scholarly journals Elliptic curves over function fields with a large set of integral points

2013 ◽  
Vol 161 (4) ◽  
pp. 351-370
Author(s):  
Ricardo P. Conceição
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Large Set ◽  
Integral Points

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 Integral points on elliptic curves over function fields of positive characteristic

1998 ◽  
Vol 58 (3) ◽  
pp. 353-357
Author(s):  
Amílcar Pacheco
Keyword(s):  
Elliptic Curves ◽  
Finite Subset ◽  
Upper Bound ◽  
Function Field ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Function Fields ◽  
Closed Field ◽  
Integral Points ◽  
Variable Function

Let K be a one variable function field of genus g defined over an algebraically closed field k of characteristic p > 0. Let E/K be a non-constant elliptic curve. Denote by MK the set of places of K and let S ⊂ MK be a non-empty finite subset.Mason in his paper “Diophantine equations over function fields” Chapter VI, Theorem 14 and Voloch in “Explicit p-descent for elliptic curves in characteristic p” Theorem 5.3 proved that the number of S-integral points of a Weiertrass equation of E/K defined over RS is finite. However, no explicit upper bound for this number was given. In this note, under the extra hypotheses that E/K is semi-stable and p > 3, we obtain an explicit upper bound for this number for a certain class of Weierstrass equations called S-minimal.

scholarly journals Elliptic Curves with Large Rank over Function Fields

2002 ◽  
Vol 155 (1) ◽  
pp. 295 ◽  
Author(s):  
Douglas Ulmer
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Large Rank

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

Erratum: "Prime numbre races for elliptic curves over function fields"

2021 ◽  
Vol 54 (5) ◽  
pp. 1353-1362
Author(s):  
Byungchul CHA ◽  
Daniel FIORILLI ◽  
Florent JOUVE
Keyword(s):  
Elliptic Curves ◽  
Function Fields

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.

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.

Sign in / Sign up

Export Citation Format

Share Document