“橢圓曲線有理點問題研究”年度報告

田野 王崧

摘要：在該計劃第一年，我們按照年度計劃，在理論基礎準備部分取得相當進展。其中我們在橢圓曲線算術、同余數及千禧問題BSD猜想上取得重要成果。我們利用現代數論、算術代數幾何、表示論、自守形式的系列結果，證明了對任意給定的正整數k，存在無窮多個沒有平方因子的恰巧有k個奇素因子的同余數，并發展了系列新的方法工具（如二次扭轉歐拉系）。這些成果有助于我們更加深入理解橢圓曲線的算術理論，并為下一步研究提供了充足的理論基礎和方法準備。另外我們在解析數論、密碼編碼相關問題上取得一定進展，改進了Green-Tao關于F_2^n和集的一個結果。而且，在代數簇有理點，Brauer群方面取得系列進展，利用Brauer-Manin障礙技術給出了虛二次數域平方和問題的充要條件。另外，在橢圓曲線、代數簇有理點、自守形式、p-adic分析以及經典數論等其它的基礎準備方面均取得一定的進展。

關鍵詞：橢圓曲線；BSD猜想；光滑數

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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