Algebraic Theory of Indefinite Theta Functions

Jacobi's theta function Θ(q):=1+2q+2q4+2q9+…, and more generally the theta functions associated to positive-definite quadratic forms, have the property that they are modular forms of half-integral weight. The usual proof of this fact is completely analytic in nature, using the Poisson summation formula. However, Θ is also related to diffusion of heat on a uniform circle-shaped material: it is a restriction of the fundamental solution to the heat equation on a circle. By algebraically characterizing the heat equation as a specific flat connection on a certain bundle on a modular curve, we produce a completely algebraic technique for proving modularity of theta functions. More specifically, we produce a refinement of the algebraic theory of theta functions due to Moret-Bailly, Faltings--Chai, and Candelori. As a consequence of the algebraic nature of our theory and the fact that it applies to indefinite quadratic forms / non-ample line bundles (which the prior algebraic theory does not), we also generalize the Kudla--Millson analytic theory of theta functions for indefinite quadratic forms to the case of torsion coefficients. This is joint work in progress with Akshay Venkatesh.