Fri, 30 Jan 2026
12:00
12:00
Quillen Room N3.12
Three realisations of theta functions via the Heisenberg representation
Allan Perez Murillo
(University of Bristol)
Abstract
The classical theta functions appear throughout number theory, geometry, and physics, from Riemann’s zeta function to the projective geometry of abelian varieties. Despite these appearances, theta functions admit a unifying description under the lens of representation theory.
In this talk, I will explain how the Heisenberg representation, together with the Stone–von Neumann–Mackey theorem, provides a framework that
identifies three equivalent realizations of theta functions:
- as holomorphic functions on certain symplectic spaces
- as matrix coefficients of the Heisenberg (and metaplectic) representation,
- as sections of line bundles on abelian varieties.
I will describe how these perspectives fit together and, if time permits, illustrate the equivalence through concrete one-dimensional examples. The
emphasis will be on ideas rather than technicalities. I will aim to make the talk self-contained, assuming familiarity with complex geometry and representation theory; background in Lie theory and harmonic analysis will be helpful but not essential.
Effective Altruism Oxford’s Arete Fellowship