[Paper Review] Good reduction and cyclic covers

Ariyan Javanpeykar, Daniel Loughran+1 more|arXiv (Cornell University)|Sep 3, 2020

Algebraic Geometry and Number Theory85 references11 citations

TL;DR

This paper establishes finiteness results for varieties with good reduction outside a finite set of places using cyclic covers and moduli stacks. By proving that the stack of degree-r cyclic covers over divisorial pairs is proper and \'etale over a finite \'etale cover of the base stack, the authors derive new cases of the Shafarevich conjecture, including for weighted projective surfaces, double covers of abelian varieties, and hypersurfaces in high codimension, via a stacky version of the Chevalley–Weil theorem.

ABSTRACT

We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.

Motivation & Objective

• To extend the Shafarevich conjecture to new classes of varieties beyond curves.
• To resolve the ambiguity in associating cyclic covers to hypersurfaces over non-algebraically closed fields by working in moduli stack frameworks.
• To prove that the morphism from the stack of cyclic covers to the base stack of branch divisors is proper and \'etale, enabling finiteness via integral points on stacks.
• To establish new cases of the Shafarevich conjecture for weighted projective surfaces and double covers of abelian varieties.
• To reduce the general Shafarevich conjecture for hypersurfaces to the case of Fano hypersurfaces (r ≤ n), where the geometry is simpler.

Proposed method

• Use moduli stacks of cyclic covers to uniformly parameterize varieties with good reduction, avoiding scalar ambiguity in hypersurface equations.
• Prove that the natural morphism from the stack of degree-r cyclic covers to the stack of base varieties with branch divisors is proper and \'etale, generalizing the Chevalley–Weil theorem to stacks.
• Apply a stacky version of the Chevalley–Weil theorem to deduce finiteness of integral points on moduli stacks, which implies finiteness of isomorphism classes of varieties with good reduction.
• Use descent techniques à la Fermat to relate cyclic covers to their branch loci, particularly in the case of hypersurfaces and abelian varieties.
• Work over rings of S-integers to model good reduction, and use spreading out and model reduction to apply finiteness theorems over number fields.
• Construct explicit moduli stacks for double covers of abelian varieties and show they are arithmetically hyperbolic, implying finiteness of isomorphism classes.

Experimental results

Research questions

• RQ1 Can the Shafarevich conjecture for hypersurfaces be reduced to the case of Fano hypersurfaces (r ≤ n)?
• RQ2 Do smooth weighted projective surfaces of degree 2r in P(1,1,1,r) have only finitely many S-integral points over number rings, for fixed r and finite S?
• RQ3 Is the moduli stack of double covers of abelian varieties with given geometric genus and dimension arithmetically hyperbolic?
• RQ4 Does the stack of degree-r cyclic covers over divisorial pairs admit a proper \'etale morphism to the base stack of branch divisors?
• RQ5 Can finiteness of integral points on stacks of cyclic covers be used to prove new cases of the Shafarevich conjecture?

Key findings

• The Shafarevich conjecture for smooth hypersurfaces of degree r in P^{n+1} over number fields holds for all dimensions m ≤ n if it holds for dimension n, allowing reduction to high-dimensional cases.
• The moduli stack of smooth surfaces of degree 2r in P(1,1,1,r) over an integrally closed Z-finite ring A with 2 ∈ A× has only finitely many A-isomorphism classes, proving new cases of the Shafarevich conjecture.
• For g = 2 or g ≥ 4, the moduli stack G_{g,p} of double covers of abelian varieties with geometric genus p is arithmetically hyperbolic, implying finiteness of isomorphism classes over S-integers.
• The stack of degree-r cyclic covers over a divisorial pair is proper and \'etale over a finite \'etale cover of the base stack of branch divisors, a key geometric result.
• The stack of double covers of abelian varieties is equivalent to a stack of degree-2 cyclic covers over torsors under abelian schemes, and this equivalence is proper and \'etale over the moduli stack of polarized abelian varieties.
• The morphism from the stack G_{g,p} to the moduli stack of polarized abelian varieties of type (g, 2g(p−1)) is proper and \'etale, enabling finiteness via spreading out and integral point arguments.