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Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2. (English. Russian original) Zbl 1306.53013

Trans. Mosc. Math. Soc. 2013, 281-292 (2013); translation from Tr. Mosk. Mat. O.-va 74, No. 2, 339-352 (2013).
Differential equations describing the geometry of bundles of Jacobians of algebraic curves of genus 1 and 2 are being constructed. They are defined using the coefficients of a cometric compatible with the Gauss-Manin connection of the universal bundle of such Jacobians. The cometric is defined in terms of a solution \(F\) of the linear system of differential equations 2det\(MdF/dw=MF\), where \(F\) is a 3-vector-column and \[ M=\left( \begin{matrix} (3-w)&-w/6&0 \\ 3(1+w)&0&-(1+w)/12\\ 0&6w&(3+w)\end{matrix}\right). \] The description of the general solution involves Meijer G-functions and hypergeometric functions. For a genus-2 curve the author finds differential equations defined by vector fields tangent to the discriminant of the curve. Their solutions define the coefficients of matrix equations on cometrics compatible with the Gauss-Manin connection of the universal bundle of Jacobians of curves of genus 2.

MSC:

53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
34A30 Linear ordinary differential equations and systems
34A34 Nonlinear ordinary differential equations and systems
34A26 Geometric methods in ordinary differential equations
33C20 Generalized hypergeometric series, \({}_pF_q\)
14H52 Elliptic curves
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