Periods and Gauss-Manin connection for families of p-adic Schottky groups
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 connection. 5. In the final section explicit calculations are carried out for the Tate curve, i.e. the case \(g=1\). The author finds the classical product decomposition for the principal theta function and calculates the Legendre parameter \(\lambda\) as analytic infinite product in the parameter \(\pi\), \(0<| \pi | <1\), on \({\mathcal S}_ 1\). He also obtains formulas involving the \(\wp\)-function, \(\sqrt{e_ 2-e_ 1}\) and the hypergeometric function \({}_ 2F_ 1\).
Schottky groups parametrized by a rigid analytic space
scientific article; zbMATH DE number 4008551
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