Schedule for 2017 UH Number Theory seminar

All the talks will take place in Keller 301, 2:30-3:30 unless otherwise specified.

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 18th Michelle Manes, Moduli spaces in arithmetic dynamics

I'll talk a bit about what we mean by a "moduli space" or a "moduli problem" in general, and then I'll talk about the construction & some properties of moduli spaces for self-maps of projective space. I'll talk about "dynamical modular curves," their relation to classical modular curves, and what (little) we know about them. If time permits, I'll talk about some special loci including the symmetry locus and the locus of post-critically finite maps.

Wednesday, November 1st Michelle Manes, Moduli spaces in arithmetic dynamics (cont'd)
Wednesday, November 8th David Yuen, Modularity of Abelian Surfaces and Computational Aspects
Wednesday, November 15th Malik Younsi, Universality of the Riemann zeta function and zero-free polynomial approximation

It is a remarkable fact that the Riemann zeta function is universal, in the sense that it encodes every possible behavior of analytic functions. More precisely, it was proved by Voronin in the 1970's that any continuous nonvanishing function on some appropriate disk that is analytic inside the disk can be uniformly approximated by a translate of the Riemann zeta function. It was subsequently proved by Gonek and Bagchi that Voronin's universality theorem holds if the disk is replaced by any compact set K with connected complement lying in the critical strip. Can the condition that the function is nonvanishing on the boundary of K be removed? The answer was conjectured to be positive by Andersson, who also proved that this is equivalent to a conjecture on zero-free polynomial approximation of analytic functions. In this talk, I will present the proof of this equivalence, and discuss related results.

Wednesday, November 22nd Ben Kane (University of Hong Kong), Sign changes of Fourier coefficients of cusp forms and representations of integers by quadratic polynomials

In this talk, we consider applications of sign changes of Fourier coefficients to the theory of quadratic forms and representations by quadratic polynomials. In one application, we prove a conjecture implying the halting of an algorithm used to construct a supersingular elliptic curve with a given endomorphism ring. We then discuss work in progress in using sign changes to classify pairs $(a,b,c)$ for which the weighted sums $a P_m(x) +bP_m(y) +cP_m(z)$ of three generslized $m$-gonal numbers are almost universal (i.e., they represent all but finitely many integers as $x,y,z$ run through all integers). The first part is joint work with King-Cheong Fung, and the work in progress is joint with Wai-Kiu Billy Chan, Anna Haensch, and Yuk-Kam Lau.