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Transactions of the Moscow Mathematical Society

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Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2


Author: E. Yu. Netaĭ
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2013, 281-292
MSC (2010): Primary 53C07; Secondary 34A30, 34A34, 34A26, 33C20
Published electronically: April 9, 2014
MathSciNet review: 3235801
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct some differential equations describing the geometry of bundles of Jacobians of algebraic curves of genus 1 and 2.

For an elliptic curve we produce differential equations on the coefficients of a cometric compatible with the Gauss-Manin connection of the universal bundle of Jacobians of elliptic curves. This cometric is defined in terms of a solution $ F$ of the linear system of differential equations

$\displaystyle 2 \det M \frac {d}{d w} F = M F,\;\;$$\displaystyle \text {where } F = \begin {pmatrix}f_{1,1}(w)\\ f_{1,2}(w)\\ f_... ... - {\displaystyle \frac {1 + w}{12}} \\ [2mm] 0 & 6 w & (3 + w)\end{pmatrix}. $

We describe the general solution of this system in terms of Meijer $ G$-functions and hypergeometric functions.

For a curve of genus 2 we find differential equations defined by vector fields tangent to the discriminant of the curve. Solutions of these equations 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.


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Additional Information

E. Yu. Netaĭ
Affiliation: Steklov Mathematical Institute, Moscow
Email: bunkova@mi.ras.ru

DOI: https://doi.org/10.1090/S0077-1554-2014-00223-5
Keywords: Elliptic curves, hyperelliptic curves, Gauss--Manin connection, Meijer $G$-functions, hypergeometric functions.
Published electronically: April 9, 2014
Additional Notes: This research was supported by the Russian Foundation for Basic Research (grant nos. 12-01-33058 and 11-01-00197-a) and by the grant 2010-220-01-077 of the Government of the Russian Federation, contract 11.G34.31.0005.
Article copyright: © Copyright 2014 E. Yu.Netaĭ