【作者】 张翀；

【导师】 印林生；

【作者基本信息】 清华大学 ， 数学， 2011， 博士

【摘要】 数论中有两大主题：解析与算术。Birch和Swinnerton-Dyer猜想（BSD）将解析量与算术量联系在一起。对于Q上解析秩≤1的椭圆曲线，此猜想已被Gross-Zagier[19]和Kolyvagin[28][29]的工作所基本完全证明。张寿武已将Gross-Zagier公式推广到全实域，并且田野和张寿武将Kolyvagin[28][29]和Bertolini-Darmon[4][7]关于BSD的相应工作推广到了全实域。在本文中，我们主要研究两个课题：一方面是自守表示理论中的theta对应；另一方面是算术几何理论中的Shimura曲线。它们之间是有紧密联系的。我们的目标是将Bertolini-Darmon[5]的方法推广到全实域。在解析方面，我们研究对于酉相似群的theta对应的局部和整体理论。在p-adic域时，受Roberts[41]在正交相似群和辛相似群上工作的启发，我们考虑了两种构造酉相似群的Weil表示的方法并且证明它们是等价的。在某些情形下我们证明了强Howe对偶成立。在全实数域时，我们讨论了Siegel-Weil公式、加倍积分和Rallis内积公式，这是对Harris的部分工作[20][21][22]的总结。在第三章中，为了今后的算术应用，我们考虑2维的情形，主要讨论L-函数的中心值公式。在算术方面，给定全实域上GL2的一个尖自守表示可以得到一条相关的Shimura曲线。在第四章中，我们研究这条Shimura曲线的算术性质，其中包括它的坏约化以及相关的连通分支群。基于L-函数的中心值公式，通过在Shimura曲线上构造CM点以及利用连通分支群上的monodromy配对，我们得到了一个算术中心值公式，这是将Bertolini和Darmon[5]的工作推广到全实域。作为应用，我们在第五章证明了一个关于阿贝尔簇的Mordell-Weil群有限的BSD类型定理，这同样是将Bertolini和Darmon的工作[5]推广到全实域。我们的方法与田野和张寿武[45]的方法不同。更多还原

【Abstract】 There are two main themes in number theory: analytic and arithmetic. The Birchand Swinnerton-Dyer (BSD) conjecture relates analytic invariants to arithmetic invari-ants. It has been essentially completely proved for elliptic curves overQof analyticrank≤1by the work of Gross-Zagier[19]and Kolyvagin[28][29]. Zhang has general-ized the Gross-Zagier formula to totally real field, and Tian-Zhang has generalized thework of Kolyvagin[28][29]and Bertolini-Darmon[4][7]on BSD to totally real field. Inthis thesis, we mainly study two topics: one is theta correspondence which belongs tothe theory of automorphic representations; the other is Shimura curves which belongto the theory of arithmetic geometry. There is a closed relationship between them. Ourgoal is to generalize the method of Bertolini-Darmon[5]directly to totally real field.On the analytic side, we study the local and global theory of theta correspondencefor unitary similitude groups. Over a p-adic field, inspired by the work of Roberts[41]on orthogonal similitude groups and symplectic similitude groups, we consider two ap-proaches to construct Weil representations for unitary similitude groups and show thatthey are essentially the same. We prove strong Howe duality holds in certain situations.Over a totally real field, we discuss the Siegel-Weil formula, double integrals and Ral-lis inner product formula, which is a summarization of part of Harris’ work[20][21][22].In Chapter3, for arithmetic application, we focus on the case of dimension two andmainly discuss the central value formula of L-functions.On the arithmetic side, there is a Shimura curve associated to a given cuspidalautomorphic representation of GL2over totally real field. In Chapter4, we study thearithmetic of this Shimura curve, including its bad reduction and the groups of con-nected components associated to it. Based on the central value formula of L-functions,by constructing CM points on the Shimura curve and using the monodromy pairingson the groups of connected components, we obtain an arithmetic central value formula,which is a generalization of Bertolini-Darmon[5]to totally real field. As an application, we prove a theorem of BSD type on the finiteness of Mordell-Weil group of an abelianvariety in Chapter5, which is also a generalization of Bertolini-Darmon[5]to totallyreal field. Our method is different from Tian-Zhang’s[45].更多还原