Selected Publications
- On the decoding of algebraic-geometric codes. IEEE Trans. Inform. Theory 36 (1990) 1051–1060. (with S. G. Vladut)
- Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points. Inv. Math. 134 (1998) 579–650. (with J.-L. Colliot-Thélène and Sir Peter Swinnerton-Dyer)
- Beyond the Manin obstruction. Inv. Math. 135 (1999) 399-424.
- Rational points on pencils of conics and quadrics with many degenerate fibres. Ann. of Math. 180 (2014) 381-402 (with T. Browning and L. Matthiesen)
- Finiteness theorems for K3 surfaces and abelian varieties of CM type. Compositio Math. 154 (2018) 1571-1592. (with M. Orr)
FRIAS Project
p-adic methods in the Brauer–Manin obstruction
The area of the proposed research is a modern development of Diophantine equations, a classical area of number theory. This subject belongs to arithmetic and algebraic geometry, and concerns rational points of algebraic varieties over local and global fields, the Brauer group and the Brauer–Manin obstruction. I would like to investigate new p-adic methods that apply fundamental results of K. Kato to the evaluation of elements of the Brauer group at local points when the torsion of a Brauer element is divisible by the residual characteristic. These methods have applications to the computation of the Brauer–Manin set. One concrete situation is that of diagonal surfaces in the projective space, where some work in this direction has already been done. This is related to the current research on open problems concerning rational points and reduction of K3 surfaces.