Toward higher chromatic analogs of elliptic cohomology II

Abstract

Let $p$ be a prime and $f$ a positive integer, greater than 1 if $p = 2$. We construct liftings of the Artin-Schreier curve $C(p, f)$ in characteristic $p$ defined by the equation $y^e = x - x^p$ (where $e = p^f - 1)$ to a curve $\tilde{C}(p, f)$ over a certain polynomial ring $R^\prime$ in characteristic 0 which shares the following property with $C(p, f)$. Over a certain quotient of $R^\prime$, the formal completion of the Jacobian $J( \tilde{C}(p, f))$ has a 1-dimensional formal summand of height $(p - 1)f$. Along the way we show how Honda’s theory of commutative formal group laws can be extended to more general rings and prove a conjecture of his about the Fermat curve.