{"title":"The Calogero-Franc¸oise Integrable System: Algebraic Geometry, Higgs Fields, and the Inverse Problem","authors":"S. Rayan, Thomas Stanley, J. Szmigielski","doi":"10.1017/9781108773287.015","DOIUrl":"https://doi.org/10.1017/9781108773287.015","url":null,"abstract":"We review the Calogero-Francoise integrable system, which is a generalization of the Camassa-Holm system. We express solutions as (twisted) Higgs bundles, in the sense of Hitchin, over the projective line. We use this point of view to (a) establish a general answer to the question of linearization of isospectral flow and (b) demonstrate, in the case of two particles, the dynamical meaning of the theta divisor of the spectral curve in terms of mechanical collisions. Lastly, we outline the solution to the inverse problem for CF flows using Stieltjes' continued fractions.","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"229 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124533465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Normal Forms for Kummer Surfaces","authors":"A. Clingher, Andreas Malmendier","doi":"10.1017/9781108773355.006","DOIUrl":"https://doi.org/10.1017/9781108773355.006","url":null,"abstract":"We determine normal forms for the Kummer surfaces associated with abelian surfaces of polarization of type $(1,1)$, $(1,2)$, $(2,2)$, $(2,4)$, and $(1,4)$. Explicit formulas for coordinates and moduli parameters in terms of Theta functions of genus two are also given. The normal forms in question are closely connected to the generalized Riemann identities for Theta functions of Mumford's.","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125330821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian","authors":"Y. Zarhin","doi":"10.1017/9781108773355","DOIUrl":"https://doi.org/10.1017/9781108773355","url":null,"abstract":"Let $f(x)$ be a degree $(2g+1)$ monic polynomial with coefficients in an algebraically closed field $K$ with $char(K)ne 2$ and without repeated roots. Let $mathfrak{R}subset K$ be the $(2g+1)$-element set of roots of $f(x)$. Let $mathcal{C}: y^2=f(x)$ be an odd degree genus $g$ hyperelliptic curve over $K$. Let $J$ be the jacobian of $mathcal{C}$ and $J[2]subset J(K)$ the (sub)group of its points of order dividing $2$. We identify $mathcal{C}$ with the image of its canonical embedding into $J$ (the infinite point of $mathcal{C}$ goes to the identity element of $J$). Let $P=(a,b)in mathcal{C}(K)subset J(K)$ and $M_{1/2,P}subset J(K)$ the set of halves of $P$ in $J(K)$, which is $J[2]$-torsor. In a previous work we established an explicit bijection between $M_{1/2,P}$ and the set of collections of square roots $$mathfrak{R}_{1/2,P}:={mathfrak{r}: mathfrak{R} to Kmid mathfrak{r}(alpha)^2=a-alpha forall alphainmathfrak{R}; \u0000prod_{alphainmathfrak{R}} mathfrak{r}(alpha)=-b}.$$ The aim of this paper is to describe the induced action of $J[2]$ on $mathfrak{R}_{1/2,P}$ (i.e., how signs of square roots $mathfrak{r}(alpha)=sqrt{a-alpha}$ should change).","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"14 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120864917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Jacobi Inversion Formulae for a Curve in Weierstrass Normal Form","authors":"J. Komeda, S. Matsutani","doi":"10.1017/9781108773355.013","DOIUrl":"https://doi.org/10.1017/9781108773355.013","url":null,"abstract":"We consider a pointed curve $(X,P)$ which is given by the Weierstrass normal form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +cdots + A_{r-1}(x) y+ A_{r}(x)$ where $x$ is an affine coordinate on $mathbb{P}^1$, the point $infty$ on $X$ is mapped to $x=infty$, and each $A_j$ is a polynomial in $x$ of degree $le j s/r$ for a certain coprime positive integers $r$ and $s$ ($r<s$) so that its Weierstrass non-gap sequence at $infty$ is a numerical semigroup. Since it can be regarded as a natural generalization of Weierstrass normal form in the Weierstrass elliptic function theory, it has natural properties. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian based on results of Jorgenson (Israel J. Math (1992) 77 pp 273-284).","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125540840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Arithmetic Analogues of Hamiltonian Systems","authors":"A. Buium","doi":"10.1017/9781108773355.002","DOIUrl":"https://doi.org/10.1017/9781108773355.002","url":null,"abstract":"The paper reviews various arithmetic analogues of Hamiltonian systems and presents some new facts suggesting ways to relate/unify these examples.","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121445907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tropical Markov Dynamics and Cayley Cubic","authors":"K. Spalding, A. Veselov","doi":"10.1017/9781108773287.016","DOIUrl":"https://doi.org/10.1017/9781108773287.016","url":null,"abstract":"We study the tropical version of Markov dynamics on the Cayley cubic, introduced by V. E. Adler and one of the authors. We show that this action is semi-conjugated to the standard action of SL2(Z) on a torus, and thus is ergodic with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.","PeriodicalId":110875,"journal":{"name":"Integrable Systems and Algebraic Geometry","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133647066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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