Hilbert theta series and invariants of genus 2 curves

In this paper, we compute pull-backs of Siegel theta functions to the Hilbert moduli space and consider their application to generating genus 2 curves for cryptography. We express invariants of genus 2 curves such as the Gundlach invariants and Rosenhain invariants in terms of these Hilbert theta functions. A result of independent interest is a simple formula in terms of these functions for the Eisenstein series of weight 2, which is not the pull-back of any Siegel modular form of level 1. We present an algorithm to compute minimal polynomials for the invariants, including a description of CM points and how to compute them, along with numerical examples.