“椭圆曲线有理点问题研究”年度报告

田野 王崧

摘要：在该计划第一年，我们按照年度计划，在理论基础准备部分取得相当进展。其中我们在椭圆曲线算术、同余数及千禧问题BSD猜想上取得重要成果。我们利用现代数论、算术代数几何、表示论、自守形式的系列结果，证明了对任意给定的正整数k，存在无穷多个没有平方因子的恰巧有k个奇素因子的同余数，并发展了系列新的方法工具（如二次扭转欧拉系）。这些成果有助于我们更加深入理解椭圆曲线的算术理论，并为下一步研究提供了充足的理论基础和方法准备。另外我们在解析数论、密码编码相关问题上取得一定进展，改进了Green-Tao关于F_2^n和集的一个结果。而且，在代数簇有理点，Brauer群方面取得系列进展，利用Brauer-Manin障碍技术给出了虚二次数域平方和问题的充要条件。另外，在椭圆曲线、代数簇有理点、自守形式、p-adic分析以及经典数论等其它的基础准备方面均取得一定的进展。

关键词：椭圆曲线；BSD猜想；光滑数

Annual report on study of rational point of elliptic curves

Abstract：During the first year of our project， we made quite a progress according to the annual plan in the preparation of theoretic foundation. In particular， we made an important achievement on elliptic curves， the congruent number problem， and the BSD conjecture， one of the millennium. By using a series of results from modern number theory， arithmetic algebraic geometry， representation theory and automorphic forms， we proved that given any potitive integer k， there are infinite many square-free non-congruent numbers with exactly k odd prime factors， and we also developed some techniques and methods such as quadratic twist Euler system. The short version of this result was published on PNAS in 2012， and the long version has been submitted. This result help us to understand arithmetic of ellpitic curves more deeply， and provide ample theoretic basis and perparation of our research methods. Next， we made some progress on analytic number theory，and problems related to encryption and coding theory， improved a result of Green-Tao on sumsets on F_2^n. Moreover， we made a series progress on rational points on varieties and Brauer groups， formulated and proved a necessary and sufficient condition on square sum problems on imaginary quadratic number fields by using the Brauer-Manin obstruction method.We also made some progress on theoretic basis perparation on other various subjects such as elliptic curves， analytic number theory， rational points on algebraic varieties，automorphic forms，p-adic analysis and classical number theory.

Keywords：elliptic curves，BSD conjuncture，smooth number

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