| Summary: | We consider N-parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Physics Letters A 211 19, 1999 Physics Letters A 258 263 ) to correspond to N−parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function λ is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor, Θ1. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157). Key to this is the identification of PDE satisfied on Θ1. We then consider the general case for N > 3. Again, the mapping function λ is calculated explicitly by inverting a second kind Abelian integral on the stratum Θ1 of the genus g Jacobi variety. This is done using a method based on the result of Jorgenson (1992 Isr. J Math. 77 273). The approach used for the hyperelliptic reductions is then applied to a genus 4 trigonal reduction and the function λ is calculated explicitly. A secondary result of this calculation is an expansion of the leading terms of the σ function for this family of curves.
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