|Wednesday, September 6th|| Luca Candelori, Periods of modular curves
The periods of a Riemann surface X are complex numbers obtained by integrating holomorphic differential 1-forms against classes of loops in integral homology. The study of these periods is classical and can be traced back to the work of Abel and Jacobi in the 19th century. For X a modular curve corresponding to a sub-group G of the modular group, the holomorphic 1-forms correspond to weight 2 cusp forms on G, thus computing periods naively involves computing the Fourier expansions of weight 2 cusp forms. We present here a new method to compute periods without the Fourier expansions. This method works equally well for congruence and non-congruence subgroups G, and in certain cases it can be used to establish whether the periods of X are algebraic. This is joint work with T. Hartland, C. Marks, D. Yepez.
|Wednesday, September 13th||Luca Candelori, Periods of modular curves (cont'd)|
|Wednesday, September 20th||Kiran Kedlaya (UCSD), Modular forms and ternary quadratic forms: revisiting a method of Birch
In 1991, Birch proposed a conjectural method for computing Hecke operators on modular forms using lattices and quadratic forms. We report on work of Hein, Tornaria, and Voight which establishes the conjecture of Birch and provides a blisteringly fast C++ implementation of the method for forms of weight 2 and squarefree level (which we are currently working on importing into Sage).
|Monday, September 25th, Keller 402||Alina Bucur (UCSD), Statistics for points on curves over finite fields
A curve is a one dimensional space cut out by polynomial equations. In particular, one can consider curves over finite fields, which means the polynomial equations should have coefficients in some finite field and that points on the curve are given by values of the variables in the finite field that satisfy the given polynomials. A basic question is how many points such a curve has, and for a family of curves one can study the distribution of this statistic. We will give concrete examples of families in which this distribution is known or predicted, and give a sense of the different kinds of mathematics that are used to study different families. Our main focus will be the family of cyclic prime degree covers, which can be approached both via the original combinatorial/analytic approach and via maps from the idele class group. This is joint work with Chantal David, Brooke Feigon, Nathan Kaplan, Matilde Lalin, Ekin Ozman and Melanie Matchett Wood.
|Wednesday, October 4th||Michelle Manes, Moduli spaces in arithmetic dynamics|