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}
}
  • Lei Yang
  • Published 14 December 2006
  • Mathematics
  • arXiv: Number Theory
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… 
1 Citations
Modularity experiments on $S_4$-symmetric double octics
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
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
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)
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
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
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
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
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
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
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
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.
...
1
2
3
4
5
...