"Integrable Systems, Geometry, and Abelian Functions"
日時：2005 年 5 月 26 日（木）10:50 から 27 日（金）17:30
場所：首都大学東京 (東京都立大学)，理学研究科棟６階６１０室又は６１８室
科学研究費補助金 基盤研究(C)(2)
研究代表者：大西良博
研究課題名：『行列式公式・Bernoulli-Hurwitz 数の Abel 函数版の研究』
研究課題番号： 16540002
の研究計画の一環として，下記の研究集会を行ひます．
連絡先：
大西良博（岩手大学）onishi at iwate-u.ac.jp
マーティンゲスト (首都大学東京) martin at comp.metro-u.ac.jp
プログラム
5 月 26 日（木）
10:50-11:00 挨拶
11:00-12:00 大西良博（岩手大学）"A generalization of the determinantal expression
of Frobenius-Stickelberger to Abelian functions"
ABSTRACT: The original detarminantal formula of Frobenius-Stickeberger (1877) is an expression of a quotient of sigma (or theta) functions by a determinant of Weierstrass $\wp$-function and its higher derivatives (see Whittaker-Watson, p.458). I recently generalized their formula to the case of hyperelliptic functions (more information : web page) . This formula is very explicit and quite different from the "trisecant" and "general formula" of Fay (LNM 352, pp.33-34). I will talk about such a generalization. Moreover, I will report further extension of this determinantal formula along the line above to functions on any "Purely Trigonal Curve", namely, a curve defined by $y^3=f(x)$, where $f(x)$ is a polynomial of $x$ whose degree is coprime to 3. If possible, I hope to mention the case of purely $d$-gonal curves.
13:45-14:45 Victor Z. Enol'skii（Concordia 大学，Heriot-Watt 大学）"Algebro-geometric
integration of charge 3 monopole equations"
15:00-16:00 J. Chris Eilbeck （Heriot-Watt 大学）"Some computations
involving theta-functions"
ABSTRACT: I will discuss two topics involving the efficient evaluations of theta functions connected with algebraic curves and integrable systems. These are of interest on both practical and theoretical grounds. One is the use of the Richelot transformation to evaluate genus two hyperelliptic integrals, a generalization of the Algebraic-Geometric Mean of Gauss. The other is the study of reducible period matrices, when the algebraic curve is a cover of one or more of lower genus. In this case the higher genus theta function can be written as a sum of products of lower genus (often g=1) theta functions.
TEA
16:30-17:30 松本圭司（北海道大学）"Automorphic functions for the Whitehead-link"
ABSTRACT: We construct automorphic functions on the real 3-dimensional hyperbolic space H^3 for the Whitehead-link-complement group W\subset GL_2(Z[i]) and for a few groups commensurable with W. These automorphic functions give embeddings of the orbit spaces of H^3 under these groups, and arithmetical characterizations of them.
5 月 27 日（金）
11:00-12:00 軍司 圭一 (東京大学) "The defining equations of the universal
abelian surfaces with level three structure"
ABSTRACT: In this talk, we consider the universal principally polarized abelian surface, and embed it into the 8 dimensional projective space by the third power of the line bundle which gives a principal polarization. Then we can write down the defining equations explicitly, 9 quadratic equations and 3 cubic equations. Moreover, we give the coefficients of the equations by using Siegel modular forms of level 3.
13:45-14:45 松谷茂樹（キヤノン（株））"Weierstrass al functions and their applications"
ABSTRACT: In this talk, I will give an introduction to the hyperelliptic al functions which was discovered by Weierstrass as generalization of Jacobi sn, cn, dn functions. They give solutions of the Neumann dynamical system, the MKdV equation and the sine-Gordon equation. It is known that the hyperelliptic solutions of the sine-Gordon equation are associated with differential geometric objects and are closely connected with geometrical properties of the hyperelliptic curve itself. Thus I will mainly talk about the application of the al function to the sine-Gordon equation. Using the al function, we can prove for it to be a solution only by very primitive residual computations and combinatorial tricks, without using any theta functions. As this story can be generalized to one on certain subvarieties in the Jacobi variety associated with the curve, I will mention the topics if possible.
15:00-16:00 中屋敷 厚 （九州大学）"Differential structure of abelian functions"
ABSTRACT: The Weierstrass elliptic pe function together with its derivatives and 1 form a linear basis of the space of meromorphic functions on a elliptic curve which have poles at a fixed point. In this talk I will discuss the substitute for this fact in higher dimensions. We study the case of a principally polarized abelian variety whose theta divisor is non-singular. The character method, which was previously used for Jacobians of spectral curves, is extensively used. The talk is based on joint work with Koji Cho.
