Title: Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts.

Abstract: The problem of classifying congruences between automorphic forms has attracted a considerable amount of attention due not only to its inherent interest, but also because of the arithmetic applications of many of these congruences.  For instance, congruences between automorphic forms have played a crucial role in recent progress on the Birch and Swinnerton-Dyer conjecture.  In this talk we will discuss a sufficient condition for a prime to be a congruence prime for an automorphic form on $\te$U(n,n)($𝔸_F$) where $F$/$ℚ$ is a totally real field. This sufficient condition is given in terms of the divisibility of a certain special $L$-value of the automorphic form.  We then apply this result to the case of the Hermitian Ikeda lift.  This work is joint with Kris Klosin.

Title: Random Hypersurfaces and Embedding Curves in Surfaces

Abstract: We'll present two new applications of Poonen's closed point sieve over finite fields.  The first is that the obvious local obstruction to embedding a curve in a smooth surface is the only global obstruction.  The second is a proof of a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

Title: Arithmetic intersections on Hilbert modular surfaces and the Jacquet-Langlands correspondence

Abstract: In this talk I will review joint work with D. Eriksson (Chalmers Univ.) and S. Sankaran (McGill Univ.). The Riemann-Roch theorem in Arakelov geometry relates the determinant of the cohomology of a hermitian vector bundle on a proper arithmetic variety to some arithmetic intersection numbers. There are cases of interest to which the formula does not apply, like the Hilbert modular varieties we study here. It is not clear how to prove such a result, not even the right statement. Instead, we try to conjecture a sensible formula that is compatible with the Riemann-Roch formula on a compact Shimura curve, through the Jacquet-Langlands correspondence. In particular, we need to give a meaning to holomorphic analytic torsion, and relate it to the one of a compact Shimura curve. This complements very well with classical results of Shimura for norms of Petersson forms, needed as well in our discussion. I will try to motivate and present the main lines of this work.

Title: Holomorphic and phantom holomorphic projection

Abstract: For Siegel modular forms, Sturm-type formulas describe the holomorphic part of a nonholomorphic form generally when its weight is larger than twice the genus or in the  case of weight two for the classical genus one. We establish this holomorphic projection in case of genus two and scalar weight four. Applying the same method for weight three produces additional nonholomorphic terms which we call phantoms. We show that their occurrence is not a coincidence of our choices (Poincaré$\'e$$$ series) but a property of Sturm's operator. (These results are partially joint with Rainer Weissauer.)