72 research outputs found
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
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
ABC inequalities for some moduli spaces of log-general type
We prove a new bound for the Arakelov-Faltings height of an abelian variety
over a function field of characteristic zero and relate it to inequalities of
ABC-type as conjectured by Buium and Vojta.Comment: 7 page
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
- …