Computational Algebra Seminar
Joint work with Andrew Sutherland (MIT).
We make a practical comparison of several algorithms for computing the zeta functions of curves of genus up to 3 over finite fields, and computing L-series of such curves over number fields. These include optimized enumeration of points, generic group algorithms improving upon the Shanks baby step-giant step approach, and p-adic cohomology (work of David Harvey). If time permits, I may also touch upon some observations regarding exceptional distributions of Frobenius eigenvalues (i.e., higher-genus analogues of the CM exception to the Sato–Tate law).
Bogomolov and Tschinkel showed that many nonsingular hypersurfaces of low degree and low dimension are densely covered by rational points (after allowing a finite extension of the base field). One of their methods involved building elliptic fibrations with good properties. I will sketch the ideas behind this and then show how to adapt methods of Cheltsov and Park to show the same in other concrete contexts. To some extent, this is about taking an ideal and working out how to push some of its variables into the coefficient field so that the rest resemble the ideal of an elliptic curve; sometimes you have to add new variables in a careful way to make it work, which I will illustrate with examples.