M ar 2 00 4 Computing the level of a modular rigid Calabi - Yau threefold Luis

6 M ar 2 00 9 A modular quintic Calabi - Yau threefold of level 55 Edward

In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold which is modular, in the sense that the L-function of its middle-dimensional cohomology is associated to a classical modular form of weight 4 and level 55.

Computing the Level of a Modular Rigid Calabi-Yau Threefold

Existence of Nonelliptic mod Galois Representations for Every > 5 Luis

In [Shepherd-Barron and Taylor 97] it is shown that for = 3 and 5 every odd, irreducible, two-dimensional Galois representation of Gal(Q̄/Q) with values in F and determinant the cyclotomic character is “elliptic,” i.e., it agrees with the representation given by the action of Gal(Q̄/Q) on the -torsion points of an elliptic curve defined over Q. In this note we will show that this is false for eve...

Modular Calabi–yau Threefolds of Level Eight

If X̃ is a Calabi–Yau threefold defined over Q, and p is a good prime (i.e., a prime such that the reduction of X̃ mod p is nonsingular) then the map Frob∗p : H i ét(X̃,Ql) −→ H i ét(X̃,Ql) on l–adic cohomology induced by the geometric Frobenius morphism gives rise to l–adic Galois representations ρl,i : Gal(Q/Q) −→ GLbi(Ql). If a Calabi–Yau threefold X̃ is rigid (i.e., h(X̃) = 0 or equivalently b(X̃)...

A Modular Non-Rigid Calabi-Yau Threefold

We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Gal...