[Paper Review] The Grothendieck-Katz Conjecture for certain locally symmetric varieties

Benson Farb, Mark Kisin|ArXiv.org|Jul 8, 2008

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper proves the Grothendieck-Katz $p$-curvature conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties $\mathcal{A}_g$ when $g > 1$, by applying Margulis's normal subgroup theorem and rigidity results from discrete subgroups of Lie groups. The key result establishes that vanishing $p$-curvature modulo almost all primes implies finite monodromy, hence algebraic solutions, for a broad class of connections on these varieties.

ABSTRACT

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz $p$-Curvature Conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties ${\cal A}_g$ when $g > 1.$

Motivation & Objective

• To establish the Grothendieck-Katz $p$-curvature conjecture for locally symmetric varieties with $\mathbf{Q}$-rank $\geq 1$ and $\mathbf{R}$-rank $\geq 2$.
• To extend known cases of the conjecture beyond solvable monodromy or Picard-Fuchs equations to non-abelian fundamental groups.
• To apply rigidity theorems from the theory of arithmetic lattices to differential Galois theory and $p$-curvature invariants.
• To show that for such varieties, vanishing $p$-curvature modulo almost all primes implies finite monodromy, hence algebraic solutions.

Proposed method

• Utilizes Margulis’s normal subgroup theorem for irreducible lattices in semisimple Lie groups to constrain monodromy representations.
• Reduces the connection $(V, \nabla)$ modulo primes $\mathfrak{p}$ to obtain $p$-curvature maps $\psi_p(V/\mathfrak{p}V, \nabla)$ on finite fields.
• Applies Katz’s result that vanishing $p$-curvature implies regular singularities and finite local monodromy around boundary divisors.
• Leverages Margulis superrigidity to realize representations of arithmetic groups as subrepresentations of automorphic bundles from symplectic groups.
• Uses André’s extension of Katz’s result on Picard-Fuchs equations to show finiteness of monodromy in Tannakian categories generated by such systems.
• Reduces the problem to showing that the monodromy representation factors through a finite group by embedding into a symplectic group and using the connectedness of $\tilde{G}^{\operatorname{der}}$.

Experimental results

Research questions

• RQ1 Does the Grothendieck-Katz $p$-curvature conjecture hold for locally symmetric varieties with non-abelian fundamental groups?
• RQ2 Can rigidity theorems from arithmetic groups be used to prove finiteness of monodromy from $p$-curvature vanishing?
• RQ3 For which locally symmetric varieties does vanishing $p$-curvature modulo almost all primes imply algebraic solutions?
• RQ4 Is the Tannakian Galois group of a connection on such varieties finite when $p$-curvatures vanish?

Key findings

• The Grothendieck-Katz $p$-curvature conjecture holds for all connections on locally symmetric varieties $M = \Gamma \backslash G(\mathbb{R})^+ / K$ when each $\mathbf{Q}$-simple factor of $G$ has $\mathbf{Q}$-rank $\geq 1$ and $\mathbf{R}$-rank $\geq 2$.
• The conjecture is proven for the moduli space of abelian varieties $\mathcal{A}_g$ when $g > 1$, a case with non-abelian fundamental group.
• Vanishing $p$-curvature for almost all primes implies that the monodromy representation has finite image, so the connection becomes trivial on a finite étale cover.
• The monodromy representation factors through a finite group after replacing the arithmetic group $\Gamma$ by a finite index subgroup, due to the Zariski density of $\Gamma$ in $\tilde{G}^{\operatorname{der}}$.
• For $G$ of type $A, B, C, D$ with $\mathbf{R}$-rank $\geq 2$, the conjecture holds via embedding into $\mathrm{Sp}_{2g}$ and applying André’s result on motivic Galois groups.
• The proof relies on the fact that $\tilde{G}^{\operatorname{der}}$ is connected and semisimple, so scalar actions on representations imply triviality of the monodromy.