Diophantine Equations and Algebraic Points on Curves

研究者研究曲线上点的丢番图性质和涉及丢番图方程解的相关问题。丢番图方程是一个具有整数系数的多项式方程。人们通常对这些有理数或整数方程的解感兴趣。研究者使用p进方法Chabauty-Coleman来研究Thue方程的解的数量，这些方程的形式为F(x,y) = m，其中m是整数，F是具有整数系数且没有重复根的齐次多项式。研究人员已经证明，在某些假设下，具有x和y的素数的整数解（x,y）的数量可以用F的程度来限定，现在正在努力削弱这些假设，锐化他的界限，并扩展他的方法来解决其他类似的问题。研究者还研究了与曲线上代数点的Vojta猜想有关的各种问题，该猜想暗示了丢芬图数论中的各种重要结果，其中最著名的是abc猜想。此外，他还利用算术几何的技巧来研究二次域的类群、abc猜想的结果和有限域上的曲线。研究者研究与两个或多个变量多项式方程的整数和有理数解有关的问题。这些问题是数论这一数学分支中最古老的问题之一。事实上，他们的名字“丢番图方程”来源于亚历山大的丢番图，一位生活在公元前2世纪的希腊数学家。在过去的几年里，数学家们已经成功地完全解决了许多丢番图方程，这些方程在几百年前首次被研究：安德鲁·怀尔斯解决了著名的费马方程（形式为“x的n次方加上y的n次方等于z的n次方”，其中x，y和z是正整数，n是大于等于3的整数），这个方程最初是由法国数学家皮埃尔·费马在17世纪研究的，其他人也解决了丢芬图自己提出的问题。当代对丢番图方程的研究产生了许多新的问题和猜想；例如，人们已经推测，两个变量的多项式方程的有理数解的数量只取决于方程的程度。近年来，数论已被证明具有比历史和理论更重要的意义。它提供的理论思想导致了激光唱机采用的纠错码的发展，并在密码学和数据加密领域找到了应用，这对国家安全和电子商务都很重要。

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.