An exceptional isomorphism between modular varieties
An exceptional isomorphism between modular varieties
复制标题
模块化品种之间的特殊同构
DOI：
10.1007/978-1-4612-0457-2_5
复制
发表时间：
1991
影响因子：
0.7
通讯作者：
B. Geemen
中科院分区：
文献类型：
--
作者：
T. Ekedahl;B. Geemen
本文的目的是给一个模块之间的对应关系的模块化建设的存在被怀疑在[Ge-Ny]。所讨论的模簇一方面是Siegel模三重X°，即二维阿贝尔簇的模空间，它是具有8级结构的模空间的商，另一方面是e → Y 0（8）的自积（在基曲线上）Vo，即具有8阶循环子群的泛（光滑）椭圆曲线。这些簇分别具有光滑紧化X和V，如（上述）所示。同前）二维伽罗瓦表示\（{G_Q} = Gall（\overline Q /Q）\; \to \;Aut（{H^3}（X，{Q_t}））\）是\（{H^3}（X，{Q_t}）\）的子表示。这一事实可以用椭圆模尖点形式到Siegel模形式的提升理论来解释，但是关于代数圈和代数上同调群中伽罗瓦不变子空间的Tate猜想预言了X × V上存在一个代数圈Z，从而导致伽罗瓦表示H3（X）→ H3（V）的注入。利用显式方程，J. Stienstra发现了一个支配有理映射X → V，其图形给出了所需的循环。在本文中，我们实际上给出了堆栈的同构（参见。Cor 2.10），这也诱导了所需的包含。然而，我们的构造似乎并不直接适用于更高的能级，而提升理论是相当普遍的。
This article aims to give a modular construction of a correspondence between modular varieties whose existence was suspected in [Ge-Ny]. The modular varieties in question are on the one hand a Siegel modular threefold X°, that is a moduli space of 2 dimensional abelian varieties, which is a quotient of the moduli space with level 8 structure and on the other hand the self-product (over the base curve) V o of e → Y 0(8), the universal (smooth) elliptic curve with a cyclic subgroup of order 8. These varieties have smooth compactifications X and V respectively and it is shown in (loc. cit.) that the 2 dimensional Galois representation \( {G_Q} = Gall(\overline Q /Q)\; \to \;Aut({H^3}(X,{Q_t})) \) is a subrepresentation of \( {H^3}(X,{Q_t}) \). This fact can be explained by the theory of lifting of elliptic modular cusp forms to Siegel modular forms, but the Tate conjecture relating algebraic cycles and Galois invariant subspaces in the etale cohomology groups predicts the existence of an algebraic cycle Z on X × V inducing the injection of Galois representations H 3(X) → H 3(V). Using explicit equations, J. Stienstra found a dominant rational map X → V whose graph gives the desired cycle. In this paper we give in fact an isomorphism of stacks (cf. Cor 2.10) which also induces the desired inclusion. It seems however that our construction does not directly work for higher levels, whereas the lifting theory is quite general.