Torsion points on Jacobians of quotients of Fermat curves and \(p\)-adic soliton theory
type: Item
instance of: scholarly article
title: Torsion points on Jacobians of quotients of Fermat curves and \(p\)-adic soliton theory
zbMATH Open document ID: 0838.14020
DOI: 10.1007/BF01231542
author: Greg W. Anderson
published in: Inventiones Mathematicae
publication date: 1996-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144245
review text: In this paper the author introduces a \(p\)-adic analog of the theory of Sato Grassmannian and of the action of loop groups, inspired by the algebro-geometric and complex-analytic constructions described in the papers of Mumford and Segal-Wilson [cf. \textit{D. Mumford}, Proc. Intl. Symp. Algebraic Geometry, Kyoto 1977, 115-153 (1977; Zbl 0423.14007) and \textit{G. Segal} and \textit{G. Wilson}, Publ. Math., Inst. Hautes Étud. Sci. 61, 5-65 (1985; Zbl 0592.35112)]. As an application of that \(p\)-adic theory, the author proves theorem 1 quoted below. We fix some notations. Let \(\ell\) be an odd prime number and let \(K\) be a field of characteristic zero containing a primitive \(\ell\)-th root of unity \(\zeta\). Let \(X/K\) be the smooth projective model of the plane curve \(y^\ell = x^a(1 - x)^b\), where \(a,b\) are integers such that \(\ell > a \geq b > 0\) and \(a + b = \ell + 1\). Let \(\gamma\) be the unique automorphism of \(X\) such that \(\gamma^* x = x\) and \(\gamma^* y = \zeta y\) and denote by \(\infty\) the unique point of \(X\) at which the coordinate functions \(x\) and \(y\) have poles. Let \(J\) be the Jacobian of \(X\), equipped with the \(\mathbb{Z} [\zeta]\)-module structure for which \(\zeta\) acts by \(\gamma^*\) and let \(\Theta \subset J\) be the set of isomorphism classes of line bundles of degree 0 such that \(h^0 ({\mathcal L} ((g - 1) \infty)) > 0\). Let \(p \equiv 1 \bmod \ell\) be a prime number and let \({\mathfrak p} \subset \mathbb{Z} [\zeta]\) be a prime above \(p\). For each ideal \({\mathfrak a} \subset \mathbb{Z} [\zeta]\), let \(J_{\mathfrak a}\) denote the \({\mathfrak a}\)-torsion subgroup of \(J\). We can finally state the announced Theorem 1. \(\Theta \cap (J_{\mathfrak p} + J_{(1 - \zeta)}) \subseteq J_{(1 - \zeta)}\). In the following, we trace a sketch of the \(p\)-adic \(\tau\)-formalism introduced by the author, by comparing it with the ``classical'' theory on the subject (cf. the papers quoted above). Let \(K\) be a …
reviewed by: Maurizio Candilera
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H40
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 703865
zbMATH Keywords: Jacobians of quotients of Fermat curves
zbMATH Keywords: \(p\)-adic soliton
zbMATH Keywords: \(p\)-adic \(\tau\)-formalism
zbMATH Keywords: Sato \(\tau\)-function
zbMATH Keywords: Krichever pair
zbMATH Keywords: homothety classes
zbMATH Keywords: Theta divisor
zbMATH Keywords: Laurent series
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 703865