The Weierstrass Sigma Function in Higher Genus and Applications to Integrable Equations

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Shigeki Matsutani

The Weierstrass Sigma Function in Higher Genus and Applications to Integrable Equations

Shigeki Matsutani

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English

Springer Nature
26 March 2025

Groups & group theory; Algebraic geometry; Mathematical physics

Series: Springer Monographs in Mathematics

This book’s area is special functions of one or several complex variables. Special functions have been applied to dynamics and physics. Special functions such as elliptic or automorphic functions have an algebro-geometric nature. These attributes permeate the book. The “Kleinian sigma function”, or “higher-genus Weierstrass sigma function” generalizes the elliptic sigma function. It appears for the first time in the work of Weierstrass. Klein gave an explicit definition for hyperelliptic or genus-three curves, as a modular invariant analogue of the Riemann theta function on the Jacobian (the two functions are equivalent). H.F. Baker later used generalized Legendre relations for meromorphic differentials, and brought out the two principles of the theory: on the one hand, sigma uniformizes the Jacobian so that its (logarithmic) derivatives in one direction generate the field of meromorphic functions on the Jacobian, therefore algebraic relations among them generate the ideal of the Jacobian as a projective variety; on the other hand, a set of nonlinear PDEs (which turns out to include the “integrable hierarchies” of KdV type), characterize sigma. We follow Baker’s approach.

There is no book where the theory of the sigma function is taken from its origins up to the latest most general results achieved, which cover large classes of curves. The authors propose to produce such a book, and cover applications to integrable PDEs, and the inclusion of related al functions, which have not yet received comparable attention but have applications to defining specific subvarieties of the degenerating family of curves. One reason for the attention given to sigma is its relationship to Sato's tau function and the heat equations for deformation from monomial curves.

The book is based on classical literature and contemporary research, in particular our contribution which covers a class of curves whose sigma had not been found explicitly before.

By: Shigeki Matsutani
Imprint: Springer Nature
Country of Publication: Singapore
Edition: 2024 ed.
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9789819781621
ISBN 10: 9819781620
Series: Springer Monographs in Mathematics
Pages: 501
Publication Date: 26 March 2025
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Hardback
Publisher's Status: Active

Shigeki Matsutani is currently Professor at Kanazawa University. He previously held positions at National Institute of Technology Sasebo College.