Musings on $\Q(1/4)$: Arithmetic spin structures on elliptic curves

TL;DR

This paper introduces arithmetic spin structures on elliptic curves over finite fields, identifying a unique class that may serve as a geometric realization of the fractional Tate motive (1/4).

Contribution

It defines and characterizes arithmetic spin structures on elliptic curves, linking them to a candidate for the motive (1/4) in algebraic geometry.

Findings

Unique isogeny class of elliptic curves over _{p^2} with a spin structure

Provides a geometric object of weight 1/2 as a candidate for (1/4)

Establishes a connection between spin structures and fractional motives

Abstract

We introduce and study arithmetic spin structures on elliptic curves. We show that there is a unique isogeny class of elliptic curves over $\F_{p^2}$ which carries a unique arithmetic spin structure and provides a geometric object of weight 1/2 in the sense of Deligne and Grothendieck. This object is thus a candidate for $\Q(1/4)$.