Diophantine Equations and Algebraic Points on Curves

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

The investigator studies diophantine properties of points on curves and related questions involving solutions to diophantine equations. A diophantine equation is simply a polynomial equation with integer coefficients. One is generally interested in solutions to these equations that are rational numbers or integers. The investigator uses the p-adic method Chabauty-Coleman to study the number of solutions to Thue equations, which are equations of the form F(x,y) = m, where m is an integer and F is a homogeneous polynomial with integer coefficients and without repeated roots. The investigator has shown that the number of integer solutions (x,y), with x and y coprime, can be bounded in terms of the degree of F under certain hypotheses and is now working on weakening these hypotheses, sharpening his bounds, and extending his methods to attack other similar problems. The investigator also looks at various questions related to Vojta's conjecture for algebraic points on curves, which implies a variety of important results in diophantine number theory, most notably the abc conjecture. In addition, he uses techniques from arithmetic geometry to study class groups of quadratic fields, consequences of the abc conjecture, and curves over finite fields. The investigator studies problems having to do with whole number and rational solutions to polynomial equations in two or more variables. Such problems are among the oldest in the branch of mathematics known as number theory. Indeed, they derive their name "diophantine equations" from Diophantus of Alexandria, a Greek mathematician who lived in the 2nd century BC. Over the past few years, mathematicians have succeeded at completely solving many diophantine equations that were first examined hundreds of years ago: Andrew Wiles solved the famed Fermat equation (which has the form "x to the nth power plus y to the nth power equals z to the nth power", where x,y, and z are positive whole numbers and n is a whole number greater than or equal to 3) that was initially studied by French mathematician Pierre Fermat in the 1600's, and others have solved problems posed by Diophantus himself. Contemporary investigations of diophantine equations have given rise to a host of new questions and conjectures; for example, it has been conjectured that the number of solutions in rational numbers to a polynomial equation in two variables depends only on the degree of the equation. Number theory has proven to be of more than historical and theoretical interest in recent years. It provided the theoretical ideas that led to the development of error-correcting codes employed by compact disc players and has found applications in the areas of cryptography and data encryption, which are important both for national security and for e-commerce.