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Lang-Trotter Questions on the Reductions of Abelian Varieties

dc.contributor.advisorWashington, Lawrence Cen_US
dc.contributor.authorBloom, Samuelen_US
dc.date.accessioned2018-07-17T06:01:12Z
dc.date.available2018-07-17T06:01:12Z
dc.date.issued2018en_US
dc.identifierhttps://doi.org/10.13016/M20R9M71Q
dc.identifier.urihttp://hdl.handle.net/1903/20904
dc.description.abstractLet A be a geometrically simple g-dimensional abelian variety over the rationals. This thesis investigates the behavior of the reductions A_p of A modulo its primes p of good reduction. Questions about these reductions are called "questions of Lang-Trotter type" after the 1976 memoir of S. Lang and H. Trotter. This thesis studies two aspects of the reductions A_p in particular: the "Frobenius fields," End(A_p) tensor Q, when A_p is simple and ordinary, and the primality (or failure thereof) of the number of rational points, #A_p(F_p). Our questions and conjectures generalize the study of the "fixed-field" Conjecture of Lang-Trotter and the Koblitz Conjecture on elliptic curves, and our work generalizes the work by previous authors to this higher-dimensional context: through sieve-theoretic arguments and the use of explicit error bounds for the Chebotarev Density Theorem, we produce various conditional and unconditional upper and lower bounds on the number of primes p at which A_p has a specified behavior.en_US
dc.language.isoenen_US
dc.titleLang-Trotter Questions on the Reductions of Abelian Varietiesen_US
dc.typeDissertationen_US
dc.contributor.publisherDigital Repository at the University of Marylanden_US
dc.contributor.publisherUniversity of Maryland (College Park, Md.)en_US
dc.contributor.departmentMathematicsen_US
dc.subject.pqcontrolledMathematicsen_US
dc.subject.pquncontrolledabelian varietyen_US
dc.subject.pquncontrolledelliptic curveen_US
dc.subject.pquncontrolledfinite fieldsen_US
dc.subject.pquncontrolledLang-Trotter Conjectureen_US
dc.subject.pquncontrolledsieve theoryen_US


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