Abstract: Wiles' work on modularity of elliptic curves over the rationals, used as a starting point that odd, irreducible represenations $G_Q \rightarrow GL_2 (F_3)$ arise from cohomological cusp forms (i.e. new forms of weight $K \geq 2$).
Abstract: I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real multiplication points on the Drinfeld p-adic upper half-plane. Numerical experiments carried out with Vonk and James Rickards supporting the conjecture will be described.
Abstract: Euler systems are compatible families of cohomology classes for a global Galois represenation, which plan an important role in studying Selmer groups. I will outline the construction of a new Euler system, for the Galois representation associated to a cohomological cuspidal automorphic representation on the symplectic group GSp(4). This is joint work with Chris Skinner and Sarah Zerbes.
Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel.
Abstract: In his ladmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. We use deformation theory of pseudorepresentations to study the corresponding Hecke algebra. We will discuss how this method can be used to refine Mazur's results, quantifying the number of Eisenstein congruences. We'll also discuss some partial results in the composite-level case. This is joint work with Carl Wang-Erickson.