University of Bristol, UK
1:00pm-2:00pm Ulrich Derenthal (Leibniz Universität Hannover)
Manin's conjecture for certain spherical threefolds
2:15pm-3:15pm Nils Bruin (Simon Fraser University)
Visibility of 4-covers of elliptic curves
3:45pm-4:45pm Adam Morgan (King's College, London)
Parity of 2-Selmer ranks of abelian varieties over quadratic extensions
Ulrich Derenthal: Manin's conjecture for certain spherical threefolds
Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on Fano varieties. Spherical varieties admit a combinatorial description by Luna data and colored fans. In this talk, we discuss Manin's conjecture for some singular spherical threefolds. Its rational points are counted via universal torsors, which can be explicitly described using Brion's work on Cox rings of spherical varieties. This is joint work with Giuliano Gagliardi.
Nils Bruin: Visibility of 4-covers of elliptic curves
Mazur observed that in many cases where an elliptic curve E has a non-trivial element C in its Tate-Shafarevich group, one can find another elliptic curve E' such that ExE' admits an isogeny that kills C. For elements of order 2 and 3 one can prove that such an E' always exists. However, for order 4 this leads to a question about rational points on certain K3-surfaces. We show how to explicitly construct these surfaces and give some results on their rational points.
Adam Morgan: Parity of 2-Selmer ranks of abelian varieties over quadratic extensions
For an abelian variety A over a number field K, a consequence of the Birch and Swinnerton-Dyer conjecture is the 2-parity conjecture: the global root number agrees with the parity of the 2-infinity Selmer rank. It is a standard result that the root number may be expressed as a product of local terms and we show that, over any quadratic extension of K, the same holds true for the parity of the 2-infinity Selmer rank. Using this we prove several new instances of the 2-parity conjecture for general principally polarised abelian varieties by comparing the local contributions arising. Somewhat surprisingly, the local comparison relies heavily on results from the theory of quadratic forms in characteristic 2.
The talks will take place in the seminar room on the 4th floor of Howard House. Coffee and tea will be available in the 4th floor common room at 3:15pm.
For a map of how to find Howard House click here and for information about getting to University of Bristol go here. If you require accommodation then here are some local hotels.