Homology, Homotopy and Applications

Toward higher chromatic analogs of elliptic cohomology II

Douglas C. Ravenel

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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.

Article information

Source
Homology Homotopy Appl., Volume 10, Number 3 (2008), 335-368.

Dates
First available in Project Euclid: 1 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.hha/1251832478

Mathematical Reviews number (MathSciNet)
MR2475628

Zentralblatt MATH identifier
1178.55004

Subjects
Primary: 55N34: Elliptic cohomology
Secondary: 14H40: Jacobians, Prym varieties [See also 32G20] 14H50: Plane and space curves 14L05: Formal groups, $p$-divisible groups [See also 55N22] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90]

Keywords
Formal group law elliptic cohomology algebraic curve

Citation

Ravenel, Douglas C. Toward higher chromatic analogs of elliptic cohomology II. Homology Homotopy Appl. 10 (2008), no. 3, 335--368. https://projecteuclid.org/euclid.hha/1251832478


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