Thibaut Misme

Towards an automatized 2-descent on Jacobians of general curves.

Descent procedures are an essential tool to compute rational points on curves. At present, our ability to perform them are very restricted. In this talk, I will present a forthcoming algorithm whose motivation is the possibility to perform an automatized 2-descent on Jacobians of general curves of moderate genus and coefficients. I will begin the talk by defining the necessary background, and present the algorithm which does the following: given a one-equation (singular) model of a smooth algebraic curve and a rational point, it returns an algebraic representation of the Theta Characteristics using Makdisi’s techniques and some of their invariants. In particular, it includes a dictionary of their parity and associated quadratic forms, their Galois action as their etale algebra, the equations of their Theta Hyperplanes (when it makes sense) and some specific functions. If time remains, I will describe how the algorithm works.