Chow motives of genus one fibrations

Let \(f: X \rightarrow C\) $f:X\to C$ be a genus 1 fibration from a smooth projective surface, i.e. its generic fiber is a regular genus 1 curve. Let \(j: J \rightarrow C\) $j:J\to C$ be the Jacobian fibration of f. In this paper, we prove that the Chow motives of X and J are isomorphic. As an application, combined with our concomitant work on motives of quasi-elliptic fibrations, we prove Kimura finite-dimensionality for smooth projective surfaces not of general type with geometric genus 0. This generalizes Bloch–Kas–Lieberman’s result to arbitrary characteristic.