[Paper Review] Prime-to-p étale covers of algebraic groups

Michel Brion, Tamás Szamuely|arXiv (Cornell University)|Sep 13, 2011

Algebraic Geometry and Number Theory4 references3 citations

TL;DR

This paper establishes that every prime-to-p étale Galois cover of a connected algebraic group over an algebraically closed field of characteristic p ≥ 0 admits a central isogeny structure, proving the prime-to-p fundamental group is commutative and bounded by Z_{(p')}^{2g+r}, where g is the dimension of the abelian variety quotient and r the rank of the unipotent radical. The result extends to homogeneous spaces with a refined bound depending on stabilizer subgroups.

ABSTRACT

Let G be a connected algebraic group over an algebraically closed field of characteristic p (possibly 0), and X a variety on which G acts transitively with connected stabilizers. We show that any étale Galois cover of X of degree prime to p is also homogeneous, and that the maximal prime-to-p quotient of the étale fundamental group of X is commutative. We moreover obtain an explicit bound for the number of topological generators of the said quotient. When G is commutative, we also obtain a description of the prime-to-p torsion in the Brauer group of G.

Motivation & Objective

• To generalize classical results on topological groups to algebraic groups in positive characteristic.
• To determine the structure and topological rank of the prime-to-p étale fundamental group π₁^{(p')} for connected algebraic groups.
• To extend the theory of étale covers to homogeneous spaces with connected stabilizers.
• To provide a foundational tool for studying Picard and Brauer groups of commutative algebraic groups.
• To establish a precise bound on the number of topological generators of π₁^{(p')}(G) in terms of geometric invariants.

Proposed method

• Use of Chevalley's structure theorem to decompose G as an extension of an abelian variety A by a linear group G_aff.
• Application of the Barsotti–Weil formula to classify extensions of A by a torus T via homomorphisms c: X*(T) → Pic⁰(A).
• Leveraging the fact that μ_n-torsors over commutative algebraic groups admit a commutative group scheme structure via pullback from central extensions.
• Reduction of Pic(G) to Pic(A) modulo unipotent radicals, using the isomorphism Pic(G) ≅ Pic(G/U).
• Use of the Néron–Severi group and ℓ-adic cohomology to compute the ℓ-primary torsion in the Brauer group.
• Elementary construction of the group structure on μ_n-torsors via Rosenlicht’s lemma and trivialization of line bundle powers.

Experimental results

Research questions

• RQ1 What is the structure of the prime-to-p étale fundamental group π₁^{(p')}(G) for a connected algebraic group G over an algebraically closed field of characteristic p ≥ 0?
• RQ2 Can every prime-to-p étale Galois cover of G be given a central isogeny structure making it a group homomorphism?
• RQ3 How does the topological rank of π₁^{(p')}(G) depend on the geometric invariants of G, such as the dimension of its abelian variety quotient and the rank of its reductive part?
• RQ4 What is the structure of π₁^{(p')}(X) for a G-homogeneous space X with connected stabilizers?
• RQ5 How can the Brauer group of a commutative algebraic group be described in terms of its torsion subgroups and geometric invariants?

Key findings

• The prime-to-p fundamental group π₁^{(p')}(G) is commutative and is a quotient of Z_{(p')}^{2g + r}, where g = dim A and r = rank(G_aff).
• The bound 2g + r is sharp, achieved when G is a product of an abelian variety of dimension g and a torus of rank r.
• For homogeneous spaces X with connected stabilizers, π₁^{(p')}(X) is a quotient of Z_{(p')}^{2(g - g_H) + (r - r_H)}, where g_H and r_H are the dimension and rank of the stabilizer subgroup H.
• The Brauer group Br(G) has ℓ-primary torsion isomorphic to (Q_ℓ/Z_ℓ)^{(2g + r)(2g + r - 1)/2 - ρ}, where ρ is the Néron–Severi rank of the abelian variety quotient A.
• The exact sequence 0 → Pic(G)/nPic(G) → Hom(Λ²(_nG), μ_n) → _nBr(G) → 0 holds for n prime to the characteristic, extending Grothendieck’s formula to commutative algebraic groups.
• The group structure on μ_n-torsors arises from a central extension of G by μ_n, constructed via trivialization of line bundle powers and Rosenlicht’s lemma.