Event Details

Speaker: Dan Fretwell (Lancaster). Title: Ternary lattices, elliptic curves, modular forms and Ramanujan graphs.

Abstract: How does one classify all quadratic spaces over the rationals, up to isometry? You consider the possible completions, i.e. the corresponding real and p-adic quadratic spaces (by Hasse-Minkowski). How does one classify all lattices over the integers, up to isometry? This is famously an open problem…going all the way back to work of Euler, Gauss, and Legendre. Completions only get us so far and the class set Cl(L) measures the failure (the set of lattices passing all local isometry tests with L, up to global isometry).

Seminal work of Kneser tells us how to compute Cl(L) using his theory of p-neighbours (used later by Borcherds to classify unimodular lattices in 24 dimensions). A p-neighbour move allows us to navigate Cl(L) in a non-trivial way, and naturally one wonders how the p-neighbours of a lattice distribute among the classes in Cl(L). To this end we can construct families G_p of directed graphs that capture this information and study the adjacency matrices T_p. Prior work of Birch has showed that these adjacency matrices have interesting spectra, relating to elliptic curves and modular forms (which eventually feeds back into proving that these graphs are Ramanujan graphs!).

In this talk, I will give a survey of the above ideas and finish by featuring new results (joint with Samadrito Bhattacharyya and Giovanna De Lauri).