The algebraic functional equation of Riemann's theta function

TL;DR

This paper develops an algebraic analog of Riemann's theta function functional equation by constructing a theta multiplier line bundle over the moduli stack of abelian schemes and establishing its duality with the determinant bundle.

Contribution

It introduces a new algebraic framework for the functional equation of Riemann's theta function using moduli stacks and Picard group computations.

Findings

Defined a theta multiplier line bundle over the moduli stack.

Proved the dual of this bundle is isomorphic to the determinant bundle.

Recovered the classical functional equation over the complex numbers.

Abstract

We give an algebraic analog of the functional equation of Riemann's theta function. More precisely, we define a `theta multiplier' line bundle over the moduli stack of principally polarized abelian schemes with theta characteristic and prove that its dual is isomorphic to the determinant bundle over the moduli stack. We do so by explicitly computing with Picard groups over the moduli stack. This is all done over the ring R=Z[1/2,i]: passing to the complex numbers, we recover the classical functional equation.