Philadelphia Area Number Theory Seminar

Tea and snacks beforehand, starting at 3pm.

Let K be a number field and let V be a "nice" p-adic representation of the absolute Galois group of K. The Iwasawa main conjectures attach arithmetic significance to special values of the p-adic L-function attached to V. Euler systems are a powerful tool for proving such conjectures; constructing Euler systems is difficult, though, and few examples exist. We will discuss applications of a new method for constructing Euler systems, first pioneered by Sangiovanni and Skinner. To explain the key ideas we focus on a simple example (based on joint work with Shang and Skinner): we reconstruct the two simplest Euler systems, the cyclotomic and elliptic units (corresponding to V=\Q_p(1), and K=\Q or K=an imaginary quadratic field, respectively). The Euler system classes are defined as extensions of Galois representations within the étale cohomology of the modular curve relative to a collection of points (cusps or CM points, respectively). We use Eisenstein series to construct special classes in cohomology; these Eisenstein series are naturally connected to the relevant p-adic L-functions in both cases. Afterwards, we discuss new applications of this method, including forthcoming work of the speaker to construct an Euler system for the “triple product” within the cohomology of the Siegel threefold.

Let K be a field with a nonarchimedean valuation, and let C be a curve over K defined by an equation of the form y^p = f(x), where p is any prime (which is allowed to be the residue characteristic of K).  The shape of a semistable model of such a curve can be determined from the cluster data of the roots of the polynomial f.  I will explain a way to encode such cluster data as a metric graph and, using this framework, provide a criterion for C to have a special geometric property called split degenerate reduction.  Meanwhile, this property is equivalent to C being uniformizable as a certain subset of the projective line modulo the action of a group of fractional linear transformations.  I will use this uniformization to demonstrate another perspective on the cluster data of a superelliptic curve with split degenerate reduction.

Isogeny graphs of supersingular elliptic curves have broad application, from the study and computation of modular forms to post-quantum cryptography. This is in part because the family of q-isogeny graphs of supersingular elliptic curves in characteristic p (with prime p varying, for a fixed prime q) is Ramanujan. One tool for studying a graph is its Ihara zeta function, defined as an Euler product over the primes of the graph. Defining the zeta function formally requires a graph in the sense of Serre and Bass, i.e. a directed graph equipped with a fixed-point free involution on the edge set. In general, isogeny graphs fail to be graphs in this sense. In this talk, I will discuss joint work with Lau, Orvis, Scullard, and Zobernig in which we introduce abstract isogeny graphs along with their zeta functions; these graphs capture the combinatorial structure of supersingular isogeny graphs (with level structure) . I will survey some of our results, including an analogue of Ihara’s determinant formula, showing in particular that the zeta function is rational. We use this formula and the Eichler-Shimura relation to give a formula relating the zeta function of a q-isogeny graph with level-H structure (for certain H, including B0(N) and B1(N)) to the Hasse-Weil zeta functions of two associated modular curves over the finite field Fq, generalizing results of Hashimoto, Sugiyama, and Lei-Muller.

Following Kneser, we survey the notion of neighboring lattices and how the study of the neighboring relation on lattices gives rise to modular forms. We proceed to study natural maps between spaces of modular forms induced by theta series, and relate it to classical questions in number theory.