Some Arithmetic Problems On Shimura Curves

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].