Derived categories of curves of genus one and torsors over abelian varieties

TL;DR

This paper establishes a Fourier-Mukai equivalence between the derived categories of genus one curves without rational points and their Jacobians, extending the result to torsors over higher-dimensional abelian varieties.

Contribution

It generalizes the known derived category equivalence from genus one curves to torsors over higher-dimensional abelian varieties.

Findings

Derived categories of genus one curves and their Jacobians are equivalent via Fourier-Mukai transforms.

The equivalence extends to torsors over higher-dimensional abelian varieties.

The class in the Brauer group determines the twisted derived category equivalence.

Abstract

Suppose $C$ is a smooth projective curve of genus 1 over a perfect field $F$, and $E$ is its Jacobian. In the case that $C$ has no $F$-rational points, so that $C$ and $E$ are not isomorphic, $C$ is an $E$-torsor with a class $\delta(C)\in H^1(\text{Gal}(\bar F/F), E(\bar F))$. Then $\delta(C)$ determines a class $\beta \in \text{Br}(E)/\text{Br}(F)$ and there is a Fourier-Mukai equivalence of derived categories of (twisted) coherent sheaves $\mathcal D(C) \xrightarrow{\cong} \mathcal D(E, \beta^{-1})$. We generalize this result to higher dimensions; namely, we prove it also for torsors over abelian varieties.