Study on arithmetic invariants attached to automorphic forms

(1)对于Hecke同余子群上的椭圆尖点形式f和虚二次域K的Hecke特征标W，证明了当水平自由平方时，附加在(f，W)上的某个CM-周期的平方可以用附加在(f，W)上的L函数的中心值来表示.(2)我们提出了关于U(2，1)上尖点形式F的傅立叶-雅可比系数与附着在F上的自同构函数的中心值之间的关系的猜想.当F是全纯Eisenstein级数或酉Kudla提升时，证明了这个猜想.文(1)中的结果主要用于证明后一个结果。(3)设f和f‘分别是GL(2)上的自同构形和四元数代数B的乘群。设L(f，f‘)是由(f，f’)构造的Sp(1，1)上的theta提升。证明了L(f，f‘)是Hecke本征形，如果f和f’都是Hecke本征形。我们还证明了L(f，f‘)的傅里叶系数是用f和f’的CM周期表示的。这是与Hiro-aki Narita共同完成的工作。(4)构造了一个由仿射Hecke簇参数表示的p-进无穷族的Hilbert模形式。当基场的次数F为偶数时，构造了固定有限斜率的权参数Hilbert-Hecke特征形式的p-正解析无限族。这是山上敦志的作品。

(1) For an elliptic cusp form f on a Hecke congruence subgroup and a Hecke character W of an imaginary quadratic field K, I showed that the square of a xertain CM-period attached to (f,W) is expressed in terms of the central value of the L-function attached to (f,W), when the level off is square free.(2) We proposed a conjecture concerning a relation between the Fourier-Jacobi coefficients of a cusp form F on U(2,1) and the central values of automorphic L-functions attaced to F. The conjecture are proved when F is a holomorphic Eisenstein series or a unitary Kudla lift. The results in (1) are essentially used in the proof of the latter result. This is a joint work with Takashi Sugano.(3) Let f and f' be automorphic forms on GL(2) and the multiplicative group of a quaternion algebra B, respectively. Let L(f,f') be the theta lift on Sp(1,1) constructed from (f,f'). We showed that L(f,f') is a Hecke eigenform if so are f and f'. We also showed that the Fourier coefficients of L(f,f') are expressed in terms of CM-periods of f and f'. This is a joint work with Hiro-aki Narita.(4) A p-adic infinite family of Hilbert modular forms parameterized by an Affinoid Hecke variety is constructed. When the degree of the base field F is even, a p-adically analytic infinite family of Hilbert Hecke eigenforms of a fixed finite slope parameterized by weights is constructed. This is a work of Atsushi Yamagami.