Minimal Degree Liftings of Hyperbolic Curves

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 1, Page 1--47.

Finotti, Luis R. A.
Minimal Degree Liftings of Hyperbolic Curves
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

The main goal of this paper is to analyze the properties of lifts of hyperelliptic curves $y_0^2 = f(x_0)$ over perfect fields of characteristic $p>2$ (to hyperelliptic curves over the ring of Witt vectors) that have lifts of points whose coordinate functions have minimal degrees. It is shown that, when trying to minimize the degrees of the $\wvx$-coordinate, the $(n+1)$-th entry, say $F_n$, can be taken to be a polynomial in $x_0$ such that $(dp^n-(d-2))/2 \leq \deg F_n \leq (dp^n+(d-2))/2$, where $d= \deg f(x_0)$. Besides upper and lower bounds for the degrees, other topics discussed include a necessary condition to achieve the lower bounds and lifting the Frobenius. Computational aspects are also considered and the case of elliptic curves is analyzed in more detail. An explicit formula for derivatives of coordinate functions of the elliptic Teichm\"uller lift is proved, namely $dF_n/dx_0=0$, if $p=2$, and $dF_n/dx_0 = \hi^{(p^n-1)/(p-1)} \,y_0^{p^n-1}- \sum_{i=0}^{n-1} F_i^{(p^{n-i}-1)} \, dF_i/dx_0$, if $p \geq 3$, where $\hi$ is the Hasse invariant of the curve. Finally, we establish a connection between minimal degree liftings and Mochizuki's theory of ``canonical liftings'' in the case of genus 2 curves.

Keywords: elliptic curves, canonical liftings, elliptic Teichm\"uller lift, Witt vectors, minimal degree liftings, Mochizuki liftings

Mathematics Subject Classification (1991): Primary 11G20; Secondary 11T71
Mathematical Reviews Number: MR2044910

Received: 2002-11-07