Multiplicity of Galois representations in the higher weight sheaf cohomology associated to Shimura curves.
Item
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Title
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Multiplicity of Galois representations in the higher weight sheaf cohomology associated to Shimura curves.
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Identifier
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AAI9630527
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identifier
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9630527
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Creator
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Yang, Lei.
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Contributor
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Adviser: Bruce Jordan
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Date
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1996
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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Let T{dollar}\sb{lcub}N{rcub}{dollar} be the Hecke algebra attached to {dollar}S\sb{lcub}k{rcub}(\Gamma\sb0(N),\epsilon){dollar} where {dollar}\epsilon{dollar} is a Dirichlet character. For a maximal ideal m of {dollar}T\sb{lcub}N{rcub}{dollar} with residue characteristic {dollar}\ell,{dollar} assume that the modular Galois representation {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} of Gal(Q/Q) attached to m is irreducible. We generalize Ribet's theorem regarding the multiplicities of {dollar}\rho\sb{lcub}bf m{rcub}{dollar} in jacobians of Shimura curves to higher even weights. The two main results of this paper are as follows: (1) Let p,q be two distinct primes not equal to {dollar}\ell,{dollar} and M a positive integer prime to {dollar}pq\ell.{dollar} We show that for a Shimura curve {dollar}V\sb{lcub}B{rcub}(M){dollar} arising from an Eichler order of conductor M in the indefinite quaternion algebra B of discriminant pq over Q, if {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} is ramified at at least one of the primes dividing the discriminant, say p, the multiplicity of {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} in {dollar}H\sp1(V\sb{lcub}B{rcub}(M)\times\bar{lcub}\bf Q{rcub},\ \bar\vartheta\sb{lcub}\ell{rcub}\lbrack{lcub}\bf m{rcub}\rbrack{dollar} is 1, unless {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} is unramified at q and Frob{dollar}\sb{lcub}q{rcub}{dollar} acts as a scalar in {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} whose square is {dollar}\epsilon(q).{dollar} In this exceptional case, the multiplicity is 2. We show that this exceptional case does occur. (2) We give an upper bound for the multiplicity in the case when the discriminant of B is arbitrary and {dollar}\rho\sb{lcub}{lcub}\bf m{rcub}{rcub}{dollar} is ramified at at least half of the primes dividing the discriminant. We prove that higher multiplicities also exist in this case.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
