On special values of certain L-functions
On special values of certain L-functions
复制标题
DOI：
10.1353/ajm.2014.0032
复制
发表时间：
2014-09
影响因子：
1.7
通讯作者：
M. Furusawa;Kazuki Morimoto
中科院分区：
文献类型：
--
作者：
M. Furusawa;Kazuki Morimoto
我们证明了完全确定的二次型的特殊正交群的自同构表示所对应的张量积$L$，以及完全实数域上与原始尖形对应的${\rm GL}(2)$的尖形表示，在Deligne意义上的临界点处的特殊值的代数性结果。我们还证明了特殊值${\rm Gal}({\overline{\Bbb Q}}/{\Bbb Q})$作用下的互反律。在附录中，第二作者计算了这些$L$-函数的Deligne周期，假设相应动机存在，并且自同构转移到特殊正交群的拟分裂形式。我们的结果符合著名的Deligne关于动机$L$-函数特殊值的猜想。
We prove an algebraicity result concerning special values at critical points, in the sense of Deligne, of tensor product $L$-functions associated to automorphic representations of special orthogonal groups for quadratic forms which are totally definite, and, cuspidal representations of ${\rm GL}(2)$ corresponding to primitive cusp forms, over totally real number fields. We also prove the reciprocity law, i.e., the equivariance under the action of ${\rm Gal}({\overline{\Bbb Q}}/{\Bbb Q})$, for the special values. In the appendix, the second author calculates the Deligne periods for such $L$-functions, assuming the existence of corresponding motives and the automorphic transfer to a quasi-split form of the special orthogonal group. Our result conforms with the celebrated conjecture of Deligne on special values of motivic $L$-functions.