
Ilija Vrećica
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Function Field ◽  
Function Fields ◽  
Resonance Method ◽  
Dirichlet Characters ◽  
Field Setting

In this paper, we investigate the existence of large values of [Formula: see text], where [Formula: see text] varies over non-principal characters associated to prime polynomials [Formula: see text] over finite field [Formula: see text], as [Formula: see text], and [Formula: see text]. When [Formula: see text], we provide a lower bound for the number of such characters. To do this, we adapt the resonance method to the function field setting. We also investigate this problem for [Formula: see text], where now [Formula: see text] varies over even, non-principal, Dirichlet characters associated to prime polynomials [Formula: see text] over [Formula: see text], as [Formula: see text]. In addition to resonance method, in this case, we use an adaptation of Gál-type sums estimate.

Introduction

2019 ◽  
pp. 1-61
Author(s):  
Dennis Gaitsgory ◽  
Jacob Lurie
Keyword(s):  
Finite Field ◽  
Function Field ◽  
Algebraic Curve ◽  
Mass Formula ◽  
Essential Feature ◽  
Function Fields ◽  
Rational Functions ◽  
Algebraic Stack ◽  
Field Setting

This introductory chapter sets out the book's purpose, which is to study Weil's conjecture over function fields: that is, fields K which arise as rational functions on an algebraic curve X over a finite field F q. It reformulates Weil's conjecture as a mass formula, which counts the number of principal G-bundles over the algebraic curve X. An essential feature of the function field setting is that the objects that we want to count (in this case, principal G-bundles) admit a “geometric” parametrization: they can be identified with Fq-valued points of an algebraic stack BunG(X). This observation is used to reformulate Weil's conjecture yet again: it essentially reduces to a statement about the ℓ-adic cohomolog of BunG(X), reflecting the heuristic idea that it should admit a “continuous Künneth decomposition”.

DIOPHANTINE DEFINABILITY OVER HOLOMORPHY RINGS OF ALGEBRAIC FUNCTION FIELDS WITH INFINITE NUMBER OF PRIMES ALLOWED AS POLES

1998 ◽  
Vol 09 (08) ◽  
pp. 1041-1066 ◽  
Author(s):  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Finite Field ◽  
Infinite Number ◽  
Function Field ◽  
Function Fields ◽  
Algebraic Function ◽  
Algebraic Function Field ◽  
Algebraic Function Fields ◽  
Finite Set ◽  
Infinite Set

Let K be an algebraic function field over a finite field of constants of characteristic greater than 2. Let W be a set of non-archimedean primes of K, let [Formula: see text]. Then for any finite set S of primes of K there exists an infinite set W of primes of K containing S, with the property that OK,S has a Diophantine definition over OK,W.

On locally repeated values of arithmetic functions over Fq[T]

2018 ◽  
Vol 70 (2) ◽  
pp. 451-472 ◽  
Author(s):  
Zeév Rudnick ◽  
Ron Peled
Keyword(s):  
Number Theory ◽  
Finite Field ◽  
Analytic Number Theory ◽  
Arithmetic Functions ◽  
Frequency Of Occurrence ◽  
Common Theme ◽  
Divisor Function ◽  
Field Limit ◽  
Euler Totient Function ◽  
Consecutive Integers

AbstractThe frequency of occurrence of ‘locally repeated’ values of arithmetic functions is a common theme in analytic number theory, for instance in the Erdős–Mirsky problem on coincidences of the divisor function at consecutive integers, the analogous problem for the Euler totient function and the quantitative conjectures of Erdős, Pomerance and Sarkőzy and of Graham, Holt and Pomerance on the frequency of occurrences. In this paper, we introduce the corresponding problems in the setting of polynomials over a finite field, and completely solve them in the large finite field limit.

scholarly journals ON THE GROWTH OF LINEAR RECURRENCES IN FUNCTION FIELDS

2020 ◽  
pp. 1-10
Author(s):  
CLEMENS FUCHS ◽  
SEBASTIAN HEINTZE
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Linear Recurrences ◽  
Field Case ◽  
Power Sum ◽  
Linear Recurrence Sequence

Abstract Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals FINITE GROUPS AS GALOIS GROUPS OF FUNCTION FIELDS WITH INFINITE FIELD OF CONSTANTS

2010 ◽  
Vol 88 (3) ◽  
pp. 301-312
Author(s):  
C. ÁLVAREZ-GARCÍA ◽  
G. VILLA-SALVADOR
Keyword(s):  
Finite Groups ◽  
Function Field ◽  
Function Fields ◽  
Galois Groups ◽  
Infinite Field ◽  
Separable Extension ◽  
Separable Extensions

AbstractLetE/kbe a function field over an infinite field of constants. Assume thatE/k(x) is a separable extension of degree greater than one such that there exists a place of degree one ofk(x) ramified inE. LetK/kbe a function field. We prove that there exist infinitely many nonisomorphic separable extensionsL/Ksuch that [L:K]=[E:k(x)] andAutkL=AutKL≅Autk(x)E.

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