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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 detM d dw F = MF, where F = ⎛ ⎝ f1,1(w) f1,2(w) f2,2(w) ⎞ ⎠ , M = ⎛ ⎜⎜⎜⎝ (3− w) − 6 0 3(1 + w) 0 − + w 12 0 6w (3 + w) ⎞ ⎟⎟⎟⎠ . 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.

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