منابع مشابه
Pseudoprime Reductions of Elliptic Curves
Let E be an elliptic curve over Q without complex multiplication, and for each prime p of good reduction, let nE(p) = |E(Fp)|. For any integer b, we are studying in this paper elliptic pseudoprimes to the base b. More precisely, let QE,b(x) be the number of primes p 6 x such that bE ≡ b (modnE(p)), and π E,b (x) be the number of compositive nE(p) such that b nE(p) ≡ b (modnE(p)) (also called el...
متن کاملPseudoprime reductions of elliptic curves
Let b 2 be an integer and let E/Q be a fixed elliptic curve. In this paper, we estimate the number of primes p x such that the number of points nE(p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of nE(p).
متن کاملReductions of Points on Elliptic Curves
Let E be an elliptic curve defined over Q. Let Γ be a subgroup of rank r of the group of rational points E(Q) of E. For any prime p of good reduction, let Γ̄ be the reduction of Γ modulo p. Under certain standard assumptions, we prove that for almost all primes p (i.e. for a set of primes of density one), we have |Γ̄| ≥ p f(p) , where f(x) is any function such that f(x) → ∞, at an arbitrary slow ...
متن کاملExponents of Modular Reductions of Families of Elliptic Curves
For a basic background on elliptic curves, we refer to [11]. For a prime p > 3, we denote by Fp the finite field of p elements, which we identify with the set of integers {0,±1, . . . ,±(p− 1)/2}. When p ∤ 4a+27b, the set Ea,b(Fp), consisting of the Fp-rational points of Ea,b together with a point at infinity O, forms an abelian group under an appropriate composition rule called addition, and t...
متن کاملGroup Order Formulas for Reductions of CM Elliptic Curves
We give an overview of joint work with Karl Rubin on computing the number of points on reductions of elliptic curves with complex multiplication, including some of the history of the problem.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Canadian Journal of Mathematics
سال: 2012
ISSN: 0008-414X,1496-4279
DOI: 10.4153/cjm-2011-044-x