Tate Cycles on Abelian Varieties with Complex Multiplication

TL;DR

This paper establishes an effective bound for identifying Tate cycles on CM Abelian varieties over large number fields, linking Frobenius actions at small primes to the global Tate cycle structure.

Contribution

It provides an explicit bound to verify Tate cycles via Frobenius actions and shows the Tate cycle space on special fibers matches the generic fiber for density 1 primes.

Findings

Effective bound C for Tate cycle verification

Frobenius action at primes v of norm ≤ C suffices

Tate cycle spaces on special fibers and generic fiber coincide for density 1 primes

Abstract

We consider Tate cycles on an Abelian variety $A$ defined over a sufficiently large number field $K$ and having complex multiplication. We show that there is an effective bound $C = C(A,K)$ so that to check whether a given cohomology class is a Tate class on $A$, it suffices to check the action of Frobenius elements at primes $v$ of norm $ \leq C$. We also show that for a set of primes $v$ of $K$ of density 1, the space of Tate cycles on the special fibre $A_v$ of the N\'eron model of $A$ is isomorphic to the space of Tate cycles on $A$ itself.