Sparsity of Rational and Algebraic Points
高紫阳教授
2023-08-11 15:40-17:10
国科大S302教室
2023代数与数论暑期学校 数学前沿讲座时间：8月11日（周五）15:40-17:10
地点：国科大S302教室
报告人：高紫阳教授（Leibniz Universität Hannover）
题目：Sparsity of Rational and Algebraic Points
摘要：It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.
In my talk, I will explain the historical and recent developments of this problem according to the different grades. Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.
