Jacobi inversion on strata of the Jacobian of the \(C_{rs}\) curve \(y^r = f(x)\). (English) Zbl 1160.14018

Matsutani, Shigeki; Previato, Emma

J. Math. Soc. Japan 60, No. 4, 1009-1044 (2008).

Let \(X\) be a smooth projective complex curve of genus \(g\). For fixed base point \(p_{0} \in X\) and basis \(\omega_{1}, \dots, \omega_{g}\) for \(H^{0}(X, K)\), the abelian integrals \((\int_{p_{0}}^{p} \omega_{1}, \dots, \int_{p_{0}}^{p} \omega_{g})\) define \(w:X \to \mathbb C^{g}\). If \(\Lambda \subset \mathbb C^{g}\) is the period lattice and \(\kappa: \mathbb C^{g} \to \mathbb C^{g}/\Lambda = J(X)\) is the projection to the Jacobian of \(X\), the map \(u=\kappa \circ w\) extends linearly to \(u:{\roman {Div}}(X) \to J(X)\). Jacobi’s inversion theorem says that the restriction to \({\roman {Div}}^{g}(X)\) induces an isomorphism \(\psi: {\roman {Pic}}^{g}(X) \to J(X)\) and the Jacobi inversion problem [C. G. J. Jacobi, J. Reine Angew. Math. 30, 183–184 (1846; ERAM 030.0861cj)] asks an explicit inverse to \(\psi\) for effective divisors: given \(u=u(P_{i}) \in J(X)\), can one give formulas for coordinates of the \(P_{i}\) in terms of \(u\)? Meanwhile G. Frobenius and L. Stickelberger gave nice formulas which express the group addition on elliptic curves in terms of quotients of sigma functions on the curve and the determinant of a matrix involving derivatives of the Weierstrass \(\wp\)-function [“Zur Theorie der elliptischen Functionen,” Borchardt J. LXXXIII. 175–179 (1877; JFM 09.0347.02)]. In the case \(g=2\), H. F. Baker used Klein’s analog of Weierstrass’ \(\sigma\) function to generalize the Frobenius-Stickelberger formula and also solved the Jacobi inversion problem [Am. J. Math. 20, 301–384 (1898; JFM 29.0394.03); An introduction to the theory of multiply-periodic functions. (Cambridge): University Press. (1907; JFM 38.0478.05)].
In the paper under review, the authors generalize these results to \(C_{r,s}\) curves: these are smooth curves \(X \subset \mathbb C^{2}\) with equation \[ y^{r}=x^{s}+\lambda_{s-1} x^{s-1}+ \dots + \lambda_{1} x + \lambda_{0} \] satisfying \(r<s\) and \((r,s)=1\) and can be smoothly completed by adding just one point at \(\infty\). The authors define monomials \(\phi_{k}(x,y)\) in terms of vanishing order at \(\infty\), which act as analogs of the derivatives of the Weierstrass \(\wp\)-function. Given \((P_{1}, \dots P_{k}) \in S^{k}(X - \{\infty\})\) avoiding the singular locus, they introduce functions \(\mu_{k}(x,y)=\sum \mu_{k,i} \phi_{k}(x,y)\) which vanish at the \(P_{i}\) and have a single pole of minimal order at \(\infty\). Using \(\infty\) for the base point of the map \(w\) and setting \(v = \pm w(P_{1}, \dots P_{k})\), their main theorem (Theorem 5.1) expresses the coefficients \(\mu_{k,i}\) in terms of rational expressions of \(\sigma\) functions in \(v\), which generalize formulas of V. Z. Enolski and J. Gibbons for the case \(r=2\) [Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, preprint, (2002)]. These expressions sometimes evaluate to \(0/0\), when they must be defined in terms of limits, which leads to results on the order of vanishing of the \(\sigma\) functions (Corollary 5.7). When \(g=2\), their theorem recovers formulas of D. Grant [J. Reine Angew. Math. 392, 125–136 (1988; Zbl 0646.14033)] and J. Jorgenson [Isr. J. Math. 77, 273–284 (1992; Zbl 0790.30038)]. When \(k=g\), they achieve Jacobian inversion on the strata \({\mathcal W}^{g} \subset J(X)\) (the image of the natural map \(u: S^{g} (X-\{\infty\}) \to J(X)\)) in the sense that \(\mu_{k}\) vanishes at the \(P_{i}\), explaining the title of the paper.

