April 6 |
Cristian Popescu (UCSD) TBA |
April 13 |
Chantal David (Concordia University and MSRI) We investigate in this talk the average root number (i.e. sign of the functional equation) of non-isotrivial one-parameter families of elliptic curves (i.e elliptic curves over Q(t), or elliptic surfaces over Q). For most one-parameter families of elliptic curves, the average root number is predicted to be 0. Helfgott showed that under Chowla's conjecture and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative reduction over Q(t). We then build non-isotrivial families of elliptic curves with no place of multiplicative reduction, and compute the average root number of the families. Some families have periodic root number, giving a rational average, and some other families have an average root number which is expressed as an infinite Euler product. We then prove several density results for the average root number of non-isotrivial families of elliptic curves, over Z and over Q (the previous density results found in the literature were for isotrivial families). We also exhibit some surprising examples, for example, non-isotrivial families of elliptic curves with rank r over Q(t) and average root number -(-1)^{r}, which were not found in previous literature. Joint work with S. Bettin and C. Delaunay. |
April 20 |
Philippe Michel (EPFL and MSRI)
In a series of recent works Blomer, Fouvry, Kowalski, Milicevic, Sawin and myself have been able to solve the vexing problem of evaluating asymptotically the second moment of the central L-values of character twists (of large prime conductor) of a fixed modular form; the solution combines the spectral theory of modular forms, bounds for bilinear sums of Kloosterman sums and advanced methods in l-adic cohomology. We will describe the proof and especially the second and third ingredient which is joint work with E. Kowalski an W. Sawin. |
April 27 |
TBA TBA |
May 4 |
TBA TBA |
May 11 |
Frank Thorne (University of South Carolina and MSRI) One important technical ingredient in many arithmetic statistics papers is upper bounds for finite exponential sums which arise as Fourier transforms of characteristic functions of orbits. This is typical in results obtaining power saving error terms, treating "local conditions", and/or applying any sort of sieve. In my talk I will explain what these exponential sums are, how they arise, and what their relevance is. I will outline a new method for explicitly and easily evaluating them, and describe some pleasant surprises in our end results. I will also outline a new sieve method for efficiently exploiting these results, involving Poisson summation and the Bhargava-Ekedahl geometric sieve. For example, we have proved that there are "many" quartic field discriminants with at most eight prime factors. This is joint work with Takashi Taniguchi. |
May 18 |
Tim Browning (University of Bristol and MSRI) I will discuss recent joint work with Pankaj Vishe, in which we are able to say something about the naive moduli space of rational curves on arbitrary smooth hypersurfaces of sufficiently low degree, by invoking methods from analytic number theory. |
May 25 |
Anne Carter (UCSD) TBA |
June 1 |
TBA TBA |
June 8 |
TBA TBA |