[math-fun] hyperelliptic curves and AGM?
Victor Miller's cite of PhD thesis by Robert Carls was pretty useless to me -- seems to be a hellish marriage of number theory and topology (??!) without what I wanted, which is complex or real analysis. It seems amazing that they've somehow translated Gauss's AGM into the p-adic number theory world and now use it to do things like count points on finite field versions of "curves"... but at the same time, for mundane analysis purposes this seems useless. However, google then revealed these papers: Takayuki Kato and Keiji Matsumoto: http://www.math.sci.hokudai.ac.jp/~matsu/pdf/FD.pdf we define a (generalized) arithmetic-geometric mean among four terms and express it by Lauricella’s hypergeometric function FD of three variables. Kenji Koike & Hironori Shiga: http://mitizane.ll.chiba-u.jp/metadb/up/irtoroku2/06001.pdf A 3-variable AGM-like iteration yields a certain hyperelliptic integral, also an Appell 2-variable hypergeometric fn. There also are other papers by these same authors. They seem to in some of them be claiming a more general theory. This all seems an impressive breakthrough in analysis which probably will (or should) lead to various new "superfast" algorithms. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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.
* Warren Smith <warren.wds@gmail.com> [Dec 30. 2011 15:41]:
[...]
Thanks for the pointers.
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.
Nicely said. I am still looking for _one_ explicit algorithm in any of those papers I see (about 10 years now). cheers, jj
Look at the Borchardt mean: Define f(a,b,c,d) = (1/2) (sqrt( a b) + sqrt( c d )) Then the iteration for genus 2 is (a,b,c,d) -> ((1/4)(a+b+c+d), f(a,b,c,d),f(a,c,b,d),f(a,d,b,c)) which converges much like the AGM Victor On Fri, Dec 30, 2011 at 10:46 AM, Joerg Arndt <arndt@jjj.de> wrote:
* Warren Smith <warren.wds@gmail.com> [Dec 30. 2011 15:41]:
[...]
Thanks for the pointers.
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.
Nicely said. I am still looking for _one_ explicit algorithm in any of those papers I see (about 10 years now).
cheers, jj
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
This paper ( http://www.emis.de/journals/HOA/IJMMS/Volume12_2/245.pdf ) looks like something good to read. Victor On Fri, Dec 30, 2011 at 10:46 AM, Joerg Arndt <arndt@jjj.de> wrote:
* Warren Smith <warren.wds@gmail.com> [Dec 30. 2011 15:41]:
[...]
Thanks for the pointers.
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.
Nicely said. I am still looking for _one_ explicit algorithm in any of those papers I see (about 10 years now).
cheers, jj
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
* Victor Miller <victorsmiller@gmail.com> [Dec 31. 2011 21:36]:
This paper ( http://www.emis.de/journals/HOA/IJMMS/Volume12_2/245.pdf ) looks like something good to read.
Victor
Thanks it's already in my bib. The papers cited in my chapter 31.4 "AGM-type algorithms for hypergeometric functions" (which possibly adds nothing really new) I find quite readable, especially {Robert S.\ Maier: {A generalization of Euler's hypergeometric transform}, % Trans. Amer. Math. Soc. 358 (2006), 39-57. arXiv:math/0302084v4 [math.CA], \bdate{14-March-2006}. URL: \url{http://arxiv.org/abs/math.CA/0302084}.} {Robert S.\ Maier: {Algebraic hypergeometric transformations of modular origin}, arXiv:math/0501425v3 [math.NT], \bdate{24-March-2006}. URL: \url{http://arxiv.org/abs/math.NT/0501425}.} {Robert S.\ Maier: {On rationally parametrized modular equations}, arXiv:math.NT/0611041v4, \bdate{7-July-2008}.} URL: \url{http://arxiv.org/abs/math.NT/0611041}.} {J.\ M.\ Borwein, P.\ B.\ Borwein: {On the Mean Iteration $(a,b)\leftarrow\big(\frac{a+3b}{4},\frac{\sqrt{ab}+b}{2}\big)$}, Mathematics of Computation, vol.53, no.187, pp.311-326, \bdate{July-1989}. J.\ M.\ Borwein, P.\ B.\ Borwein: {A cubic counterpart of Jacobi's Identity and the AGM}, Transactions of the American Mathematical Society, vol.323, no.2, pp.691-701, \bdate{February-1991}. {J.\ M.\ Borwein, P.\ B.\ Borwein, F.\ Garvan: {Hypergeometric Analogues of the Arithmetic-Geometric Mean Iteration}, Constructive Approximation, vol.9, no.4, pp.509-523, \bdate{1993}. URL: \url{http://www.math.ufl.edu/~frank/publist.html}.} {Frank Garvan: {Cubic modular identities of Ramanujan, hypergeometric functions and analogues of the arithmetic-geometric mean iteration}, Contemporary Mathematics, vol.166, pp.245-264, \bdate{1993}. URL: \url{http://www.math.ufl.edu/~frank/publist.html}.} {Kenji Koike, Hironori Shiga: {A three terms Arithmetic-Geometric mean}, Journal of Number Theory, vol.124, pp.123-141, \bdate{2007}.} I didn't cite any paper of Raimundas Vid\={u}nas because all are over my head (as almost all papers that might offer substantial material). cheers, jj
participants (3)
-
Joerg Arndt -
Victor Miller -
Warren Smith