Reviewer: Scott Nollet (Fort Worth)

MSC:

14H40	Jacobians, Prym varieties
14H55	Riemann surfaces; Weierstrass points; gap sequences
14K20	Analytic theory of abelian varieties; abelian integrals and differentials

Keywords:

sigma functions; inversion of abelian integrals; \(C_{r; s

Citations:

Zbl 0646.14033; Zbl 0790.30038; JFM 09.0347.02; JFM 29.0394.03; JFM 38.0478.05; ERAM 030.0861cj

References:

[1]	E. Arbarello, M. Cornalba, P. A. Griffiths and J. Harris, Geometry of Algebraic Curves, I , Springer-Verlag, New York, 1985. · Zbl 0559.14017
[2]	S. Arita, An addition algorithm in Jacobian of \(C_{ab}\) curves, Discrete Appl. Math., 130 (2003), 13-31. · Zbl 1037.14010
[3]	H. F. Baker, Abelian functions, Abel’s theorem and the allied theory of theta functions, Cambridge University Press, Cambridge, 1995, reprint of the 1897 original. · Zbl 0848.14012
[4]	H. F. Baker, On the hyperelliptic sigma functions, Amer. J. Math., 20 (1898), 301-384. · JFM 29.0394.03
[5]	H. F. Baker, An introduction to the theory of multiply-periodic functions, Cambridge University Press, Cambridge, 1907. · JFM 38.0478.05
[6]	S. Baldwin and J. Gibbons, Higher genus hyperelliptic reduction of the Benney equations, J. Phys. A, 37 (2004), 5341-5354. · Zbl 1168.37321
[7]	S. Baldwin and J. Gibbons, Genus 4 trigonal reduction of the Benney equations, J. Phys. A, 39 (2006), 3607-3639. · Zbl 1091.35065
[8]	Ch. Birkenhake and P. Vanhaecke, The vanishing of the theta function in the KP direction: a geometric approach, Compositio Math., 135 (2003), 323-330. · Zbl 1042.14017
[9]	V. M. Buchstaber, V. Z. Enolskiĭ and D. V. Leĭkin, Hyperelliptic Kleinian functions and applications, in Solitons, geometry and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, 179 , Amer. Math. Soc., Providence, RI, 1997, pp.,1-33. · Zbl 0911.14020
[10]	V. M. Bukhshtaber, D. V. Leĭkin and V. Z. Ènol’skiĭ, Rational analogues of Abelian functions, Funct. Anal. Appl., 33 (1999), 83-94. · Zbl 1056.14049
[11]	V. M. Bukhshtaber, D. V. Leĭkin and V. Z. Ènol’skiĭ, \(\sigma\)-functions of \((n, s)\)-curves, Russian Math. Surveys, 54 (1999), 628-629. · Zbl 1081.14519
[12]	V. M. Bukhstaber, D. V. Leĭkin and V. Z. Ènol’skiĭ, Uniformization of Jacobi manifolds of trigonal curves and nonlinear differential equations, Funct. Anal. Appl., 34 (2000), 159-171. · Zbl 0978.58012
[13]	J. L. Burchnall and T. W. Chaundy, Commutative ordinary differential operators, Proc. Roy. Soc. London Ser. A, 118 (1928), 557-583. · JFM 54.0439.01
[14]	J. C. Eilbeck, V. Z. Enolskii and D. V. Leykin, 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, pp.,121-138. · Zbl 1003.14008
[15]	V. Z. Enolski, \(\theta\)-derivatives, draft (2002).
[16]	V. Z. Enolski and J. Gibbons, Addition theorems on the strata of the theta divisor of genus three hyperelliptic curves, draft (2002).
[17]	B. Enriquez and V. Rubtsov, Commuting families in skew fields and quantization of Beauville’s fibration, Duke Math. J., 119 (2003), 197-219. · Zbl 1052.53065
[18]	J. D. Fay, Theta functions on Riemann Surfaces, Lecture Notes in Math., 352 , Springer-Verlag, Berlin-New York, 1973. · Zbl 0281.30013
