The isogeny conjecture for A-motives

Abstract

We prove the isogeny conjecture for A-motives over finitely generated fields K of transcendence degree ≤1. This conjecture says that for any semisimple A-motive M over K, there exist only finitely many isomorphism classes of A-motives M′ over K for which there exists a separable isogeny M′→M. The result is in precise analogy to known results for abelian varieties and for Drinfeld modules and will have strong consequences for the ${\mathfrak {p}}$ -adic and adelic Galois representations associated to M. The method makes essential use of the Harder–Narasimhan filtration for locally free coherent sheaves on an algebraic curve.