Abstract. I will discuss the existence of imaginary quadratic fields with class numbers prime to l and given splitting conditions at a finite set of primes.

Abstract. I will describe joint work with Manjul Bhargava (Princeton) and Tom Fisher (Cambridge) in which we determine the probability that random equation from certain families has a solution either locally (over the reals or the p-adics), everywhere locally, or globally. Three kinds of equation will be considered: quadratics in any number of variables, ternary cubics and hyperelliptic quartics.

Abstract. An irreducible algebraic variety V over a field K will be said to be diophantine-stable for a field extension L/K if V has no new L-rational points; that is, if V(L)=V(K). If V contains a curve birationally equivalent to the projective line over K, then V fails to be diophantine-stable for any nontrivial extension L/K. What happens if V doesn't contain such a curve? I will talk about results related to diophantine-stability obtained jointly with Karl Rubin. One application of our work is that for K a number field, the only smooth irreducible projective curve over K that is not diophantine-stable for any nontrivial extension L/K is the projective line over K.

Abstract. We state a new collection of reciprocity laws with coefficients in a hyperbolic curve and give examples of associated explicit formulas.

Abstract. I will start by explaining the relevance of Hecke algebras for the representation theory of p-adic groups. After briefly describing the homological Auslander-Gorenstein condition for noncommutative rings I will state a theorem that Iwahori-Hecke algebras are Auslander-Gorenstein. The main part of the talk will be devoted to the discussion of consequences of this property for the module theory of Hecke algebras. This is joint work with R. Ollivier.

Abstract. After giving a short introduction to the theory of hermitian modular forms (modular forms associated to unitary groups), we will discuss algebraic and p-adic properties of special values of L-functions attached to a Hecke eigenform. For the study of these special values there exist two general methods, the "Doubling-Method" and the "Rankin-Selberg method". In this talk we will focus on the construction of p-adic measures using the second method.

Abstract. Heegner proved that any prime congruent to 5, 7 modulo 8 or twice prime congruent to 6 modulo 8 is a congruent number. With general Gross-Zagier formula and Waldspuger formula, we are able to extend Heegner's result to the case with many prime factors by an induction method.

Abstract. Analyzing known elementary relations between U(p) operators and Picard/Albanese functoriality of the Jacobians of each tower of modular curves of p-power level, we get fairly exact control of the ordinary part of the limit Barsotti-Tate groups and the (p-adically completed) limit Mordell-Weil groups with respect to the weight Iwasawa algebra. Computing Galois cohomology of these controlled Galois modules, we hope to get good control of the (ordinary part of) limit Selmer groups and limit Tate-Shafarevich groups. We will describe the ideas and hopefully some of exact control (to the extent I can prove by the time of the conference).

Abstract. Let p be a prime number, E an elliptic curve defined over Q and set K = Q(E[p]). When does the group H^1(K/Q,E[p]) vanish? Very often it does, but not always. After explaining the list of all cases, I will include one application of the non-vanishing.

Abstract. I will explain a very explicit, conjectural relationship between first homology groups of modular curves modulo Eisenstein ideals and second K-groups of cyclotomic integer rings, in a form the mildly refines the published conjecture. Taken up the modular and cyclotomic towers, this conjecture can be viewed as a refinement of the main conjecture of Iwasawa theory. This Iwasawa-theoretic version of the original conjecture has been proven by Fukaya and Kato up to torsion when the relevant Kubota-Leopoldt p-adic L-function has no multiple zeros. I will discuss my work on a refinement of their method. Finally, I will very briefly hint at work of Fukaya, Kato, and myself on higher-dimensional analogues of the conjecture.

Abstract. In a recent work, Hida proved, under some mild assumptions, that the image of the Galois representation associated to a non CM p-adic family of p-ordinary Hecke eigenforms contains a congruence subgroup of non zero level; he also relates the best level to p-adic L-functions. In a joint work with A. Conti and A. Iovita, we generalise these results, under similar assumptions to the case of (non CM) finite slope p-adic families. The new ingredient replacing ordinarity is relative Sen theory.

Abstract. In this talk I first talk on the Galois module structure of ideal class groups and Weil etale cohomology groups, using Rubin-Stark elements, and on their relation with the zeta elements in the equivariant Tamagawa number conjecture for the Tate motive. Using these elements, I discuss Iwasawa theory over general number fields. This is a joint work with D. Burns and T. Sano. Aside from mathematics, I also plan to mention with concrete examples how deep John's understanding of Japanese culture is.

Abstract. I will talk about congruences needed to construct p-adic L-functions for elliptic curves without complex multiplication.