TEA
16:30-17:30 Emma Previato（Boston 大学）"On the Hitchin system in genus
2"
ABSTRACT: The Hitchin system of the title is an algebraically completely integrable
system whose integral manifolds are Prymians of dimension 3g-3, defined on the
holomorphic-symplectic manifold T*SU_X(2,xi), the cotangent bundle to the moduli
space of vector bundles of rank 2 and fixed odd determinant xi, over a Riemann
surface X of genus g>1. Further work by Hitchin (1990) implemented geometric
quantization and provided a link of the system with the KZ (Knizhnik-Zamolodchikov)
equations when the Riemann surface varies, by showing that an analog of the
rank-1 heat equation holds over these moduli spaces. These results have not
been rendered in terms of explicit functions except in genus 2, and in fact
only in an extended sense (even-determinant case).
In this talk, which is intended to be introductory and partly expository, I
will detail the mentioned features and their interconnection; more precisely:
(1) explicit Hamiltonians of the Hitchin system in genus 2, even-determinant
case (joint work with B. van Geemen, and work by K. Gawcedzky and P. Trang-Ngoc-Bich)
including a 'strange duality' of projective-geometric nature; (2) interpretation
of the integrals for the genus 2, odd-determinant case (W.M. Oxbury, unpublished
D.Phil. thesis, Oxford, 1987); (3) geometric quantization in genus 2, even-determinant
case (B. van Geemen and A. de Jong), as well as a(nother?) heat connection for
hyperelliptic Riemann surfaces of any genus, even-determinant case; (4) KZ equations
in genus 2 (K. Gawcedzky and P. Trang-Ngoc-Bich); (5) ad hoc reduction of Hitchin
to Neumann (rational-parameter Lax equations) in genus 2, even determinant (K.
Gawcedzky and P. Trang-Ngoc-Bich); (6) Lax representation for the Hitchin system
in terms of Tyurin parameters (I.M. Krichever); (7) singular cases (N. Nekrasov;
A. Chervov and D. Talalaev; B. Enriques and V. Rubtsov).
This background will serve to formulate a current research program, articulated
as follows: (1) description of SU_X(2,L) for X hyperelliptic, L even/odd (S.
Ramanan, A. Beauville) and of SU_X(2,L) for X non-hyperelliptic of genus 3,
L even (Coble); (2) construction of a geometric-quantization coordinate space
in the cases given in (1), and of the class of functions for which the generalized
heat equations are to be found.
ENGLISH ANNOUNCEMENT
"Integrable Systems, Geometry, and Abelian Functions"
Tokyo Metropolitan University, Department of Mathematics
26-27th May 2005, Faculty of Science Building, Room 610 or 618
For further information please contact
Yoshihiro Onishi (Iwate University) onishi at iwate-u.ac.jp
Martin Guest (Tokyo Metropolitan University) martin at comp.metro-u.ac.jp
PROGRAM
26th May (Thu.)
10:50-11:00 Opening
11:00-12:00 Yoshihiro Onishi (Iwate University) "A generalization
of the determinantal expression of Frobenius-Stickelberger to Abelian functions"
ABSTRACT: The original detarminantal formula of Frobenius-Stickeberger (1877) is an expression of a quotient of sigma (or theta) functions by a determinant of Weierstrass $\wp$-function and its higher derivatives (see Whittaker-Watson, p.458). I recently generalized their formula to the case of hyperelliptic functions (more information : web page) . This formula is very explicit and quite different from the "trisecant" and "general formula" of Fay (LNM 352, pp.33-34). I will talk about such a generalization. Moreover, I will report further extension of this determinantal formula along the line above to functions on any "Purely Trigonal Curve", namely, a curve defined by $y^3=f(x)$, where $f(x)$ is a polynomial of $x$ whose degree is coprime to 3. If possible, I hope to mention the case of purely $d$-gonal curves.
13:45-14:45 Victor Z. Enol'skii (Concordia University, Heriot-Watt University)
"Algebro-geometric integration of charge 3 monopole equations"
15:00-16:00 J. Chris Eilbeck (Heriot-Watt University) "Some computations
involving theta-functions"
TEA
16:30-17:30 Keiji Matsumoto (Hokkaido University) "Automorphic
functions for the Whitehead-link"
ABSTRACT: We construct automorphic functions on the real 3-dimensional hyperbolic space H^3 for the Whitehead-link-complement group W\subset GL_2(Z[i]) and for a few groups commensurable with W. These automorphic functions give embeddings of the orbit spaces of H^3 under these groups, and arithmetical characterizations of them.
