The Jacobi theta function near the real line

The Jacobi elliptic theta function is a function of two complex variables, \tau (with positive imaginary part) and z. As \tau approaches the real line, the function may blow up. In a joint work with Tariq Osman (Queen’s University), we find a uniform bound for generalised Weyl sums using homogeneous dynamics. As a consequence, we find affine curves z=z(\tau) along which the blow up in the Jacobi theta function can be uniformly controlled.