We prove that the L-function of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

Abstract. We prove the compatibility of local and global Langlands correspondences for GLn, which was proved up to semisimplification in [HT]. More precisely, for the ndimensional l-adic representation Rl(Π) of the Galois group of a CM-field L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation Π, which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of Rl(Π) to the decomposition group of a prime v of L not dividing l corresponds to Πv by the local Langlands correspondence.

Abstract not found

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

Abstract. Let k be a field finitely generated over Q and let X be a smooth, separated and geometrically connected curve over k. Fix a prime ℓ. A representation ρ: π1(X) → GLm(Zℓ) is said to be geometrically strictly rationally perfect if any open subgroup of ρ(π1(Xk)) has finite abelianization. Typical examples of such representations are those arising from the action of π1(X) on the generic ℓ-adic Tate module Tℓ(Aη) of an abelian scheme A over X or, more generally, from the action of π1(X) on the ℓ-adic etale cohomology groups Hi (Yη, Qℓ), i ≥ 0 of the geometric generic fiber of a smooth proper scheme Y over X. Let G denote the image of ρ. Any k-rational point x on X induces a splitting x: Γk: = π1(Spec(k)) → π1(X) of the canonical restriction epimorphism π1(X) → Γk so one can define the closed subgroup Gx: = ρ ◦ x(Γk) ⊂ G. The main result of this note is the following uniform open image theorem. Under the above assumptions, for any geometrically strictly rationally perfect representation ρ: π1(X) → GLm(Zℓ) the set Xρ of all x ∈ X(k) such that Gx is not open in G is finite and there exists an integer Bρ ≥ 1 such that [G: Gx] ≤ Bρ, x ∈ X(k) � Xρ.

In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.

Abstract. Let N ≥ 23 be a prime number. In this paper, we prove a conjecture of Coleman, Kaskel, and Ribet about the Q-valued points of the modular curve X0(N) which map to torsion points on J0(N) via the cuspidal embedding. We give some generalizations to other modular curves, and to noncuspidal embeddings of X0(N) into J0(N). 1.

Abstract. We construct several examples of higher-dimensional Calabi-Yau manifolds and prove their modularity. 1.

dedicated to Yuri Manin on his seventieth birthday

Abstract. Let A be an abelian variety defined over a number field K and let ˆ h be the canonical height function on A ( ¯ K) attached to a symmetric ample line bundle L. We prove that there exists a constant C = C(A, K, L)> 0 such that ˆ h(P) ≥ C for all nontorsion points P ∈ A(K ab), where K ab is the maximal abelian extension of K.