[19]	J. Fay, On the even-order vanishing of Jacobian theta functions, Duke Math. J., 51 (1984), 109-132. · Zbl 0583.14017
[20]	G. Frobenius and L. Stickelberger, Zur Theorie der elliptischen Functionen, J. Reine Angew. Math., 83 (1877), 175-179. · JFM 09.0347.02
[21]	R. Garnier, Sur une classe de systems différentiels Abéliens déduits de la théorie des équations linéaires, Rend. Circ. Mat. Palermo, 43 (1918), 155-191. · JFM 47.0404.01
[22]	K. Gawȩdzki and P. Tran-Ngoc-Bich, Self-duality of the \(SL_2\) Hitchin integrable system at genus 2, Comm. Math. Phys., 196 (1998), 641-670. · Zbl 0923.14014
[23]	B. van Geemen and E. Previato, On the Hitchin system, Duke Math J., 85 (1996), 659-683. · Zbl 0879.14010
[24]	D. Grant, A generalization of Jacobi’s derivative formula to dimension two, J. Reine Angew. Math., 392 (1988), 125-136. · Zbl 0646.14033
[25]	C. G. J. Jacobi, Note sur les fonctions Abéliennes, J. Reine Angew. Math., 30 (1846), 183-184. · ERAM 030.0861cj
[26]	J. Jorgenson, On directional derivatives of the theta function along its divisor, Israel J. Math., 77 (1992), 273-284. · Zbl 0790.30038
[27]	G. Kempf, Toward the inversion of abelian integrals, I, Ann. of Math. (2), 110 (1979), 243-273. · Zbl 0452.14011
[28]	G. Kempf, Toward the inversion of abelian integrals, II, Amer. J. Math., 101 (1979), 184-202. · Zbl 0452.14012
[29]	Y. Kodama and B. G. Konopelchenko, Singular sector of the Burgers-Hopf hierarchy and deformations of hyperelliptic curves, J. Phys. A, 35 (2002), L489-L500. · Zbl 1039.37062
[30]	S. Lang, Introduction to algebraic and abelian functions, Second edition, Graduate Texts in Mathematics, 89 , Springer-Verlag, New York-Berlin, 1982. · Zbl 0513.14024
[31]	S. Matsutani and E. Previato, A generalized Kiepert formula for \(C_{ab}\) curves, toappear in Israel J. Math., (preprint Institut Mittag-Leffler (ISSN 1103-467X ISRN IM-L-R-03-05/06-SE+fall)). · Zbl 1186.14025
[32]	S. Miura, Algebraic geometric codes on certain plane curves, Trans. IEICE, J75 -A (1992), 1735-1745.
[33]	D. Mumford, Tata Lectures on Theta II Jacobian theta functions and differential equations, Progress in Mathematics, 43 , Birkhäuser Boston, Inc., Boston, MA, 1984. · Zbl 0549.14014
[34]	Y. Ônishi, Determinant expressions for hyperelliptic functions, Proc. Edinburgh Math. Soc., 48 (2005), 1-38. · Zbl 1148.14303
[35]	Y. Ônishi, Determinant expressions in Abelian functions for purely trigonal curves of degree four, to appear in International J. Math., (2008): math.NT/0503696.
[36]	E. Previato, Generalized Weierstrass \({\wp}\)-functions and KP flows in affine space, Comment. Math. Helv., 62 (1987), 292-310. · Zbl 0638.14024
[37]	G. F. B. Riemann, Ueber das Verschwinden der \(\vartheta\)-functionen, J. Reine Angew. Math., 65 (1866), 161-172. · ERAM 065.1693cj
[38]	G. Segal and G. Wilson, Loop groups and equations of \(K\)d\(V\) type, Inst. Hautes Études Sci. Publ. Math., 61 (1985), 5-65. · Zbl 0592.35112
[39]	K. Weierstrass, Zur theorie der abelschen functionen, J. Reine Angew. Math., 47 (1854), 289-306. · ERAM 047.1271cj
[40]	E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, Cambridge University Press, 1927. · JFM 53.0180.04