scholarly journals Average Frobenius distribution for elliptic curves defined over finite Galois extensions of the rationals

2011 ◽  
Vol 150 (3) ◽  
pp. 439-458 ◽  
Author(s):  
KEVIN JAMES ◽  
ETHAN SMITH
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Fixed Number ◽  
Prime Ideals ◽  
Galois Extensions ◽  
Trace Of Frobenius

AbstractLet K be a fixed number field, assumed to be Galois over ℚ. Let r and f be fixed integers with f positive. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree f prime ideals of K with trace of Frobenius equal to r. Except in the case f = 2, we show that ‘on average,’ the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

scholarly journals Average Frobenius distribution for the degree two primes of a number field

2013 ◽  
Vol 154 (3) ◽  
pp. 499-525 ◽  
Author(s):  
KEVIN JAMES ◽  
ETHAN SMITH
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Prime Ideals ◽  
Trace Of Frobenius

AbstractLet K be a number field and r an integer. Given an elliptic curve E, defined over K, we consider the problem of counting the number of degree two prime ideals of K with trace of Frobenius equal to r. Under certain restrictions on K, we show that “on average” the number of such prime ideals with norm less than or equal to x satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang–Trotter conjecture and extends the work of several authors.

scholarly journals Primitive divisors of sequences associated to elliptic curves with complex multiplication

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Matteo Verzobio
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Complex Multiplication ◽  
Dedekind Domain ◽  
Torsion Point ◽  
Integral Ideal ◽  
Primitive Divisors ◽  
Primitive Divisor ◽  
The Ideal

AbstractLet P and Q be two points on an elliptic curve defined over a number field K. For $$\alpha \in {\text {End}}(E)$$ α ∈ End ( E ) , define $$B_\alpha $$ B α to be the $$\mathcal {O}_K$$ O K -integral ideal generated by the denominator of $$x(\alpha (P)+Q)$$ x ( α ( P ) + Q ) . Let $$\mathcal {O}$$ O be a subring of $${\text {End}}(E)$$ End ( E ) , that is a Dedekind domain. We will study the sequence $$\{B_\alpha \}_{\alpha \in \mathcal {O}}$$ { B α } α ∈ O . We will show that, for all but finitely many $$\alpha \in \mathcal {O}$$ α ∈ O , the ideal $$B_\alpha $$ B α has a primitive divisor when P is a non-torsion point and there exist two endomorphisms $$g\ne 0$$ g ≠ 0 and f so that $$f(P)= g(Q)$$ f ( P ) = g ( Q ) . This is a generalization of previous results on elliptic divisibility sequences.

scholarly journals Selmer Groups of Elliptic Curves with Complex Multiplication

2004 ◽  
Vol 56 (1) ◽  
pp. 194-208
Author(s):  
A. Saikia
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Simple Formula ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Selmer Groups ◽  
Selmer Group ◽  
Finitely Generated ◽  
Ring Of Integers

AbstractSuppose K is an imaginary quadratic field and E is an elliptic curve over a number field F with complex multiplication by the ring of integers in K. Let p be a rational prime that splits as in K. Let Epn denote the pn-division points on E. Assume that F(Epn) is abelian over K for all n ≥ 0. This paper proves that the Pontrjagin dual of the -Selmer group of E over F(Ep∞) is a finitely generated free Λ-module, where Λ is the Iwasawa algebra of . It also gives a simple formula for the rank of the Pontrjagin dual as a Λ-module.

scholarly journals Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem

2017 ◽  
Vol 14 (01) ◽  
pp. 255-288
Author(s):  
Evan Chen ◽  
Peter S. Park ◽  
Ashvin A. Swaminathan
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Riemann Hypothesis ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Explicit Result ◽  
Generalized Riemann Hypothesis ◽  
Quadratic Nonresidue ◽  
Trace Of Frobenius

