Hasse principle for intersections of two quadrics via Kummer surfaces
Adam Morgan (University of Cambridge)
Wednesday 15th January 16:00-17:00 Maths 311B
Abstract
I will discuss recent work with Skorobogatov establishing the Hasse principle for a broad class of degree 4 del Pezzo surfaces, conditional on finiteness of Tate--Shafarevich groups of abelian surfaces. A corollary of this work is that the Hasse principle holds for smooth complete intersections of two quadrics in P^n for n\geq 5, conditional on the same conjecture. This was previously known by work of Wittenberg assuming both finiteness of Tate--Shafarevich groups of elliptic curves and Schinzel's hypothesis (H).
I will also discuss forthcoming work with Lyczak which, again under the Tate--Shafarevich conjecture, shows that the Brauer--Manin obstruction explains all failures of the Hasse principle for certain degree 4 del Pezzo surfaces about which nothing was known previously.
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