Halphen's transform and middle convolution

2009/03/21 by Stefan Reiter, Reiter, Stefan
Mathematics · #20H10 #33C05 #33E10 #Algebraic Geometry (math.AG) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.AG #math.CA #msc:20H10 #msc:33C05 #msc:33E10

paper · pdf · doi:10.48550/arxiv.0903.3654

27 pages

arxiv created 2011/05/13 · arxiv updated 2011/05/16

Abstract

We show that the Halphen transform of a Lamé equation can be written as the symmetric square of the Lamé equation followed by an Euler transform. We use this to compute a list of Lamé equations with arithmetic Fuchsian monodromy group. It contains all those Lamé equations where the quaternion algebra A over k associated to the arithmetic Fuchsian group is a quaternion algebra A over ℚ. Further we classify all geometric braid group orbits in \rmSL₂(\ZZ)⁴ with the possible exception of three orbits.

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