Barry Mazur asked about the nature of the coordinate functions that give the j-invariant of the canonical lifting of an ordinary elliptic curve, as a Witt vector, in terms of the j-invariant the ordinary elliptic curve. Tate asked about the possibility to extend these functions to supersingular values. We will discuss these two problems and the necessary improvements in computations with Witt vectors that were necessary to obtain data and give the answers. The slides for this talk can be found here.

       The SIC-POVM conjecture from quantum information theory posits the existence of d² equiangular lines in each complex dimension d, saturating a known upper bound. In this talk, I will discuss how extensive computer calculations by Appleby-Appleby-Zauner and Scott-Grassl indicate that such equiangular sets can always be obtained as orbits of finite Heisenberg groups acting on vectors over ray class fields of real quadratic fields. The symmetries of the solutions also point toward a generalization of Shimura reciprocity in this setting.

       In part 1, we will motivate the definition of modular symbols for cusp forms of weight 2 by considering special values of L-functions of elliptic curves over the rationals. In part 2, we will generalize this definition to higher weights, define Hecke actions, and state the Eichler-Shimura Theorem. The slides for this talk can be found here.

       Let K be a number field and let G be a finite group. In the locally free class group Cl(O_K[G]) we can consider the subset of classes realized as Galois modules by the ring of integers of tamely ramified G-extensions. The study of the structure of this subset goes under the name of realizable classes problem. In this talk, after a general presentation of the problem, I will give an overview of the known results and I shall point out the main recent open questions on this topic. No prior knowledge about locally free class groups and realizable classes will be assumed.