27th May (Fri.)
11:00-12:00 Keiichi Gunji (Tokyo University) "The defining equations
of the universal abelian surfaces with level three structure"
ABSTRACT: In this talk, we consider the universal principally polarized abelian surface, and embed it into the 8 dimensional projective space by the third power of the line bundle which gives a principal polarization. Then we can write down the defining equations explicitly, 9 quadratic equations and 3 cubic equations. Moreover, we give the coefficients of the equations by using Siegel modular forms of level 3.
13:45-14:45 Shigeki Matsutani (Canon Inc.) "Weierstrass al functions
and their applications"
ABSTRACT: In this talk, I will give an introduction to the hyperelliptic al functions which was discovered by Weierstrass as generalization of Jacobi sn, cn, dn functions. They give solutions of the Neumann dynamical system, the MKdV equation and the sine-Gordon equation. It is known that the hyperelliptic solutions of the sine-Gordon equation are associated with differential geometric objects and are closely connected with geometrical properties of the hyperelliptic curve itself. Thus I will mainly talk about the application of the al function to the sine-Gordon equation. Using the al function, we can prove for it to be a solution only by very primitive residual computations and combinatorial tricks, without using any theta functions. As this story can be generalized to one on certain subvarieties in the Jacobi variety associated with the curve, I will mention the topics if possible.
15:00-16:00 Atsushi Nakayashiki (Kyushu University) "Differential
structure of abelian functions"
ABSTRACT: The Weierstrass elliptic pe function together with its derivatives and 1 form a linear basis of the space of meromorphic functions on a elliptic curve which have poles at a fixed point. In this talk I will discuss the substitute for this fact in higher dimensions. We study the case of a principally polarized abelian variety whose theta divisor is non-singular. The character method, which was previously used for Jacobians of spectral curves, is extensively used. The talk is based on joint work with Koji Cho.
TEA
16:30-17:30 Emma Previato (Boston University) "On the Hitchin system
in genus 2"
ABSTRACT: The Hitchin system of the title is an algebraically completely integrable
system whose integral manifolds are Prymians of dimension 3g-3, defined on the
holomorphic-symplectic manifold T*SU_X(2,xi), the cotangent bundle to the moduli
space of vector bundles of rank 2 and fixed odd determinant xi, over a Riemann
surface X of genus g>1. Further work by Hitchin (1990) implemented geometric
quantization and provided a link of the system with the KZ (Knizhnik-Zamolodchikov)
equations when the Riemann surface varies, by showing that an analog of the
rank-1 heat equation holds over these moduli spaces. These results have not
been rendered in terms of explicit functions except in genus 2, and in fact
only in an extended sense (even-determinant case).
In this talk, which is intended to be introductory and partly expository, I
will detail the mentioned features and their interconnection; more precisely:
(1) explicit Hamiltonians of the Hitchin system in genus 2, even-determinant
case (joint work with B. van Geemen, and work by K. Gawcedzky and P. Trang-Ngoc-Bich)
including a 'strange duality' of projective-geometric nature; (2) interpretation
of the integrals for the genus 2, odd-determinant case (W.M. Oxbury, unpublished
D.Phil. thesis, Oxford, 1987); (3) geometric quantization in genus 2, even-determinant
case (B. van Geemen and A. de Jong), as well as a(nother?) heat connection for
hyperelliptic Riemann surfaces of any genus, even-determinant case; (4) KZ equations
in genus 2 (K. Gawcedzky and P. Trang-Ngoc-Bich); (5) ad hoc reduction of Hitchin
to Neumann (rational-parameter Lax equations) in genus 2, even determinant (K.
Gawcedzky and P. Trang-Ngoc-Bich); (6) Lax representation for the Hitchin system
in terms of Tyurin parameters (I.M. Krichever); (7) singular cases (N. Nekrasov;
A. Chervov and D. Talalaev; B. Enriques and V. Rubtsov).
This background will serve to formulate a current research program, articulated
as follows: (1) description of SU_X(2,L) for X hyperelliptic, L even/odd (S.
Ramanan, A. Beauville) and of SU_X(2,L) for X non-hyperelliptic of genus 3,
L even (Coble); (2) construction of a geometric-quantization coordinate space
in the cases given in (1), and of the class of functions for which the generalized
heat equations are to be found.
This conference is supported by the Ministry of Education, Science, Sports and
Culture under Grants-in-Aid for Scientific Research.