Density of Selmer ranks in families of even Galois representations

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This thesis concerns the arithmetic statistics of even Galois representations. The first chapter shows how level-raising leads to the study of the splitting of primes in number fields that depend on the prime. In addition, the parity of Galois representations is placed in the context of reciprocity in the Langlands program. The second chapter concerns an even, reducible residual Galois representation in even characteristic. By thickening the image with cohomology classes, all lifts of the representation are ensured to be irreducible. The global reciprocity law of Galois cohomology is applied to lift the representation to mod 8, and smooth quotients of the local deformation rings at the primes where the representation is ramified are found. By using the generic smoothness of the local deformation rings at trivial primes and the Wiles-Greenberg formula, a balanced global setting is created, in the sense that the Selmer group and the dual Selmer group have the same rank. The Selmer group is computed explicitly and shown to have rank three. Finally, the distribution over primes of the ranks of Selmer groups in a family of even representations obtained by allowing ramification at auxiliary primes is studied. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192.

Denne afhandling vedrører aritmetisk statistik for lige Galoisrepræsentationer. Det første kapitel viser, hvordan niveauhævning af Galoisrepræsentationer leder til studiet af spaltningen af primtal i tallegemer, som afhænger af primtallet selv. Desuden sættes pariteten af Galoisrepræsentationer i kontekst med reciprocitet i Langlandsprogrammet. Det andet kapital omhandler en lige, reducibel residuel Galoisrepræsentation i lige karakteristik. Ved at virke på billedet med kohomologiklasser sikres det, at alle løft af repræsentationen er irreducible. Den globale reciprocitetslov fra Galoiskohomologi anvendes til at løfte repræsentationen til mod 8, og glatte kvotienter af de lokale deformationsringe findes ved de primtal, hvor repræsentationen er forgrenet. Den generiske glathed af de lokale deformationsringe ved trivielle primtal samt Wiles-Greenberg-formlen benyttes til at skabe en balanceret global ramme, hvor Selmergruppen og den duale Selmergruppe har samme rang. Selmergruppen udregnes eksplicit og vises at have rang tre. Slutteligt studeres fordelingen af Selmerrangen i en familie af lige Galoisrepræsentationer opnået ved at tillade forgrening ved auxiliære primtal. Uendeligheden af primtal for hvilke Selmerrangen øges med én bevises, og tætheden af sådanne primtal vises at være 1/192.

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92 pages


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Density theorems; Galois cohomology; Galois representations; Langlands reciprocity; Level-raising; Selmer groups


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Ramakrishna, Ravi

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Zywina, David
Stillman, Michael

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Ph. D., Mathematics

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Doctor of Philosophy

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Government Document




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Attribution-NonCommercial-NoDerivatives 4.0 International


dissertation or thesis

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