[Paper Review] On uniformity conjectures for abelian varieties and K3 surfaces

Martin Orr, Alexei N. Skorobogatov+1 more|Project Muse (Johns Hopkins University)|Sep 5, 2018

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes logical equivalences among major uniformity conjectures for abelian varieties and K3 surfaces over number fields of bounded degree. It proves that Coleman’s conjecture on endomorphism algebras implies both Shafarevich’s conjecture on Néron–Severi lattices and Várilly-Alvarado’s conjecture on the finiteness of the Galois-invariant Brauer group, unifying key conjectures in arithmetic geometry via monodromy and Galois representation techniques.

ABSTRACT

We discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Neron-Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group.

Motivation & Objective

• To investigate logical relationships among uniformity conjectures for abelian varieties and K3 surfaces over number fields of bounded degree.
• To prove that Coleman’s conjecture on endomorphism algebras implies Shafarevich’s conjecture on Néron–Severi lattices.
• To establish that Coleman’s conjecture implies the uniform boundedness of the Galois-invariant geometric Brauer group for K3 surfaces.
• To show that the conjecture on the finiteness of the Brauer group modulo constants (Br(AV)) implies Várilly-Alvarado’s conjecture on the boundedness of Br(X)/Br₀(X) for K3 surfaces.
• To unify several major conjectures in arithmetic geometry through monodromy and Galois representation techniques.

Proposed method

• Uses the Mumford–Tate conjecture to relate the dimension of ℓ-adic monodromy groups to the geometry of K3 surfaces.
• Applies the Tate conjecture to relate Galois-invariant cohomology classes to algebraic cycles in Néron–Severi groups.
• Employs the Kummer exact sequence to relate integral cohomology to Néron–Severi lattices over ℤℓ.
• Analyzes the action of Galois groups on Néron–Severi groups and Brauer groups via finite-index subgroups and their fixed submodules.
• Utilizes the finiteness of H¹(k, Pic(X̄)) for K3 surfaces over fields finitely generated over ℚ to bound Brauer group quotients.
• Establishes equivalence between various formulations of Shafarevich’s conjecture by showing that the Galois-invariant subgroup of the Néron–Severi lattice determines the full lattice up to isomorphism.

Experimental results

Research questions

• RQ1 Does Coleman’s conjecture on the endomorphism algebra of abelian varieties imply Shafarevich’s conjecture on the Néron–Severi lattice of K3 surfaces?
• RQ2 Can the uniform boundedness of the Galois-invariant Brauer group of K3 surfaces be deduced from Coleman’s conjecture?
• RQ3 Is the conjecture that |Br(Ā)Γ| is uniformly bounded for abelian varieties of bounded dimension and degree equivalent to other uniformity conjectures?
• RQ4 To what extent are the different variants of Shafarevich’s conjecture—using NS(X̄), NS(X̄)Γ, or Pic(X)—logically equivalent?
• RQ5 How do monodromy representations and Galois actions on cohomology relate to the finiteness of Brauer group quotients in one-parameter families of K3 surfaces?

Key findings

• Coleman’s conjecture on End(Ā) implies Shafarevich’s conjecture on the Néron–Severi lattice of K3 surfaces, via the equivalence of their discriminant bounds.
• The conjecture that |Br(Ā)Γ| is uniformly bounded for abelian varieties of bounded dimension and degree implies Várilly-Alvarado’s conjecture on the boundedness of |Br(X)/Br₀(X)| for K3 surfaces.
• All variants of Shafarevich’s conjecture—using NS(X̄), NS(X̄)Γ, or Pic(X)—are logically equivalent, as the Galois-invariant subgroup determines the full lattice up to isomorphism.
• The set of points in a family of K3 surfaces where the ℓ-adic monodromy group is not open is independent of ℓ, due to the Mumford–Tate conjecture and dimension invariance.
• When the monodromy group Gℓ,x is open in Gℓ, the Néron–Severi lattice of the fiber Yx is isomorphic to that of the generic fiber Yη over ℤℓ, via Tate and Kummer theory.
• The finiteness of H¹(k, Pic(X̄)) for K3 surfaces over fields finitely generated over ℚ ensures that Br(X)/Br₀(X) is finite, supporting the Brauer group uniformity conjectures.