Abelian varieties with no power isogenous to a Jacobian

2024/01/12 by Fortman, Olivier de Gaay, Schreieder, Stefan · 2 citations
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2401.06577

Abstract

For a curve of genus at least four which is either very general or very general hyperelliptic, we classify all ways in which a power of its Jacobian can be isogenous to a product of Jacobians of curves. As an application, we show that, for a very general principally polarized abelian variety of dimension at least four, or the intermediate Jacobian of a very general cubic threefold, no power is isogenous to a product of Jacobians of curves. This confirms some cases of the Coleman-Oort conjecture. We further deduce from our results some progress on the question whether the integral Hodge conjecture fails for such abelian varieties.

Cited by

• Powers of abelian varieties over \mathbb Q(t) not isogenous to a Jacobian
• On direct summands of products of Jacobians over arbitrary fields

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• The Existence of an Abelian Variety over the Algebraic Numbers isogenous\n to no Jacobian
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