[Paper Review] Galois modules, ideal class groups and cubic structures

G. Pappas|ArXiv.org|Jun 20, 2003

Algebraic Geometry and Number Theory29 references3 citations

TL;DR

This paper establishes a deep connection between cyclotomic ideal class groups and the Galois module structure of coherent cohomology groups of unramified Galois covers of arithmetic varieties over ℤ. By introducing hypercubic structures on line bundles over group schemes, the author proves that obstructions to virtual normal integral bases in cohomology are annihilated by products of Bernoulli numbers and orders of even K-groups of ℤ, linking this to the Kummer–Vandiver conjecture.

ABSTRACT

We establish a connection between the theory of cyclotomic ideal class groups and the theory of "geometric" Galois modules and obtain results on the Galois module structure of coherent cohomology groups of Galois covers of varieties over Z. In particular, we show that an invariant that measures the obstruction to the existence of a virtual normal integral basis for the coherent cohomology of such covers is annihilated by a product of certain Bernoulli numbers with orders of even K-groups of Z. We also show that the existence of such a normal integral basis is closely connected to the truth of the Kummer-Vandiver conjecture for the prime divisors of the degree of the cover. Our main tool is a theory of "hypercubic structures" for line bundles over group schemes.

Motivation & Objective

• To connect the theory of cyclotomic ideal class groups with the structure of Galois modules in coherent cohomology of unramified Galois covers over ℤ.
• To develop a theory of hypercubic structures for line bundles over group schemes as a central tool.
• To determine annihilators of the projective equivariant Euler characteristic in K₀(ℤ[G]) using Bernoulli numbers and K-groups.
• To investigate the relationship between the existence of normal integral bases and the Kummer–Vandiver conjecture.
• To extend classical results on normal integral bases in number fields to higher-dimensional arithmetic varieties.

Proposed method

• Introduces hypercubic structures on line bundles over finite flat group schemes to encode higher-order multiplicative data.
• Uses multiextensions of abelian sheaves and reflection homomorphisms to relate cohomological invariants to arithmetic data.
• Constructs the determinant of cohomology with a hypercubic structure to define characteristic classes in K-theory.
• Applies Grothendieck–Riemann–Roch and derived category techniques to compute Euler characteristics in K₀(ℤ[G]).
• Employs arithmetic Bertini theorems (via Rumely’s integral point theorems) to construct geometric G-torsors over ℤ with desired properties.
• Reduces the problem of Galois module structure to invariants in the class group Cl(ℤ[G]) and relates them to special values of L-functions via Bernoulli numbers.

Experimental results

Research questions

• RQ1 How do cyclotomic ideal class groups relate to the Galois module structure of coherent cohomology groups of Galois covers over ℤ?
• RQ2 What is the role of hypercubic structures in encoding obstructions to the existence of virtual normal integral bases?
• RQ3 In what way are the obstructions to normal integral bases annihilated by products of Bernoulli numbers and orders of even K-groups of ℤ?
• RQ4 How is the Kummer–Vandiver conjecture connected to the existence of normal integral bases in the cohomology of abelian covers?
• RQ5 Can geometric G-torsors with specified cohomological properties be constructed over ℤ using arithmetic Bertini theorems?

Key findings

• The obstruction to a virtual normal integral basis in the cohomology of a G-torsor over a ℤ-variety is annihilated by a product of certain Bernoulli numbers and orders of even K-groups of ℤ.
• For abelian G and dim(Y) ≤ 4 or Albanese-type covers, 2·χᴾ(𝒪_X) is the class of a free ℤ[G]-module in K₀(ℤ[G]).
• If G has odd order, then χᴾ(𝒪_X) itself is the class of a free ℤ[G]-module in K₀(ℤ[G]).
• The existence of a normal integral basis in cohomology is deeply tied to the truth of the Kummer–Vandiver conjecture for prime divisors of |G|.
• The hypercubic structure on the determinant of cohomology provides a canonical refinement of the Euler characteristic in K₀(ℤ[G])
• The construction of geometric G-torsors over ℤ is possible via arithmetic Bertini theorems, ensuring smoothness and projectivity over Spec(ℤ).