Algebraicity Criteria and Their Applications

Abstract

We use generalizations of the Borel–Dwork criterion to prove variants of the Grothedieck–Katz p-curvature conjecture and the conjecture of Ogus for some classes of abelian varieties over number fields.

The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for all but finitely many primes p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on P^1 − {0, 1, ∞}. We prove a variant of this conjecture for P^1 − {0, 1, ∞}, which asserts that if the equation satisfies a certain convergence condition for all p, then its monodromy is trivial. For those p for which the p-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of p-curvatures and certain local monodromy groups. We also prove similar variants of the p-curvature conjecture for a certain elliptic curve with j-invariant 1728 minus its identity and for P^1 − {±1, ±i, ∞}.

Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of l-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. We confirm Ogus’ conjecture for some classes of abelian varieties, under the assumption that these cycles lie in the Betti cohomology with real coefficients. These classes include abelian varieties of prime dimension that have nontrivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture. We also discuss some strengthenings of the theorem of Bost.