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

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1 Let E be an elliptic curve over Q. For a prime p of good reduction, let Ep be the reduction of E modulo p. We investigate Koblitz’s Conjecture about the number of primes p for which Ep(Fp) has prime order. More precisely, our main result is that if E is with Complex Multiplication, then there exist infinitely many primes p for which #Ep(Fp) has at most 5 prime factors. We also obtain upper bounds for the number of primes p ≤ x for which #Ep(Fp) is a prime. 1

Abstract. We study the density of integral points on punctured abelian surfaces. Linear growth rates are observed experimentally. 1

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 simpler unconditional proof of this asymptotic formula, and also to provide explicit error terms in the formula.

Introduction Let K be a global field of char p and let F q K denote the algebraic closure of F p in K. We fix an elliptic curve E/K with non-constant j-invariant and a torsion-free subgroup E(K) of rank r > 0. We write V for the open set of places v of K such that the special fiber E v is an elliptic curve and, for v in V , we let # v E v (k v ) be the image of # under reduction modulo v, where k v is the residue field of K at v. We fix a finite set of (rational) prime numbers S which is large enough to include the exceptional primes which we will define explicitly in section 2.4 and section 3), and we let G(#, S) denote the subset of v V such that # v contains the prime-to-S part of E v (k v ). For every n > 0, we write V n for the subset of v V such that deg(v) = n and let G n (#, S) = V n G(#, S). Theorem 1. Suppose r 6. There exists constants a, b satisfying 0 1 and depending only on r and S such that, for each n 1 there exists # n (#, S), depending on r, S an

Let E be an elliptic curve defined over Q. For any prime p of good reduction, let Ep be the reduction of E mod p. Denote by Np the cardinality of Ep(Fp), where Fp is the finite field of p elements. Let P (Np) be the greatest prime divisor of Np. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, P (Np)> p ϑ(p), where ϑ(p) is any function of p such that ϑ(p) → 0 as p → ∞. Moreover we show that for such E there is a positive proportion of primes p ≤ x for which P (Np)> p ϑ, where ϑ is any number less than ϑ0 = 1 − 1 2 prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have where ɛ> 0. e− 1 4 = 0.6105 · · ·. As an application of this result we |Γp |> p ϑ0−ɛ Keywords: Reduction mod p of elliptic curves, Elliptic curves over finite fields, Brun-Titchmarsh inequality in number fields, Bombieri-Vinogradov theorem in number fields, Abelian extensions of imaginary quadratic number fields. 2000 Mathematics Subject Classification. Primary 11G20, Secondary 11N37. 1

Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ) valid for the range l < log x is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are determined. RESUM E Nous considerons di#erentes generalisations de la conjecture d'Artin pour les racines primitives. Nous demontrons que pour au moins la moitie des nombres premiers p, les premiers log p nombres premiers engendrent une racine primitive. Nous demontrons une version uniforme du Theoreme de Densite de Chebotarev pour le corps Q(# l , 2 ) pour l'intervalle l < log x. On etablit une formule asymptotique uniforme pour les nombres de premiers plus petits que x tels qu' il existe une racine primitive plus petite que s. Nous determinons des minorants pour l'exposant du groupe de classe des corps quadratiques imaginaires valides pour ensembles de discriminants de densite 1. Contents

Abstract. 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 ¯ Γ 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 | ¯ Γ | ≥ p f(p), where f(x) is any function such that f(x) → ∞, at an arbitrary slow speed, as x → ∞. This provides additional evidence in support of a conjecture of Lang and Trotter from 1977. 1.

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