Toward higher chromatic analogs of elliptic cohomology II

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 $R'$ in characteristic $0$ which shares the following property with $C(p,f)$. Over a certain quotient of $R'$, 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.

Keywords

formal group law, elliptic cohomology, algebraic curve

2010 Mathematics Subject Classification

Primary 55N34. Secondary 14H40, 14H50, 14L05, 55N22.

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Published 1 January 2008