Adapt Bhatt’s proof to unramified multiplication covers

Develop an adaptation of Bhatt’s alternative proof for mock Frobenius covers of projective space to the unramified multiplication map [ℓ] colon A to A on an abelian variety.

Background

Bhatt’s alternative proof of the vanishing theorem for mock Frobenius covers relies on the ramification of the mock Frobenius morphism. The multiplication-by-ℓ map on an abelian variety is instead an unramified étale isogeny, so the existing argument does not directly apply. The paper explicitly records the unresolved difficulty of adapting that strategy to the abelian setting, although the paper itself later proves the desired vanishing by different methods.