منابع مشابه
CYCLICITY OF CM ELLIPTIC CURVES MODULO p
Let E be an elliptic curve defined over Q and with complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. We find the density of the primes p ≤ x for which E(Fp) is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new ...
متن کاملAlmost Prime Orders of CM Elliptic Curves Modulo p
Let E/Q be an elliptic curve over the rationals and, for any prime p, let us denote by |E(Fp)| the number of Fp points of the reduced curve modulo p. Consider the function ΠE,s(x) = #{p ≤ x : |E(Fp)| = Ps}, where Ps denotes a squarefree number with at most s prime divisors. In particular P1 denotes a prime number. In 1988 Koblitz, motivated by criptographic applications, (see [1]), conjectured ...
متن کاملSquare-free orders for CM elliptic curves modulo p
Let E be an elliptic curve defined over Q, of conductor N , and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes p ! N , p ≤ x , for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which ...
متن کاملCyclicity of Elliptic Curves Modulo p and Elliptic Curve Analogues of Linnik’s Problem
1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.-P. Serre and M. Ram Murty. We also consider the prob...
متن کاملAN ANALOGUE OF THE SIEGEL-WALFISZ THEOREM FOR THE CYCLICITY OF CM ELLIPTIC CURVES MOD p
Let E be a CM elliptic curve defined over Q and of conductor N . We establish an asymptotic formula, uniform in N and with improved error term, for the counting function of primes p for which the reduction mod p of E is cyclic. Our result resembles the classical Siegel-Walfisz theorem regarding the distribution of primes in arithmetic progressions.
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 2003
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-03-03283-5