pp-adic Zeros of Quintic Forms

It is shown that a quintic form over a p-adic field with at least 26 variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds 9.Comment: Minor correction

Zeros of pp-adic forms

A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.Comment: Misprints and other errors corrected, leading to a small improvemen

Introduction to Arithmetic Mirror Symmetry

We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

Quintic Forms overp-adic Fields

AbstractWe prove that a quintic form in 26 variables defined over ap-adic fieldKalways has a nontrivial zero overKif the residue class field ofKhas at least 47 elements. This is in agreement with the theorem of Ax–Kochen which states that a homogeneous form of degreedind2+1 variables defined overQphas a nontrivialQp-rational zero ifpis sufficiently large. The Ax–Kochen theorem gives no results on the bound forp. Ford=1, 2, 3 it has been known for a long time that there is a nontrivialQp-rational zero for all values ofp. Ford=4, Terjanian gave an example of a form in 18 variables overQ2having no nontrivialQ2-rational zero. This is the first result which gives an effective bound for the cased=5

Artin's Conjecture on Zeros of pp-Adic Forms

This is an exposition of work on Artin's Conjecture on the zeros of $p$-adic forms. A variety of lines of attack are described, going back to 1945. However there is particular emphasis on recent developments concerning quartic forms on the one hand, and systems of quadratic forms on the other.Comment: Submitted for publication as part of ICM 201

Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\bold Q[z]$ with rational periodic points of period $N$. Morton proved there were none with $N=4$, by showing that the genus~$2$ algebraic curve that classifies periodic points of period~4 is birational to $X_1(16)$, whose rational points had been previously computed. We prove there are none with $N=5$. Here the relevant curve has genus~$14$, but it has a genus~$2$ quotient, whose rational points we compute by performing a~$2$-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal$_{\bold Q}$-stable $5$-cycles, and show that there exist Gal$_{\bold Q}$-stable $N$-cycles for infinitely many $N$. Furthermore, we answer a question of Morton by showing that the genus~$14$ curve and its quotient are not modular. Finally, we mention some partial results for $N=6$

Quartic Forms in Many Variables

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.Comment: 8 page