scholarly journals A note on quadratic fields in which a fixed prime number splits completely

1985 ◽  
Vol 99 ◽  
pp. 63-71 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Elliptic Curve ◽  
Natural Number ◽  
Prime Number ◽  
Endomorphism Ring ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields

Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p.It is easy to see and more or less known that for any natural number n, there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

The Square Sieve and the Lang–Trotter Conjecture

2005 ◽  
Vol 57 (6) ◽  
pp. 1155-1177 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Etienne Fouvry ◽  
M. Ram Murty
Keyword(s):  
Elliptic Curve ◽  
Riemann Hypothesis ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Upper Bounds ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Frobenius Endomorphism ◽  
Square Sieve

AbstractLet E be an elliptic curve defined over ℚ and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which ℚ(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is OE(x17/18 log x), and unconditionally we show that this number is We also prove that the number of imaginary quadratic fields K, with −disc K ≤ x and of the form K = ℚ(πp), is ≫E log log log x for x ≥ x0(E). These results represent progress towards a 1976 Lang–Trotter conjecture.

scholarly journals Torsion points on elliptic curves defined over quadratic fields

1988 ◽  
Vol 109 ◽  
pp. 125-149 ◽  
Author(s):  
M. A. Kenku ◽  
F. Momose
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Prime Number ◽  
Prime Power ◽  
Quadratic Field ◽  
Rational Points ◽  
Negative Integer ◽  
Quadratic Fields ◽  
Prime Power Order ◽  
Torsion Points

Let k be a quadratic field and E an elliptic curve defined over k. The authors [8, 12, 13] [23] discussed the k-rational points on E of prime power order. For a prime number p, let n = n(k, p) be the least non negative integer such thatfor all elliptic curves E defined over a quadratic field k ([15]).

On the Étale K-Theory of an Elliptic Curve with Complex Multiplication for Regular Primes

1990 ◽  
Vol 33 (2) ◽  
pp. 145-150 ◽  
Author(s):  
Kay Wingberg
Keyword(s):  
Elliptic Curve ◽  
Prime Number ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
K Theory ◽  
Ordinary Reduction

AbstractGeneralizing a result of Soulé we prove that for an elliptic curve E defined over an imaginary quadratic field K with complex multiplication having good ordinary reduction at the prime number p > 3 which is regular for E and the extension F of K contained in K(Ep) the dimensions of the étale K-groups are equal to the numbers predicted by Bloch and Beilinson, i.e.,

scholarly journals p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

2015 ◽  
Vol 219 ◽  
pp. 269-302
Author(s):  
Kenichi Bannai ◽  
Hidekazu Furusho ◽  
Shinichi Kobayashi
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Theta Function ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Good Reduction ◽  
Ring Of Integers

AbstractConsider an elliptic curve defined over an imaginary quadratic fieldKwith good reduction at the primes abovep≥ 5 and with complex multiplication by the full ring of integersof K. In this paper, we constructp-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then provep-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

Signed-Selmer Groups over the ℤ2p-extension of an Imaginary Quadratic Field

2014 ◽  
Vol 66 (4) ◽  
pp. 826-843 ◽  
Author(s):  
Byoung Du (B. D.) Kim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Selmer Groups

AbstractLet E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ2p-extension of an imaginary quadratic field K when the prime p splits completely over K/ℚ, and prove that they enjoy a property analogous to Mazur's control theorem.Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±-p-adic L-functions of elliptic curves.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

2019 ◽  
Vol 15 (09) ◽  
pp. 1793-1800 ◽  
Author(s):  
Dongho Byeon ◽  
Taekyung Kim ◽  
Donggeon Yhee
Keyword(s):  
Elliptic Curve ◽  
Infinite Order ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Roots Of Unity ◽  
Prime Divisors ◽  
Heegner Point ◽  
Tate Group ◽  
Tamagawa Numbers

Let [Formula: see text] be an elliptic curve defined over [Formula: see text] of conductor [Formula: see text], [Formula: see text] the Manin constant of [Formula: see text], and [Formula: see text] the product of Tamagawa numbers of [Formula: see text] at prime divisors of [Formula: see text]. Let [Formula: see text] be an imaginary quadratic field where all prime divisors of [Formula: see text] split in [Formula: see text], [Formula: see text] the Heegner point in [Formula: see text], and [Formula: see text] the Shafarevich–Tate group of [Formula: see text] over [Formula: see text]. Let [Formula: see text] be the number of roots of unity contained in [Formula: see text]. Gross and Zagier conjectured that if [Formula: see text] has infinite order in [Formula: see text], then the integer [Formula: see text] is divisible by [Formula: see text]. In this paper, we show that this conjecture is true if [Formula: see text].

Special values of L-functions and height two formal groups

1989 ◽  
Vol 105 (1) ◽  
pp. 13-24
Author(s):  
Rodney I. Yager
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Maximal Order ◽  
Imaginary Quadratic Field ◽  
Fundamental Period ◽  
Formal Groups ◽  
Special Values ◽  
Image Position

Let ψ be the Grossencharacter attached to an elliptic curve E defined over an imaginary quadratic field K ⊂ of discriminant −dK, and having complex multiplication by the maximal order of K. We denote the conductor of ψ by and fix a Weierstrass model for E with coefficients in ,whose discriminant is divisible only by primes dividing 6. Let Kab be the abelian closure of K in and choose a fundamental period Ω ∈ for the above model of the curve.

scholarly journals NOTE ON THE DIVISIBILITY OF THE CLASS NUMBER OF CERTAIN IMAGINARY QUADRATIC FIELDS

2009 ◽  
Vol 51 (1) ◽  
pp. 187-191 ◽  
Author(s):  
YASUHIRO KISHI
Keyword(s):  
Class Number ◽  
Diophantine Equations ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Positive Integers

AbstractWe prove that the class number of the imaginary quadratic field $\Q(\sqrt{2^{2k}-3^n})$ is divisible by n for any positive integers k and n with 22k < 3n, by using Y. Bugeaud and T. N. Shorey's result on Diophantine equations.

scholarly journals CODIMENSION TWO CYCLES IN IWASAWA THEORY AND ELLIPTIC CURVES WITH SUPERSINGULAR REDUCTION

2019 ◽  
Vol 7 ◽  
Author(s):  
ANTONIO LEI ◽  
BHARATHWAJ PALVANNAN
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Iwasawa Theory ◽  
Imaginary Quadratic Field ◽  
Selmer Groups ◽  
Numerical Evidence ◽  
Codimension Two ◽  
Main Conjecture ◽  
Iwasawa Algebra

A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s $2$ -variable $p$ -adic $L$ -functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a $2$ -variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field $K$ (where an odd prime $p$ splits) of an elliptic curve $E$ , defined over  $\mathbb{Q}$ , with good supersingular reduction at $p$ . On the analytic side, we consider eight pairs of $2$ -variable $p$ -adic $L$ -functions in this setup (four of the $2$ -variable $p$ -adic $L$ -functions have been constructed by Loeffler and a fifth $2$ -variable $p$ -adic $L$ -function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the $\mathbb{Z}_{p}^{2}$ -extension of $K$ . We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.

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