Dissertations, Theses, and Capstone Projects

Date of Degree

6-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Lucien Szpiro

Committee Members

Aise Johan de Jong

Alexander Gamburd

Victor Kolyvagin

Andrew Obus

Subject Categories

Algebraic Geometry | Mathematics | Number Theory

Keywords

arithmetic statistics

Abstract

This thesis comes in four parts, which can be read independently of each other.

In the first chapter, we prove a generalization of Poonen's finite field Bertini theorem, and use this to show that the obvious obstruction to embedding a curve in some smooth surface is the only obstruction over perfect fields, extending a result of Altman and Kleiman. We also prove a conjecture of Vakil and Wood on the asymptotic probability of hypersurface sections having a prescribed number of singularities.

In the second chapter, for a fixed base curve over a finite field of characteristic at least 5, we asymptotically count its degree three covers of given genus, as the genus increases. This gives an algebro-geometric proof of results of Datskovsky and Wright, as well as Bhargava, Shankar, and Wang, on asymptotically counting cubic field extensions.

In the third chapter, for D a non-empty effective divisor on P^1, we show that any set of (D,S)-integral points of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let phi(z) in \Qbar(z) be a rational function of degree at least two whose second iterate is not a polynomial. We show that as we vary over points P in P^1(\Qbar) of bounded degree, the number of algebraic integers in the forward orbit of P is absolutely bounded and zero on average.

In the fourth chapter, we count algebraic numbers. Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree d and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over Z) in a homogeneously expanding star body in R^{d+1}. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one ``slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which allows us to count units, algebraic integers of given norm, trace, norm and trace, and more. We also refine the lattice point-counting arguments of Chern-Vaaler to obtain explicit error terms with better power savings, which lead to explicit versions of some results of Masser-Vaaler and Barroero. Our results can be interpreted as counting rational points and integral points on P^d.

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