Inversion of Abelian integrals on small genus curves
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 \textit{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.
scientific article; zbMATH DE number 3927085