Abstract. If G is a finite subgroup of the automorphism group of a projective curve X and D is a divisor on X stabilized by G, then under the assumption that D is nonspecial, we compute a simplified formula for the trace of the natural representation of G on Riemann-Roch space L(D). 1.

We study the locus of genus 3 curves with prescribed automorphism group G, and inclusion relations between such loci. We supply a normal form for the curves in each locus, and generators for G. Furthermore, we determine the action of G on the Weierstrass points of the curve. 1.

Abstract. Here we give the first serious consideration of a family of algebraic curves which can be characterized (essentially) either as the compact Riemann surfaces uniformized by congruence subgroups of the Fuchsian triangle groups ∆(a, b, c) or as the algebraic curves X/H, where X admits a G ∼ = P SL2(Fq)-Galois Belyi map and H is a subgroup of G. This family contains in particular the classical modular curves and XX commensurability classes of quaternionic Shimura curves. Our perspective is that all of these curves ought to be viewed, in some ways, as “generalized Shimura curves, ” notwithstanding the fact that the socalled non-arithmeticity of most of their uniformizing Fuchsian groups means that the Hecke algebra of modular correspondences will be too small to allow us to employ some of the usual automorphic techniques. Nevertheless the theory of Cohen and Wolfart provides a modular embedding of each of our curves as a cycle in a higher-dimensional quaternionic Shimura variety, and this embedding allows the theory of these curves to retain at least an automorphic

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curves of low genus over small finite fields