Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra

Details Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra

2006-12-14

arxiv.org/abs/math/0612383v1

Mathematics

Number Theory

89 pages

Scientific paper

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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