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Inversion of Abelian integrals on small genus curves. (English) Zbl 0579.14023
Let M be a smooth projective curve of genus \(g>0\). Let J be the Jacobian of M and let \(M^{(i)}\) be the i-th symmetric product. Fix a base point \(p\in M\) and let \(\phi_ i: M^{(i)}\to J\) be the Abel-Jacobi map. For \(i\geq 2g-1\), Mattuck has shown that \(\phi_ i: M^{(i)}\to J\) is a \({\mathbb{P}}^{i-g}\)-bundle. The most interesting case occurs when \(i=2g-1\), since the bundles for larger i are determined by this one [see the book of R. C. Gunning on this subject entitled ”Riemann surfaces and generalized theta functions” (1976; Zbl 0341.14013)]. The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle \(\phi_ i ?\) In this paper we present a solution to this question when M has genus 2 and when M is non-hyperelliptic of genus 3. For further background of this problem, see the cited book as well as the articles of Gunning, Kempf and Mattuck.

14H05 Algebraic functions and function fields in algebraic geometry
14H40 Jacobians, Prym varieties
Full Text: DOI EuDML
[1] Gunning, R.C.: Riemann surfaces and generalized theta functions. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0341.14013
[2] Gunning, R.C.: On generalized theta functions. Am. J. Math.104, 183-208 (1982) · Zbl 0503.14021
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