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 CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

2009 ◽  
Vol 05 (02) ◽  
pp. 229-256 ◽  
Author(s):  
A. BANDINI ◽  
I. LONGHI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Extension ◽  
Function Field ◽  
Function Fields ◽  
Global Field ◽  
Selmer Groups ◽  
Characteristic P ◽  
Main Conjecture ◽  
Fitting Ideal

Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with [Formula: see text] and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups SelE(L)l (l any prime) as L varies through the subextensions of 𝔽 via appropriate versions of Mazur's Control Theorem. In the case l = p, we let 𝔽 = ∪ 𝔽d where 𝔽d/F is a [Formula: see text]-extension. We prove that Sel E(𝔽d)p is a cofinitely generated ℤp[[ Gal (ℤd/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤp[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

scholarly journals ON THE RANK OF QUADRATIC TWISTS OF ELLIPTIC CURVES OVER FUNCTION FIELDS

2006 ◽  
Vol 02 (02) ◽  
pp. 267-288 ◽  
Author(s):  
E. KOWALSKI
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Upper Bounds ◽  
Density Theorem ◽  
Chebotarev Density Theorem ◽  
Quadratic Twists ◽  
Rank 2

We prove quantitative upper bounds for the number of quadratic twists of a given elliptic curve E/Fq(C) over a function field over a finite field that have rank ≥ 2, and for their average rank. The main tools are constructions and results of Katz and uniform versions of the Chebotarev density theorem for varieties over finite fields. Moreover, we conditionally derive a bound in some cases where the degree of the conductor is unbounded.

scholarly journals Integral points on elliptic curves over function fields

2004 ◽  
Vol 77 (2) ◽  
pp. 197-208 ◽  
Author(s):  
W. -C. Chi ◽  
K. F. Lai ◽  
K. -S. Tan
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Integral Points ◽  
New Formula

AbstractWe prove a new formula for the number of integral points on an elliptic curve over a function field without assuming that the coefficient field is algebraically closed. This is an improvement on the standard results of Hindry-Silverman.

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

Bilinear character sums and sum-product problems on elliptic curves

2010 ◽  
Vol 53 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Omran Ahmadi ◽  
Igor Shparlinski
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Character Sums ◽  
Multiplicative Character ◽  
Additive Character ◽  
Product Problem ◽  
Multiplicative Character Sums

AbstractLet E be an ordinary elliptic curve over a finite field q of q elements. We improve a bound on bilinear additive character sums over points on E, and obtain its analogue for bilinear multiplicative character sums. We apply these bounds to some variants of the sum-product problem on E.

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.

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.

scholarly journals Elliptic curves over the ring R

2019 ◽  
Vol 38 (3) ◽  
pp. 193-201 ◽  
Author(s):  
A. Boulbot ◽  
Abdelhakim Chillali ◽  
A. Mouhib
Keyword(s):  
Finite Field ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Canonical Projection ◽  
First Time ◽  
Weierstrass Equation

Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

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.

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