Paucity of rational points on fibrations with multiple fibres
Heilbronn Number Theory Seminar
26th April 2023, 4:00 pm – 5:00 pm
Fry Building, 2.04
I will report on an ongoing project with Tim Browning and Arne Smeets about the density of everywhere locally soluble members within a family of varieties. In this talk I will focus on the most general case, where components of higher multiplicity are allowed.
Using logarithmic geometry, I will derive exact conditions for a fibre to be locally soluble. 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.
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