Invariants de Hasse-Witt des r\'eductions de certaines vari\'et\'es symplectiques irr\'eductibles

TL;DR

The paper proves that for certain irreducible symplectic varieties over a number field, there exists a large set of primes where their reductions have nonzero Hasse-Witt invariants, indicating specific arithmetic properties.

Contribution

It establishes the existence of primes with nonzero Hasse-Witt invariants for reductions of certain symplectic varieties, under specific geometric conditions.

Findings

Existence of a density 1 set of primes with nonzero Hasse-Witt invariants

Applicable to varieties with Picard number at least two or even second Betti number

Results hold after a finite field extension

Abstract

Let X be an irreducible symplectic variety defined over a number field K. Assume either that X has Picard number at least two or that X has even second Betti number. We prove that there exist a finite algebraic field extension L/K and a density 1 set S of non-archimedean places of L such that the reduction of X at any place in S has nonzero Hasse-Witt invariant.