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-nonisogenous, semistable elliptic curves over [Formula: see text], having respective conductors [Formula: see text] and [Formula: see text] and both without complex multiplication. For each prime [Formula: see text], denote by [Formula: see text] the trace of Frobenius. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power [Formula: see text]-functions [Formula: see text] where [Formula: see text], we prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text] This improves and makes explicit a result of Bucur and Kedlaya. Now, if [Formula: see text] is a subinterval with Sato–Tate measure [Formula: see text] and if the symmetric power [Formula: see text]-functions [Formula: see text] are functorial and satisfy GRH for all [Formula: see text], we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime [Formula: see text] such that [Formula: see text] and [Formula: see text]

scholarly journals ON MAZUR'S CONJECTURE FOR TWISTED L-FUNCTIONS OF ELLIPTIC CURVES OVER TOTALLY REAL OR CM FIELDS

2010 ◽  
Vol 53 (1) ◽  
pp. 207-210
Author(s):  
CRISTIAN VIRDOL
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Order ◽  
Number Field ◽  
Meromorphic Continuation ◽  
Minimal Order ◽  
Finite Set ◽  
Entire Complex ◽  
Totally Real

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

Ramification of the Kummer extension generated from torsion points of elliptic curves

2015 ◽  
Vol 11 (06) ◽  
pp. 1725-1734
Author(s):  
Masaya Yasuda
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Torsion Points ◽  
Root Of Unity

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field k with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over k(ζp). In this paper, for p = 5 and 7, we study the ramification of such Kummer extensions using explicit Kummer generators directly computed by Verdure in 2006.

scholarly journals Computing the Rank of Elliptic Curves over Number Fields

2002 ◽  
Vol 5 ◽  
pp. 7-17 ◽  
Author(s):  
Denis Simon
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Number Fields ◽  
General Number

AbstractThis paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields of degree from 1 to 5.

On the K-theory of elliptic curves

1999 ◽  
Vol 1999 (507) ◽  
pp. 81-91
Author(s):  
Kevin P Knudson
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Function Field ◽  
Coordinate Ring ◽  
Infinite Field ◽  
Closed Point ◽  
K Theory

Abstract Let A be the coordinate ring of an affine elliptic curve (over an infinite field k) of the form X – {p}, where X is projective and p is a closed point on X. Denote by F the function field of X. We show that the image of H.(GL2 (A), ℤ) in H.(GL2 (F), ℤ) coincides with the image of H.(GL2 (k), ℤ). As a consequence, we obtain numerous results about the K-theory of A and X. For example, if k is a number field, we show that r2 (K2 (A) ⊗ ℚ) = 0, where rm denotes the mth level of the rank filtration.

KUMMER GENERATORS AND TORSION POINTS OF ELLIPTIC CURVES WITH BAD REDUCTION AT SOME PRIMES

2013 ◽  
Vol 09 (07) ◽  
pp. 1743-1752 ◽  
Author(s):  
MASAYA YASUDA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Fields ◽  
Torsion Subgroup ◽  
Torsion Point ◽  
Torsion Points ◽  
Root Of Unity ◽  
Fundamental Units

For a prime p, let ζp denote a fixed primitive pth root of unity. Let E be an elliptic curve over a number field K with a p-torsion point. Then the p-torsion subgroup of E gives a Kummer extension over K(ζp), and in this paper, we study the ramification of such Kummer extensions using the Kummer generators directly computed by Verdure in 2006. For quadratic fields K, we also give unramified Kummer extensions over K(ζp) generated from elliptic curves over K having a p-torsion point with bad reduction at certain primes. Many of these unramified Kummer extensions have not appeared in the previous work using fundamental units of quadratic fields.

On elliptic curves with complex multiplication and root numbers

2020 ◽  
pp. 1-16
Author(s):  
Wan Lee ◽  
Myungjun Yu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Number Field ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Root Number ◽  
Root Numbers ◽  
Quadratic Twists

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text]. Suppose that [Formula: see text] has complex multiplication over [Formula: see text], i.e. [Formula: see text] is an imaginary quadratic field. With the aid of CM theory, we find elliptic curves whose quadratic twists have a constant root number.

Sign in / Sign up

Export Citation Format

Share Document