Title: Paucity of rational points on fibrations with multiple fibres
Speaker: Julian Lyczak (University of Bath)
Abstract: This talk is based on joint work with Browning and Smeets and concerns the density of everywhere locally soluble members within a family of varieties. Our work treats the most general case, where components of higher multiplicity are allowed. Using exact conditions for a fibre to be locally soluble, coming from logarithmic geometry, we can translate the counting problem into a purely analytical question. Then using the large sieve we obtain upper bounds for the relevant counting problems for several families of examples. In some cases we can also provide lower bounds of the same order, and even asymptotics. Using these results I will discuss different ways in which the Loughran-Smeets conjecture depends on the absence of multiple fibres, and indicate the geometry involved in stating a conjecture for the general setting.
