It seems the original idea of generalizing Gauss's AGM (arithmo-geometric mean) iteration to HYPERelliptic curves, was begun by F. Richelot: Essai sur une methode generale pour determiner la valeur des integrales ultra-elliptiques, fondee sur des transformations remarquables de ces transcendentes, Com. Rend. Acad. Sci. Paris 2 (1836) 622-627 F. Richelot: De transformatione integralium Abelianorum primi ordinis commentation. Jour. fur die reine und angew. Math., 16 (1837) 221-341 G. Humbert: Sur la transformation ordinaire des fonctions abeliannes, Journal des Mathe- matiques, 7 (1901) A more modern paper: J-B.Bost & J-F. Mestre: Moyenne Arithmetico-geometrique et Periodes des Courbes de genere 1 et 2, Gazette des Mathematiciens, Soc. de Mathematique de France 38 (1988) 36-64. In Ron Donagi & Ron Livne http://arxiv.org/pdf/alg-geom/9712027 they actually for a brief moment give a semi-comprehensible explanation of what is going on: ALMOST-QUOTE: Gauss’s arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one, specifically one whose period is double the original period. A higher genus analogue should involve the explicit construction of a curve whose jacobian is isogenous to the jacobian of a given curve. The doubling of the period matrix means that the kernel of the isogeny should be a lagrangian subgroup of the group of points of order 2 in the jacobian. In genus 2 such a construction was given classically by Humbert and was studied more recently by Bost and Mestre. In this article we give such a construction for general curves of genus 3. We also give a similar but simpler construction for hyperelliptic curves of genus 3. We show that the hyperelliptic construction is a degeneration of the general one, and we prove that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of the points of order 2. We show that for g >= 4 no similar construction exists. END QUOTE. This vaguely makes it sound as though any hyperelliptic curve "period integral" if the curve has genus = 1,2 or 3 (well, g=1 is the elliptic curve case) should be computable by a superfast algorithm (?) somewhat like Gauss's AGM iteration, but with more (but still a finite set of) variables. It seems to me if so, it'd be worth writing those algorithms out highly explicitly, instead of all this ultra-abstract garbage. And running them.