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Representations of finite groups on RiemannRoch spaces,” preprint
, 2003
"... 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 RiemannRoch space L(D). 1. ..."
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Cited by 10 (8 self)
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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 RiemannRoch space L(D). 1.
On the quantum invariants for the spherical Seifert manifolds, preprint arXiv:mathph/0504082
"... ABSTRACT. We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifold S 3 /Γ where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with halfintegral weight, and we give an exact asymptotic expa ..."
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Cited by 2 (1 self)
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ABSTRACT. We study the Witten–Reshetikhin–Turaev SU(2) invariant for the Seifert manifold S 3 /Γ where Γ is a finite subgroup of SU(2). We show that the WRT invariants can be written in terms of the Eichler integral of the modular forms with halfintegral weight, and we give an exact asymptotic expansion of the invariants by use of the nearly modular property of the Eichler integral. We further discuss that those modular forms have a direct connection with the polyhedral group by showing that the invariant polynomials of modular forms satisfy the polyhedral equations associated to Γ. 1.
ALGEBRAIC CURVES UNIFORMIZED BY CONGRUENCE SUBGROUPS OF TRIANGLE GROUPS
"... 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 ∼ ..."
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Cited by 2 (1 self)
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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 nonarithmeticity 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 higherdimensional quaternionic Shimura variety, and this embedding allows the theory of these curves to retain at least an automorphic
Group action on genus 3 curves and their Weierstrass points
"... 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. ..."
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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.