Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik?s problem

Abstract

Abstract1 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(F p ) 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 problem of finding the size of the smallest prime p = p E for which the group E(F p ) is cyclic and we show that, under the generalized Riemann hypothesis,4+ε if E is without complex multiplication, and p E = O (log N ) 2+ε if E is with complex multiplication, for any 0 < ε < 1.

An analogue of the Siegel-Walfisz theorem for the cyclicity of CM elliptic curves mod p

“…The error term in the above formula has been improved significantly by Cojocaru and R. Murty in [3] under various assumptions on the Artin L-functions associated to the extensions Q(E[m])/Q. Moreover the dependence of the error term on the conductor of E has been made explicit.…”

“…Firstly, one can show that the constant c E (defined by the above series) is positive if and only if Q(E[2]) = Q (see [3,Section 6] for a proof). More precisely, if Q(E[2]) = Q, in the CM case c E ≥ 1/4, and in the non-CM case c E 1/ log log N .…”

“…In [3], Cojocaru and R. Murty, as an elliptic curve analogue of Linnik's problem, considered the problem of finding upper bounds for P (N, E). They proved, under GRH, that if E has an irrational 2-torsion point then…”

A Geometric Variant of Titchmarsh Divisor Problem

“…Our Theorem 1.4 also can be considered as a higher dimensional analogue of the cyclicity question for elliptic curves (see [20], [15], and [4]). Therefore, similar to the cyclicity question for elliptic curves, one may ask when is the density from our Theorem 1.4 positive, i.e.…”

Small exponent point groups on elliptic curves

“…For curves in extension fields, this question has been satisfactorily answered by Vlȃduţ [16]. In the situation where E is defined over Q, the question about the cyclicity of the reduction E(F p ) when p runs over the primes appears to be much harder (see [4,5,6] for recent advances and surveys of other related results). In particular, this problem is closely related to the famous Lang-Trotter conjecture.…”