scholarly journals Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

2004 ◽  
Vol 70 (1) ◽  
pp. 125-142 ◽  
Author(s):  
Noburo Ishii
Keyword(s):  
Prime Ideal ◽  
Elliptic Curve ◽  
Class Number ◽  
Quadratic Field ◽  
Complex Multiplication ◽  
Good Reduction ◽  
Absolute Value ◽  
Frobenius Endomorphism ◽  
Trace Of Frobenius ◽  
Positive Integers

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

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.

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 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.

A Titchmarsh divisor problem for elliptic curves

2015 ◽  
Vol 160 (1) ◽  
pp. 167-189 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Elliptic Curve ◽  
Complex Multiplication ◽  
Zeta Functions ◽  
Dirichlet Character ◽  
Good Reduction ◽  
Heuristic Argument ◽  
Mean Values ◽  
Average Size ◽  
Image Position ◽  
Ordinary Reduction

AbstractLet E/Q be an elliptic curve with complex multiplication. We study the average size of τ(#E(Fp)) as p varies over primes of good ordinary reduction. We work out in detail the case of E: y2 = x3 − x, where we prove that $$\begin{equation} \sum_{\substack{p \leq x \\p \equiv 1\pmod{4}}} \tau(\#E({\bf{F}}_p)) \sim \left(\frac{5\pi}{16} \prod_{p > 2} \frac{p^4-\chi(p)}{p^2(p^2-1)}\right)x, \quad\text{as $x\to\infty$}. \end{equation}$$ Here χ is the nontrivial Dirichlet character modulo 4. The proof uses number field analogues of the Brun–Titchmarsh and Bombieri–Vinogradov theorems, along with a theorem of Wirsing on mean values of nonnegative multiplicative functions.Now suppose that E/Q is a non-CM elliptic curve. We conjecture that the sum of τ(#E(Fp)), taken over p ⩽ x of good reduction, is ~cEx for some cE > 0, and we give a heuristic argument suggesting the precise value of cE. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove that this sum is ≍Ex. The proof uses combinatorial ideas of Erdős.

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 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 THE DIVISIBILITY OF THE CLASS NUMBER OF THE IMAGINARY QUADRATIC FIELD

2011 ◽  
Vol 54 (1) ◽  
pp. 149-154 ◽  
Author(s):  
ZHU MINHUI ◽  
WANG TINGTING
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Positive Integers

AbstractLet hK denote the class number of the imaginary quadratic field $K=\mathbf{Q}(\sqrt{2^{2m}-k^n})$, where m and n are positive integers, k is an odd integer with k > 1 and 22m < kn. In this paper we prove that if either 3 ∣ n and 22m − kn ≡ 5(mod 8) or n = 3 and k = (22m+2 −1)/3, then ∣ hK. Otherwise, we have n ∣ hK.

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.

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