Multivalued hyperelliptic continued fractions of generalized Halphen type

Abstract

We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperelliptic Halphen (HH) element √X 2g+2 - √Y 2/x - y, depending on parameter y, where X 2g+2 is a polynomial of degree 2g + 2 and Y 2g+2 = X 2g+2(y). We study regular and irregular HH elements, their continued fraction developments, and some basic properties of such developments such as even and odd symmetry and periodicity. There is a 2 ↔ g + 1 dynamics, which lies in the basis of the developed continued fractions theory. We give two geometric realizations of this dynamics. The first one deals with nets of polynomials and with polygons circumscribed about a conic K. The dynamics is realized as a path of polygons of g + 1 sides inscribed in a curve B of degree 2g and circumscribed about the conic K obtained by successive moves, so called flips along edges. The second geometric realization leads to a new interpretation of generalized Jacobians of hyperelliptic curves.