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

For n ≥ 3, let Γ = SLn(Z). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a factor and let U(M) be its unitary group. Let π: Γ → U(M) be a group homomorphism such that π(Γ) ′ ′ = M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π|Λ extends to a homomorphism U(L(Λ)) → U(M). This answers, in the special case of SLn(Z), a question of A. Connes discussed in [Jone00, Page 86]. The result is deduced from a complete description of the tracial states on the full C ∗ –algebra of Γ. As another application, we show that the full C ∗ –algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg. 1