The automorphism group of the modular curve
The average rank of elliptic curves I. With Appendix "The Explicit formula for elliptic curves over function fields" by Oisín McGuinness.
The Bauer Group of a Rational Surface.
The Bloch-Kato conjecture on special values of -functions. A survey of known results
This paper contains an overview of the known cases of the Bloch-Kato conjecture. It does not attempt to overview the known cases of the Beilinson conjecture and also excludes the Birch and Swinnerton-Dyer point. The paper starts with a brief review of the formulation of the general conjecture. The final part gives a brief sketch of the proofs in the known cases.
The Brauer group and the Brauer–Manin set of products of varieties
Let and be smooth and projective varieties over a field finitely generated over , and let and be the varieties over an algebraic closure of obtained from and , respectively, by extension of the ground field. We show that the Galois invariant subgroup of Br Br( has finite index in the Galois invariant subgroup of Br. This implies that the cokernel of the natural map Br Br Br is finite when is a number field. In this case we prove that the Brauer–Manin set of the product of...
The Brauer group of torsors and its arithmetic applications
Let be an algebraic variety defined over a field of characteristic , and let be an -torsor under a torus. We compute the Brauer group of . In the case of a number field we deduce results concerning the arithmetic of .
The Brauer-Manin obstruction on a general diagonal quartic surface
The Brauer-Manin obstructions for homogeneous spaces with connected or abelian stabilizer.
The canonical height and integral points on elliptic curves.
The capacity pairing.
The class group of a one-dimensional affinoid space
A curve over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space is trivial if and only if is a subspace of . As a consequence, has locally a trivial class group if and only if the stable reduction of has only rational components.
The cohomology of local systems on p-adically uniformized varieties.
The cohomology of Monsky and Washnitzer
The cohomology of p-adic symmetric spaces.
The Congruences of Clausen- von Staudt and Kummer for Bernoulli-Hurwitz Numbers.
The conjecture of Langlands and Rapoport for certain Shimura varieties of non-rational weight.
The defect of weak approximation for homogeneous spaces
The density of rational and integral points on algebraic varieties
The density of S-integral points in projective space with respect to a quadric
The dimension of some affine Deligne–Lusztig varieties