unQVNTS (Vermont)

Given a congruence subgroup of SL_2(Z), one might wonder if one can know all of the modular forms for this subgroup, by giving generators and relations for the algebra of modular forms for this subgroup. If so, one would be delighted to find out that the work of Voight and Zureick-Brown tells us how to do this! In this talk we introduce modular forms and modular curves, and the main results of their paper.

We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds on the rank of the Mordel-Weil group for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such family occurs in the congruent number problem. We consider the congruent number problem over these quadratic number fields, and subject to the finiteness of Sha, we show that there are infinitely many numbers that are not congruent over Q but become congruent over Q(sqrt(D)).

In this talk we will discuss some new results related to torsion subgroups of elliptic curves, more concretely about division fields and isogeny-torsion graphs. Let E/Q be an elliptic curve and let p be a prime. The p-th division field, denoted by Q(E[p]), is the extension of Q generated by the coordinates of all the non-trivial p-torsion points on E. In this talk we will describe the possible coincidences between division fields for different primes, that is, we will discuss when Q(E[p])=Q(E[q]) is possible. Moreover, a non-trivial intersection of division fields is called an entanglement. Here, we will describe all the possible abelian entanglements for elliptic curves over Q. (This is joint work with Harris Daniels.)
In the second part of the talk, we will define the isogeny-torsion graph attached to E/Q as a graph where each vertex is a pair consisting of an elliptic curve and its torsion subgroup over Q, and the edges represent the cyclic rational isogenies among the elliptic curves in the same isogeny class. Then, we will discuss a classification of all the possible isogeny-torsion graphs that occur for elliptic curves defined over the rationals. (This is joint work with Garen Chiloyan.)

I will discuss a number of things that we found when documenting our database of isogeny classes of abelian varieties over finite fields in the LMFDB. The talk will include a discussion of "Not-o-Tate" distributions (or "Sato-Ain't" if you prefer) and a counter-example to a conjecture of Ahmadi and Shparlinski regarding angle ranks of ordinary Jacobians (this is some stuff related to the Tate conjecture relating to algebraic cycles). Time permitting I will discuss some other weird stuff. This is joint work with Kiran Kedlaya, David Roe, and Christelle Vincent.

Let Gamma a subgroup of SL_2(Z) be a congruence subgroup. This talk will discuss the algorithms for computing q-expansions of elements in S_k(Gamma), the space of cusp forms of weight k and level Gamma. I will survey the necessary background material and discuss recent results and applications to contemporary research regarding Serre's uniformity problem.