Symplectic representation of a braid group on 3-sheeted covers of the Riemann sphere

Preface Why we called the class of two-dimensional Shimura varieties, which are not Hilbert modular, "Picard modular surfaces" ? In the mean time the name has been generally accepted, see e.g. Langlands (and others) L-R]. On the one hand Picard worked on special Fuchsian systems of diierential equations; on the other hand Shimura Shi] introduced and investigated moduli spaces of abelian varieties with prescribed division algebra of endomorphisms, which are called (complex) "Shimura varieties" after some work of Deligne. One needs a chain of conclusions in a special case in order to connect both works. Picard found ad hoc on certain Riemann surfaces ordered sets of cycles, which we will call "Picard cycles" below. Quotients of integrals along these cycles solve (completely) a special Fuchsian system of diierential equations. The basic solution consists of two multivalued complex functions of two variables. The multivalence can be described by the monodromy group of the system. By Picard-Lefschetz theory, actually described in Arnold (and others) AVH], the monodromy group acts on the homology of an algebraic curve family respecting Picard cycles. In H 95] (Lemma 2.27) we announced that the action on Picard cycles is transitive and, moreover, coincides with the action of an arithmetic unitary group U((2; 1); O); O the ring of integers of an imaginary quadratic number eld K. This is a key result. Namely, the unitary group is the modular group of the Shimura surface of (principally polarized) abelian threefolds with K-multiplication of type (2; 1). It parametrizes via Jacobians the isomorphy classes of the Riemann surfaces Picard started with. The aim of this article is to give a complete proof of the mentioned key result. It joins some actual and old mathematics. As a consequence one gets a solution of the relative Schottky problem for smooth Galois coverings of P 1 (C) (Riemann sphere) of degree 3 and genus 3. 1 Basic facts, notations and deenitions We consider smooth compact complex curves (Riemann surfaces) C of genus 3 which are three-sheeted Galois coverings of the projective line (Riemann sphere) P 1. Let g be a generator of G = Gal(C=P 1) = Z =3Z. Then G acts on the homology group H 1 (C; Z) and on the vector space H 0 (C; ! C) of regular diierential forms. We have rank Z H 1 (C; Z) = 6; dim C H 0 (C; ! C) = 3: …