Relating decision and search algorithms for rational points on curves of higher genus

For affine plane curves defined over the rationals of genus at least two, we show that a decision algorithm for the existence of solutions also yields a search algorithm for all solutions.Comment: to be published in Archive for Mathematical Logi

A vanishing theorem for Fano varieties in positive characteristic

We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.Comment: A correct proof of the original vanishing theorem (mod torsion) has been given in the meanwhile by Esnault. This note strengthens that result slightl

The motivic fundamental group of the projective line minus three points and the theorem of Siegel

In this paper, we establish a link between the structure theory of the pro-unipotent motivic fundamental group of the projective line minus three points and Diophantine geometry. In particular, we give a p-adic proof of Siegel's theorem

Why everyone should know number theory

This was an expository lecture for the graduate student colloquium at the University of Arizona on the topic of numbers.Comment: Not for separate publicatio

Diophantine geometry and non-abelian reciprocity laws I

We use non-abelian fundamental groups to define a sequence of higher reciprocity maps on the adelic points of a variety over a number field satisfying certain conditions in Galois cohomology. The non-abelian reciprocity law then states that the global points are contained in the kernel of all the reciprocity maps.Comment: Corrected some errors, including replacing fundamental group by prime-to-2 quotien

Massey products for elliptic curves of rank 1

For an elliptic curve over Q of analytic rank 1, we use the level-two Selmer variety and secondary cohomology products to find explicit analytic defining equations for global integral points inside the set of p-adic points