Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T17:21:37.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve

Part of: Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  22 June 2018

Yoshihiro Ônishi
Yoshihiro Ônishi*
Affiliation:
Department of Mathematics, Faculty of Sciences and Engineering, Meijo University, 1-501, Shiogama-guchi, Tempaku-ku, Nagoya 468-0051, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n, s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve) y3 + (μ1x + μ4)y2 + (μ2x2 + μ5x + μ8)y = x4 + μ3x3 + μ6x2 + μ9x + μ12 (where μj are constants) are given.

Keywords

abelian functionshigher genus curvesgeneralized sigma functionsHurwitz integrality

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 14K20: Analytic theory; abelian integrals and differentials 14K25: Theta functions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 995 - 1022
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ayano, T., Sigma functions for telescopic curves, Osaka J. Math. 51 (2014), 459–481.Google Scholar
2.Baker, H. F., Abelian functions – Abel's theorem and the allied theory including the theory of the theta functions (Cambridge University Press, 1897).Google Scholar
3.Bannai, K. and Kobayashi, S., Divisibilities of Eisenstein-Kronecker numbers and p-adic theta functions at supersingular primes, unpublished manuscript.Google Scholar
4.Barsotti, I., Considerazioni sulle funzioni thêta, Istituo Nazionale di alta mathematica, Sympos. Math. III (1970), 247–277.Google Scholar
5.Breen, L., Fonctions thêta et théorème du cube, Lecture Notes in Mathematics, Volume 980 (Springer-Verlag, 1983).Google Scholar
6.Buchstaber, V. M. and Leykin, D. V., Heat equations in a nonholonomic frame, Funct. Anal. Appl. 38 (2004), 88–101.Google Scholar
7.Cristante, V., Theta functions and Barsotti-Tate groups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), 181–215.Google Scholar
8.Eilbeck, J. C., Weierstrass functions for higher genus curves, 2011, http://www.ma.hw.ac.uk/Weierstrass/.Google Scholar
9.Eilbeck, J. C., Enol'skii, V. Z., Matsutani, S., Ônishi, Y. and Previato, E., Abelian functions for trigonal curves of genus three, Int. Math. Res. Not. 2007 (2007), 102–139.Google Scholar
10.Fay, J., Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Volume 352 (Springer-Verlag, 1973).Google Scholar
11.Mazur, B. and Tate, J., The p-adic sigma function, Duke Math. J. 62 (1991), 663–688.Google Scholar
12.Mazur, B., Stein, W. and Tate, J., Computation of p-adic height and log convergence, Documenta Math., Extra Volume: John H. Coates’ Sixtieth Birthday (2006), 577–614.Google Scholar
13.Miura, S., Linear code on affine algebraic curves, Trans. IEICE (Japanese) J81-A (1998), 1398–1421.Google Scholar
14.Mumford, D., Tata lectures on theta I, Progress in Mathematics, Volume 28 (Birkhäuser, 1982).Google Scholar
15.Nakayashiki, A., Sigma function as a tau function, Int. Math. Res. Not. 2010 (2010), 373–394.Google Scholar
16.Nakayashiki, A., On algebraic expressions of sigma functions for (n, s) curves, Asian J. Math. 14(2) (2010), 175–212.Google Scholar
17.Ônishi, Y., Universal elliptic functions, 2010, http://arxiv.org/abs/1003.2927.Google Scholar
18.Ônishi, Y., Theory of Abelian functions (in Japanese, Abel Kan-su-ron) (Department of Mathematics, Chuo University, 2013), http://ir.c.chuo-u.ac.jp/repository/search/item/md/-/p/4118/.Google Scholar