Computing Theta Characteristics and their Galois action on general curves of moderate genus

Source abstract

In this article, we present an algorithm that, given a smooth curve $X$ defined over $\mathbb{Q}$, returns an explicit arithmetic representation of both odd and even Theta Characteristics split by parity. It allows us to retrieve their Galois action and étale algebra. The output includes the Theta Hyperplanes of the Theta Characteristics when they are finite, their associated proof functions as well as the dictionary of their quadratic forms. The algorithm works in a rather large generality as it handles any one-equation (potentially singular) model of smooth curves. The most binding requirement remains the necessity for an explicit degree 1 rational divisor on the curve or more likely a rational point. We implemented the algorithm and give examples up to genus $5$.