三次曲面上二十七条直线的伽罗瓦表示及与Hessian多面体有关的算术
Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra
摘要：
在本文中，我们将证明三个明显不相交的对象：由三次曲面上的27条线产生的伽罗瓦表示（数论和算术代数几何），Picard模形式（自守形式），刚性Calabi-Yau三重及其算术（丢番图几何）与Hessian多面体及其不变量密切相关。我们构造了一个伽罗瓦表示，它的象是例外李代数$E_6$的Weyl群$W(E_6)$的一个真子群.本文给出了关于两种不同类型的$L$ -函数的同一性的一个猜想，该猜想可以看作是Langlands-Tunnell定理的一个高维对应.
英文摘要：
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic forms), rigid Calabi-Yau threefolds and their arithmetic (Diophantine geometry) are intimately related to Hessian polyhedra and their invariants. We construct a Galois representation whose image is a proper subgroup of $W(E_6)$, the Weyl group of the exceptional Lie algebra $E_6$. We give a conjecture about the identification of two different kinds of $L$-functions which can be considered as a higher dimensional counterpart of the Langlands-Tunnell theorem.
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