论文标题
$ \ operatoTorname {gsp} _4 $ for tater tater of total actlotal of totalle of total actal of total ofer tocter of talter of tall actal of talter领域
Tate classes and endoscopy for $\operatorname{GSp}_4$ over totally real fields
论文作者
论文摘要
内窥镜学理论预测了Shimura品种某些产品的泰特族类别的存在，并且自然要问在什么情况下可以构建代数周期，从而引起这些泰特类别。本文占据了Yoshida Lift引起的泰特班级的情况：这些是tate循环的中间程度的Shimura品种，用于组$ \ operatatorName {res} _ {f/\ Mathbb Q}（\ propatatorNAME {\ propatatorNAMe {gl} _2 \ times _2 \ times _2 \ times \ times \ times \ operemorname iSAmeName iSAmeal iSA n ofter fielf。一个特殊情况是泰特阶层的家族，反映了模块化曲线和siegel模块化三倍的中间共同体中二维Galois表示的出现。我们表明，自然代数周期完全生成了与\ emph {generic}成员在$ \ operatatorName {gsp} _ {4，f} $上关联的泰特类。在非传播情况下，我们提供了一种替代结构，这表明预测的泰特类是由霍奇周期产生的。
The theory of endoscopy predicts the existence of large families of Tate classes on certain products of Shimura varieties, and it is natural to ask in what cases one can construct algebraic cycles giving rise to these Tate classes. This paper takes up the case of Tate classes arising from the Yoshida lift: these are Tate cycles in middle degree on the Shimura variety for the group $\operatorname{Res}_{F/\mathbb Q} (\operatorname{GL}_2 \times \operatorname{GSp}_4)$, where $F$ is a totally real field. A special case is the family of Tate classes which reflect the appearance of two-dimensional Galois representations in the middle cohomology of both a modular curve and a Siegel modular threefold. We show that a natural algebraic cycle generates exactly the Tate classes which are associated to \emph{generic} members of the endoscopic $L$-packets on $\operatorname{GSp}_{4,F}$. In the non-generic case, we give an alternate construction, which shows that the predicted Tate classes arise from Hodge cycles.
