Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2
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- by
E. Yu. Netaĭ
Translated by: E. Khukhro - Trans. Moscow Math. Soc. 2013, 281-292
- DOI: https://doi.org/10.1090/S0077-1554-2014-00223-5
- Published electronically: April 9, 2014
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 \[ 2 \det M \frac {d}{d w} F = M F,\;\;\text {where } F = \begin {pmatrix} f_{1,1}(w)\\ f_{1,2}(w)\\f_{2,2}(w)\end {pmatrix}, \; M = \begin {pmatrix} (3 - w) & - {\displaystyle \frac {w}{6}} & 0 \\[1mm] 3 (1 + w) & 0 & - {\displaystyle \frac {1 + w}{12}} \\[2mm] 0 & 6 w & (3 + w)\end {pmatrix}. \] 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.
References
Bibliographic Information
- E. Yu. Netaĭ
- Affiliation: Steklov Mathematical Institute, Moscow
- Email: bunkova@mi.ras.ru
- 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.
- © Copyright 2014 E. Yu. Netaĭ
- Journal: Trans. Moscow Math. Soc. 2013, 281-292
- MSC (2010): Primary 53C07; Secondary 34A30, 34A34, 34A26, 33C20
- DOI: https://doi.org/10.1090/S0077-1554-2014-00223-5
- MathSciNet review: 3235801