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In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this integral or derive the addition theorem.

$$\arcsin(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^2}} (1)$$

In the case of the lemniscate $(2)$ we (Fagnano) can derive a doubling theorem by trying out substitutions in analogy with those which rationalize the first integral. (I learned this from Siegel - Topics in Complex Function Theory). One result (of the sort I wish I could find more) which does come nicely from this integral is via the substitution $y \mapsto iy$ we notice it has double periodicity.

$$\text{sl}^{-1}(y) = \displaystyle \int_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^4}} (2)$$

Euler and others were able to produce theorems about the elliptic integral $(3)$ by analogy with the lemniscate (I read this in Stillwell - Mathematics and its History) - Just as ideas from $\arcsin$ helped to produce theorems about the lemniscate integral. Still, these theorems are very hard earned and it appears that you have to be a master like Euler to derive them.

$$F(y,k) = \displaystyle\int_0^y\frac{\mathrm dy}{\sqrt{(1-k^2 y^2)(1-y^2)}} (3)$$


What I would really like to know is, can we derive more results about the functions from these integrals - maybe using integral manipulation techniques I just don't know about?

Another question that has been bothering me deeply is the integrands (which I believe are called invariant differentials of the lie groups for the algebraic curve) - What sort of coincidence is it that allows the integrands to be of this particular form?

Thank you!

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Some (self biased) links to get you started:

Expository:

Ziegler - talk on the AGM at the Technion's math-club

Cox: The arithmetic-geometric mean of Gauss.. Enseign. Math. (2) 30 (1984), no. 3-4.

Ritzenthaler: AGM for elliptic curves.

Textbook:

Borwein & Borwein: Pi and the AGM via Amazon (w/ look inside)

Research:

Donagi & Livne:The arithmetic-geometric mean and isogenies for curves of higher genus

Dolgachev & Lehavi: On isogenous principally polarized abelian surfaces

Lehavi & Ritzenthaler: Formulas for the arithmetic geometric mean of curves of genus 3

Humbert: Sur les fonctions abéliennes singulières (Troisième Mémoire)


And a couple of non online sources:

Richelot, De transformatione integralium Abelianorum primi ordinis comentatio. J. reine angew. Math. 16 (1837) 221–341.

J.-B. Bost and J.-F. Mestre, Moyenne arithm ́etico-g ́eom ́etrique et p ́eriodes des courbes de genre 1 et 2, Gaz. Math. 38 (1988), 36–64.

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