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Zbl 0962.14021
Donagi, Ron; Livné, Ron
The arithmetic-geometric mean and isogenies for curves of higher genus. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, No.2, 323-339 (1999). ISSN 0391-173X

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 {\it G. Humbert} (1901) and {\it J.-B. Bost} and {\it 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 $\ge 4$.
[Ch.Birkenhake (Erlangen)]

MSC 2000:: *14H40 Jacobians
14K02 Isogeny
14H51 Special divisors

Keywords: arithmetic-geometric mean; isogeny of Jacobians of curves; Prym varieties

Citations: Zbl 0682.14031

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