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

@article{Yang2006GaloisRA, title={Galois representations arising from twenty-seven lines on a cubic surface and the arithmetic associated with Hessian polyhedra}, author={Lei Yang}, journal={arXiv: Number Theory}, year={2006} }

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…

## One Citation

Modularity experiments on $S_4$-symmetric double octics

- Mathematics
- 2018

We will invest quite some computer power to find double octic threefolds that are connected to weight four modular forms.

## References

SHOWING 1-10 OF 67 REFERENCES

Hessian polyhedra, invariant theory and Appell hypergeometric partial differential equations

- Mathematics
- 2004

It is well-known that Klein's lectures on the icosahedron and the solution of equations of fifth degree is one of the most important and influential books of 19th-century mathematics. In the present…

Rational varieties: algebra, geometry and arithmetic

- Mathematics
- 1986

CONTENTS Introduction § 1. Curves: geometry and arithmetic § 2. Surfaces: geometry over a closed field § 3. Surfaces: geometry over a non-closed field § 4. k-birational invariants § 5. Torsors and…

BASE CHANGE FOR GL(2)

- Mathematics
- 1980

R. Langlands shows, in analogy with Artin's original treatment of one-dimensional p, that at least for tetrahedral p, L(s, p) is equal to the L-function L(s, ?) attached to a cuspdidal automorphic…

Rankin-Cohen Brackets and Invariant Theory

- Mathematics
- 2001

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a…

Product formulas on a unitary group in three variables

- Mathematics
- 2001

We obtain some basic partial differential operators connected with nonholomorphic automorphic forms on $\Gamma \backslash U(2, 1)/K$. We give the corresponding Eisenstein series of weight $k$ and…

Geometry and arithmetic associated to Appell hypergeometric partial differential equations

- Mathematics
- 2003

In this paper, we study the monodromy of Appell hypergeometric partial differential equations, which lead us to find four derivatives which are associated to the group GL(3). Our four derivatives…

The L-series of Certain Rigid Calabi–Yau Threefolds

- Mathematics
- 2000

The L-series of two Calabi–Yau three-folds are found. In each case it is shown that the middle cohomology has dimension 2, and that (up to Euler factors at primes of bad reduction) the Mellin…

The modularity conjecture for rigid Calabi-Yau threefolds over Q

- Mathematics
- 2000

We formulate the modularity conjecture for rigid Calabi-Yau threefolds defined over the field Q of rational numbers. We establish the modularity for the rigid Calabi-Yau threefold arising from the…

Modular Calabi-Yau Threefolds

- Mathematics
- 2005

Arithmetic on Calabi-Yau threefolds Fibre products of elliptic surfaces Quintics in $\mathbb{P}^4$ Double octics Other examples Tables, correspondences, conclusions Arrangements of eight planes…

The Geometry of some special Arithmetic Quotients

- Mathematics
- 1996

Moduli spaces of PEL structures.- Arithmetic quotients.- Projective embeddings of modular varieties.- The 27 lines on a cubic surface.- The Burkhardt quartic.- A gem of the modular universe.