The Euler top and canonical lifts

TL;DR

This paper proves a finiteness result for canonical lifts in elliptic fibrations, revealing a key difference between arithmetic and classical cases in the context of the Euler top.

Contribution

It establishes a new finiteness theorem for fibers that are canonical lifts in elliptic fibrations, motivated by an arithmetic Euler top construction.

Findings

Finiteness of fibers that are canonical lifts in elliptic fibrations.

Discrepancy between arithmetic and classical cases regarding flow extensions.

Inability to extend flows to compactifications in the arithmetic setting.

Abstract

In this note, we prove a finiteness result for fibers that are canonical lifts in a given elliptic fibration. The question was motivated by the authors' construction of an arithmetic Euler top, and it highlights an interesting discrepancy between the arithmetic and the classical case: in the former, it is impossible to extend the flows to a compactification of the phase space, viewed as an elliptic fibration over the space of action variables.