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The arithmetic-geometric mean and isogenies for curves of higher genus. (English) Zbl 0962.14021

The geometric interpretation of the computation of Gauss’s arithmetic-geometric mean is the construction of an isogeny of elliptic curves of degree 2. A generalization would be the explicit construction of an isogeny of Jacobians of curves of higher genus the kernel of which being a Lagrangian subgroup of the points of order 2. In genus 2 there are such constructions due to G. Humbert (1901) and J.-B. Bost and j.-F. Mestre [Gaz. Math., Soc. Math. Fr. 38, 36-64 (1988; Zbl 0682.14031)]. The paper under review presents such constructions for curves of genus 3 using the bigonal and trigonal constructions for Prym varieties. Moreover it is shown that there is no such construction for curves of genus \(\geq 4\).

MSC:

14H40 Jacobians, Prym varieties
14K02 Isogeny
14H51 Special divisors on curves (gonality, Brill-Noether theory)

Citations:

Zbl 0682.14031
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References:

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