scholarly journals Right triangles with algebraic sides and elliptic curves over number fields

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
E. Girondo ◽  
G. González-Diez ◽  
E. González-Jiménez ◽  
R. Steuding ◽  
J. Steuding
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Order ◽  
Number Fields ◽  
Explicit Construction ◽  
Congruent Number ◽  
Number Problem ◽  
Weil Group ◽  
Congruent Number Problem

AbstractGiven any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

scholarly journals A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

2011 ◽  
Vol 07 (08) ◽  
pp. 2237-2247 ◽  
Author(s):  
LARRY ROLEN
Keyword(s):  
Elliptic Curve ◽  
Related Problem ◽  
Fixed Angle ◽  
Root Numbers ◽  
Tate Conjecture ◽  
Congruent Number ◽  
Number Problem ◽  
Parity Conjecture ◽  
Congruent Number Problem

We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These numbers are called θ-congruent. We give an elliptic curve criterion for determining whether a given integer n is θ-congruent. We then consider the "density" of integers n which are θ-congruent, as well as the related problem giving the "density" of angles θ for which a fixed n is congruent. Assuming the Shafarevich–Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.

On the congruent number problem over integers of cyclic number fields

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Albertas Zinevičius
Keyword(s):  
Number Fields ◽  
Field Extension ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem

AbstractGiven a cyclic field extension

scholarly journals Minimal degrees of algebraic numbers with respect to primitive elements

2021 ◽  
pp. 1-16
Author(s):  
Cheol-Min Park ◽  
Sun Woo Park
Keyword(s):  
Algebraic Number ◽  
Primitive Element ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Primitive Elements ◽  
Arithmetic Properties ◽  
Congruent Number ◽  
Number Problem ◽  
Number Formula ◽  
Congruent Number Problem

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree 4 Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

scholarly journals A REFINEMENT OF KOBLITZ'S CONJECTURE

2011 ◽  
Vol 07 (03) ◽  
pp. 739-769 ◽  
Author(s):  
DAVID ZYWINA
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Arbitrary Number ◽  
Number Fields ◽  
Numerical Evidence ◽  
Fixed Positive Integer

Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of 𝔽p-points of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the |E(𝔽p)| need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often |E(𝔽p)|/t is an integer and prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture.

SELF-POINTS ON ELLIPTIC CURVES OF PRIME CONDUCTOR

2009 ◽  
Vol 05 (05) ◽  
pp. 911-932
Author(s):  
CHRISTOPHE DELAUNAY ◽  
CHRISTIAN WUTHRICH
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Cyclic Subgroup ◽  
Infinite Order ◽  
Field Of Definition ◽  
Modular Parametrization ◽  
Weil Group ◽  
Definition Of

Let E be an elliptic curve of conductor p. Given a cyclic subgroup C of order p in E[p], we construct a modular point PC on E, called self-point, as the image of (E,C) on X0(p) under the modular parametrization X0(p) → E. We prove that the point is of infinite order in the Mordell–Weil group of E over the field of definition of C. One can deduce a lower bound on the growth of the rank of the Mordell–Weil group in its PGL 2(ℤp)-tower inside ℚ(E[p∞]).

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

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