We discuss the Mordell–Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod  p information at primes of good reduction. We describe our implementation of the Mordell–Weil sieve algorithm and discuss its efficiency. Supplementary materials are available with this article.

/pdf/the-mordell-weil-sieve-proving-non-existence-of-rational-2p7n1lryb7.pdf

The Mordell-Weil sieve : proving non-existence of rational points on curves

We discuss the Mordell-Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell-Weil sieve algorithm and discuss its efficiency.

/pdf/the-mordell-weil-sieve-proving-non-existence-of-rational-2atz0903o7.pdf

The Mordell-Weil sieve: Proving non-existence of rational points on curves

http://www.math.brown.edu/~bhassett/papers/cubicK3/cubicK3May2011.pdf

Transcendental obstructions to weak approximation on general K3 surfaces

We dene a notion of Weyl CM points in the moduli space Ag;1 of g-dimensional principally polarized abelian varieties and show that the Andr e-Oort conjecture (or the GRH) implies the following statement: for any closed subvariety X $Ag;1 over Q a , there exists a Weyl special point [(B; )] 2 Ag;1(Q a ) such that B is not isogenous to the abelian variety A underlying any point [(A; )] 2 X. The title refers to the case when g 4 and X is the Torelli locus; in this case Tsimerman has proved the statement unconditionally. The notion of Weyl special points is generalized to the context of Shimura varieties, and we prove a corresponding conditional statement with the ambient spaceAg;1 replaced by a general Shimura variety.

/pdf/abelian-varieties-isogenous-to-a-jacobian-2luk3pb9ws.pdf

Abelian varieties isogenous to a Jacobian

Effectivity of Brauer–Manin obstructions on surfaces

Higher-Dimensional Geometry over Finite Fields

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Higher-Dimensional Geometry over Finite Fields