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 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.

Constructing Elliptic Curves over Ramanujan's Class Invariants

2014 ◽  
Vol 915-916 ◽  
pp. 1336-1340
Author(s):  
Jian Jun Hu
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Hilbert Polynomials ◽  
Class Invariants ◽  
Curves Over Finite Fields ◽  
Class Polynomials

The Complex Multiplication (CM) method is a widely used technique for constructing elliptic curves over finite fields. The key point in this method is parameter generation of the elliptic curve and root compution of a special type of class polynomials. However, there are several class polynomials which can be used in the CM method, having much smaller coefficients, and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding Hilbert polynomials.In this paper, we provide a method which can construct elliptic curves by Ramanujan's class invariants. We described the algorithm for the construction of elliptic curves (ECs) over imaginary quadratic field and given the transformation from their roots to the roots of the corresponding Hilbert polynomials. We compared the efficiency in the use of this method and other methods.

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.

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.

On p-adic height pairings and locally free classgroups of Hopf orders

1998 ◽  
Vol 123 (3) ◽  
pp. 447-459
Author(s):  
A. AGBOOLA
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Homogeneous Spaces ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
The Other ◽  
Module Structure ◽  
Galois Module ◽  
Ring Of Integers ◽  
The One

Let E be an elliptic curve with complex multiplication by the ring of integers [Ofr ] of an imaginary quadratic field K. The purpose of this paper is to describe certain connections between the arithmetic of E on the one hand and the Galois module structure of certain arithmetic principal homogeneous spaces arising from E on the other. The present paper should be regarded as a complement to [AT]; we assume that the reader is equipped with a copy of the latter paper and that he is not averse to referring to it from time to time.

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 Generalized division polynomials

2004 ◽  
Vol 94 (2) ◽  
pp. 161 ◽  
Author(s):  
Takakazu Satoh
Keyword(s):  
Elliptic Curve ◽  
Rational Function ◽  
Time Complexity ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Imaginary Quadratic Field ◽  
Multiplication Algorithm ◽  
Recurrence Formulas ◽  
Explicit Condition ◽  
Division Polynomials

Let $E$ be an elliptic curve with complex multiplication by the ring $O_{F}$ of integers of an imaginary quadratic field $F$. We give an explicit condition on $\alpha\in O_{F}$ so that there exists a rational function $\psi_{\alpha}$ satisfying $\div\psi_{\alpha}=\sum_{P\in\mathrm{Ker}[\alpha]}[P] - N_{F /Q}(\alpha)[{\mathcal{O}}]$ where $[\alpha ]$ is the multiplication by $\alpha$ map. We give an algorithm to compute $\psi_{\alpha}$ based on recurrence formulas among these functions. We prove that the time complexity of this algorithm is $O(N_{F/Q}(\alpha )^{2+\varepsilon})$ bit operations under an FFT based multiplication algorithm as $N_{F /Q}(\alpha )$ tends to infinity for the fixed $E$.

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 field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

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

Heights and L-Series

1990 ◽  
Vol 42 (3) ◽  
pp. 533-560 ◽  
Author(s):  
Rhonda Lee Hatcher
Keyword(s):  
Dirichlet Series ◽  
Cusp Form ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Ideal Class ◽  
Quadratic Character ◽  
Trivial Character ◽  
Image Position

Let be a cusp form of weight 2k and trivial character for Γ0(N), where N is prime, which is orthogonal with respect to the Petersson product to all forms g(dz), where g is of level L < N, dL\N. Let K be an imaginary quadratic field of discriminant — D where the prime N is inert. Denote by ∊ the quadratic character of determined by ∊(p) = (—D/p) for primes p not dividing D. For A an ideal class in K, let rA(m) be the number of integral ideals of norm m in A. We will be interested in the Dirichlet series L(f,A,s) defined by

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