Periods and Gauss-Manin connection for families of p-adic Schottky groups
type: Item
instance of: scholarly article
title: Periods and Gauss-Manin connection for families of p-adic Schottky groups
zbMATH Open document ID: 0622.14018
DOI: 10.1007/BF01458615
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164160
review text: Several new concepts and results come together in this paper: 1. The author studies families of Schottky groups parametrized by a rigid analytic space, and the corresponding family of Mumford curves. He proves (in section \(5\)) a number of results on automorphic functions for such families which generalize results in the book by the author and \textit{M. van der Put} [''Schottky groups and Mumford curves'', Lect. Notes Math. 817 (1980; Zbl 0442.14009)]. 2. The universal abelian variety \({\mathcal A}\) over the p-adic Siegel half space \({\mathcal S}\) is constructed, together with the canonical principal polarization. It comes out as the quotient of a torus \({\mathcal T}\) over \({\mathcal S}\) by a lattice \(\Lambda\). The author introduces principal theta functions on \({\mathcal T}\) and determines a basis of the \(\Lambda\)-invariant differentials on \({\mathcal T}\). This enables him to show that the first relative de Rham cohomology group of \({\mathcal A}/{\mathcal S}_{dz_ i}\) is generated by elements \(\alpha_ 1,...,\alpha_ g\) and \(\beta_ 1,...,\beta_ g\) where \(\alpha_ i\) is the class of \(dz_ i/z_ i\) and \(\beta_ i\) is the class of a cocycle defined by differentials of the form \(d\theta\) /\(\theta\) for certain principal theta functions \(\theta\). 3. The author then calculates the Gauss-Manin connection \(\nabla\) on \(H^ 1_{DR}({\mathcal A}/{\mathcal S})\) and proves that \(\nabla (\beta_ i)=0\) and \(\nabla (\alpha_ i)=\sum^{g}_{j=1}\beta_ j\otimes (dq_{ij}/q_{ij})\) where \(q_{ij}\) are the coordinates on \({\mathcal S}.\) 4. For a family of Mumford curves C/S he also finds a basis \(\alpha_ 1,...,\alpha_ g\), \(\beta_ 1,...,\beta_ g\) of \(H^ 1_{DR}(C/S):\) here \(\alpha_ i=(du_ i/u_ i)\) with a well known infinite automorphic product \(u_ i(z)\), and the \(\beta_ i\) are discussed in the author's paper [Math. Ann. 270, 381-392 (1985; Zbl 0535.14016)]. The main result of the paper is that the same formulas as above hold for the Gauss-Manin…
reviewed by: Frank Herrlich
Mathematics Subject Classification ID: 14G20
Mathematics Subject Classification ID: 14H15
Mathematics Subject Classification ID: 32G20
Mathematics Subject Classification ID: 14F40
zbMATH DE Number: 4008551
zbMATH Keywords: period map
zbMATH Keywords: Schottky groups parametrized by a rigid analytic space
zbMATH Keywords: family of Mumford curves
zbMATH Keywords: p-adic Siegel half space
zbMATH Keywords: de Rham cohomology group
zbMATH Keywords: infinite automorphic product
zbMATH Keywords: Gauss-Manin connection
zbMATH Keywords: Tate curve
zbMATH Keywords: principal theta function
author: Lothar Gerritzen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4008551