Computing the endomorphism ring of an ordinary elliptic curve over a finite field

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of , while our bound for the second algorithm depends primarily on log|DE|, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.     

On the twisted endomorphism algebra of a vector bundle on a curve

Let X be a smooth projective curve, L∈Pic (X) and E a vector bundle on X. Here we study the bigraded algebra , mainly when E is a general stable bundle on X. We give informations on the degrees of the minimal generators and study a generalization of the notion of projective normality. Entrata in Redazione il 19 gennaio 2000.     

On the Shafarevich-Tate group of an elliptic curve

The paper deals with the group of locally trivial principal homogeneous spaces over an elliptic curve. Duality is proven between the group of those spaces having a point of finite order locally with the exception of prime divisors, and the factor group of those of them which are infinitely divisible in the group of principal homogeneous spaces having a local rational point with respect to all divisors besides the singular ones.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 79–86, January, 1970.     

On lattice profile of the elliptic curve linear congruential generators

Lattice tests are quality measures for assessing the intrinsic structure of pseudorandom number generators. Recently a new lattice test has been introduced by Niederreiter and Winterhof. In this paper, we present a general inequality that is satisfied by any periodic sequence. Then, we analyze the behavior of the linear congruential generators on elliptic curves (EC-LCG) under this new lattice test and prove that the EC-LCG passes it up to very high dimensions. We also use a result of Brandstätter and Winterhof on the linear complexity profile related to the correlation measure of order $k$ to present lower bounds on the linear complexity profile of some binary sequences derived from the EC-LCG.     

On P-coherent endomorphism rings

A ring is called right P-coherent if every principal right ideal is finitely presented. Let M _R be a right R-module. We study the P-coherence of the endomorphism ring S of M _R . It is shown that S is a right P-coherent ring if and only if every endomorphism of M _R has a pseudokernel in add M _R ; S is a left P-coherent ring if and only if every endomorphism of M _R has a pseudocokernel in add M _R . Some applications are given.     

Averages of elliptic curve constants

We compute the averages over elliptic curves of the constants occurring in the Lang–Trotter conjecture, the Koblitz conjecture, and the cyclicity conjecture. The results obtained confirm the consistency of these conjectures with the corresponding “theorems on average” obtained recently by various authors.     

On the socle of an endomorphism algebra

The socle of an endomorphism algebra of a finite dimensional module of a finite dimensional algebra is described. The results are applied to the modular Hecke algebra of a finite group with a cyclic Sylow subgroup.     

On endomorphism rings of regular modules

In this paper, we study the endomorphism rings of regular modules. We give sufficient conditions on a regular projective moduleP such that End_R (P) has stable range one. Dedicated to Professor Zhou Boxun for his 80'th Birthday The author is supported by the NNSF of China (No. 19601009)     

Ramification in the division fields of an elliptic curve

We study the ramification in the division fields of an elliptic curve. Applications on the class number of quadratic fields and on elliptic curves with prime power conductor are given.