Abstract. We prove a large sieve statement for the average distribution of Frobenius conjugacy classes in arithmetic monodromy groups over finite fields. As a first application we prove a stronger version of a result of Chavdarov on the “generic ” irreducibility of the numerator of the zeta functions in a family of curves with large monodromy. 1.

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

Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x / log x) of primes p ≤ x, |Γp | ≥ p r r+2 +ɛ(p), where ɛ(p) is any function of p such that ɛ(p) → 0 as p → ∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(Fp), where Fp is a finite field of p elements. Then for any δ> 0, the set of primes p for which Np has a divisor in the range (pδ−ɛ(p), pδ+ɛ(p) ) has density zero. Moreover, the set of primes p for which |Γp | < p r r+2 −ɛ(p) has density zero. Keywords: Reduction mod p of elliptic curves, Elliptic curves over finite fields, Brun-Titchmarsh inequality, Large sieve in number fields. 2000 Mathematics Subject Classification. Primary 11G20, Secondary 11N36. 1

Pour les soixantes ans de J-M. Deshouillers

Abstract. Let K be a number field and A/K be a polarized abelian variety with absolutely trivial endomorphism ring. We show that if the Néron model of A/K has at least one fiber with potential toric dimension one, then for almost all rational primes ℓ, the Galois group of the splitting field of the ℓ-torsion of A is GSp 2g (Z/ℓ). 1.

Abstract. We consider some problems of analytic number theory for elliptic curves which can be considered as analogues of classical questions around the distribution of primes in arithmetic progressions to large moduli, and to the question of twin primes. This leads to some local results on the distribution of the group structures of elliptic curves defined over a prime finite field, exhibiting an interesting dichotomy for the occurence of the possible groups. (This paper was initially written in 2000/01, but after a four year wait for a referee report, it is now withdrawn and deposited in the arXiv). Contents

Abstract. 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. 1.

Quantum unique ergodicity for SL2(O)\H 3

Abstract. In this paper, we present an improvement of a large sieve type inequality in high dimensions and discuss its implications on a related problem. 1.

Abstract. Using properties of the Frobenius eigenvalues, we show that, in a precise sense, “most ” isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and a similar result for “most ” isogeny classes. Some global cases are also treated. 1.