Synergies and Vistas in Analytic

Châtelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. We investigate the frequency that such counter-examples arise over the rationals. This is joint work with T. Browning. Jörg Brüdern, On the Möbius function Alina Cojocaru, Frobenius fields for elliptic curves Abstract: Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Q(πp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Q(πp) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. The latter is joint work with H. Iwaniec and N. Jones. Brian Conrey, A q-analogue of the Balasubramanian, Conrey, HeathBrown conjecture Abstract: In joint work with Iwaniec and Soundararajan we use the asymptotic large sieve to obtain an asymptotic estimate for the average over q and over primitive characters modulo q of the mean square of the the central value of the Dirichlet L-function multiplied by a Dirichlet polynomial of length nearly q. As an application we can prove that in a certain average sense at least 3/5 of the zeros of Dirichlet L-functions are on the critical line. In joint work with Iwaniec and Soundararajan we use the asymptotic large sieve to obtain an asymptotic estimate for the average over q and over primitive characters modulo q of the mean square of the the central value of the Dirichlet L-function multiplied by a Dirichlet polynomial of length nearly q. As an application we can prove that in a certain average sense at least 3/5 of the zeros of Dirichlet L-functions are on the critical line. Chantal David, Elliptic curves with prescribed groups over finite fields and Cohen–Lenstra Heuristics Abstract: Let Gm,k := Z/mZ × Z/mkZ be an abelian group of rank 2 and order N = mk. When does there exist a finite field Fp and an elliptic curve E/Fp such that E(Fp) ' Gm,k? More precisely, let S(M,K) = {m 6M,k 6 K : ∃p, E/Fp with E(Fp) ' Gm,k} . It was conjectured by Banks, Pappalardi and Shparlinski that #S(M,K) = { o(MK) if K (logM)2−ε MK(1 + o(1)) if K (logM). We prove in this talk that the first part of their conjecture holds for the whole range K (logM)2−ε, and that the second part holds for the limited range K > M1/4− . We also show that #S(M,K) MK holds for a larger range. The fact that the groups Gm,k are more likely to occur when m is small is reminiscent of the Cohen–Lenstra heuristics which predict that a random abelian group G occurs with probability weighted by #G/#Aut(G). In order to see that the probability of occurrence of the groups Gm,k is really weighted by the factors #Gm,k/#Aut(Gm,k), one should count the number of times that a given group Gm,k occurs over the finite fields Fp, and not only when a given group occurs. Let G be a group as above of order N , and let Mp(G) = # {E/Fp : E(Fp) ' G} . We show that, under a suitable hypothesis for the number of primes in short intervals, we have for #G/(log (#G)) exp(G)