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

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 $\Q(1/4)$.