THE HYPERGEOMETRIC DIFFERENTIAL EQUATION_3E_2 WITH CUBIC CURVES AS SCHWARZ IMAGES

摘要

Let _3E_2 (a_1, a_2, a_3; b_1, b_2) denote the generalized hypergeometric differential equation of rank three with parameters (a_l , a_2, a_3, b_1, b_2), which is defined on the projective line P~1 (x) with regular singularities at x = 0, 1, ∞. Any set of linearly independent solutions defines a multi-valued map from P~1 — {0, 1,∞} to the projective plane P~2 called the Schwarz map. The image of this map is locally a curve, determined uniquely up to projective transformations. The condition for the image curve to be a cubic curve is written in terms of the parameters, and we list those curves and study the action of the monodromy groups on them.