Geometric Proofs of Riemann-Type Theta Relations

Source abstract

We present a geometric method for deriving theta function identities from automorphisms of positive definite quadratic forms. Given an automorphism of a quadratic form with matrix $A$, we construct a rational isometry of $I_n\otimes A$ that fixes the diagonal subspace pointwise. Elementary lattice decompositions and finite character sums then yield explicit transformation formulas for the associated theta functions. As applications, we give a geometric proof of Schultz's generalization of the Borweins' cubic theta identity and recover two related quadratic theta identities.