What are the odds that a conic bundle over a finite field is rational?

Abstract

We study rationality of conic bundles over finite fields defined over the line. In particular, for a fixed number $n$ of degenerate fibers, what is the proportion of good conic bundles that are rational over a finite field of order $q$? The Galois action on their Picard groups factors through the Weyl group $W(D_n)$, reducing rationality to a condition on signed permutation types. Using work of Colliot-Thélène and Yang together with Chebotarev density, we compute the asymptotic proportion of such conic bundles that are rational as $q$ goes to infinity.

19 pages, 1 figure