On Regularization of Second Kind Integrals

Julia Bernatska   ; Dmitry Leykin

Symmetry, integrability and geometry: methods and applications, Tome 14 (2018)

Résumé

We obtain expressions for second kind integrals on non-hyperelliptic $(n,s)$-curves. Such a curve possesses a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity serves as the basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the infinity, therefore the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for $(3,4)$, $(3,5)$, $(3,7)$, and $(4,5)$-curves. By the example of $(3,4)$-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.

Détail

Keywords: second kind integral; regularization constant; Abelian function relation; Jacobi inversion problem; addition formula.

Bibliographie

[1] Baker H. F., Abelian functions: Abel's theorem and the allied theory of theta functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995 | Zbl

[2] Buchstaber V. M., Enolskii V. Z., Leykin D. V., “Rational analogues of abelian functions”, Funct. Anal. Appl., 33 (1999), 83–94 | DOI | MR | Zbl

[3] Buchstaber V. M., Enolskii V. Z., Leykin D. V., “Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations”, Funct. Anal. Appl., 34 (2000), 159–171 | DOI | MR | Zbl

[4] Buchstaber V. M., Enolskii V. Z., Leykin D. V., Multi-dimensional sigma-functions, arXiv: 1208.0990

[5] Bukhshtaber V. M., Leykin D. V., “Heat equations in a nonholomic frame”, Funct. Anal. Appl., 38 (2004), 88–101 | DOI | MR | Zbl

[6] Bukhshtaber V. M., Leykin D. V., “Addition laws on Jacobian varieties of plane algebraic curves”, Proc. Steklov Inst. Math., 251 (2005), 49–120 | Zbl

[7] Bukhshtaber V. M., Leykin D. V., “Solution of the problem of differentiation of Abelian functions over parameters for families of $(n,s)$-curves”, Funct. Anal. Appl., 42 (2008), 268–278 | DOI | MR | Zbl

[8] Eilbeck J. C., England M., Ônishi Y., “Abelian functions associated with genus three algebraic curves”, LMS J. Comput. Math., 14 (2011), 291–326, arXiv: 1008.0289 | DOI | MR | Zbl

[9] Eilbeck J. C., Enolski V. Z., Matsutani S., Ônishi Y., Previato E., “Abelian functions for trigonal curves of genus three”, Int. Math. Res. Not., 2008 (2008), rnm 140, 38 pp., arXiv: math.AG/0610019 | DOI

[10] Eilbeck J. C., Enolskii V. Z., Leykin D. V., “On the Kleinian construction of abelian functions of canonical algebraic curves”, SIDE III – Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proc. Lecture Notes, 25, Amer. Math. Soc., Providence, RI, 2000, 121–138 | DOI | Zbl

[11] Hodge W. V. D., Atiyah M. F., “Integrals of the second kind on an algebraic variety”, Ann. of Math., 62 (1955), 56–91 | DOI | MR | Zbl

[12] Nakayashiki A., “On algebraic expressions of sigma functions for $(n,s)$ curves”, Asian J. Math., 14 (2010), 175–211, arXiv: 0803.2083 | DOI | MR

[13] Suzuki J., “Klein's fundamental 2-form of second kind for the $C_{ab}$ curves”, SIGMA, 13 (2017), 017, 13 pp., arXiv: 1701.00931 | DOI | Zbl

[14] Whittaker E. T., Watson G. N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996 | DOI | Zbl