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

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