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mH(G)-Property and Congruence of Galois Representations


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1 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China
     

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In this paper, we study the Selmer groups of two congruent Galois representations over an admissible p-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the mH(G)-property, so will the other. In the event that the mH(G)-property holds, and assuming certain further hypothesis on the decomposition of primes in the p-adic Lie extension, we compare the ranks of the π-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also study the variation of the ranks of the π-free quotient of the dual Selmer groups of specialization of a big Galois representation. We emphasis that our results do not assume the vanishing of the μ-invariant.
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  • mH(G)-Property and Congruence of Galois Representations

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Authors

Meng Fai Lim
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

Abstract


In this paper, we study the Selmer groups of two congruent Galois representations over an admissible p-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the mH(G)-property, so will the other. In the event that the mH(G)-property holds, and assuming certain further hypothesis on the decomposition of primes in the p-adic Lie extension, we compare the ranks of the π-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also study the variation of the ranks of the π-free quotient of the dual Selmer groups of specialization of a big Galois representation. We emphasis that our results do not assume the vanishing of the μ-invariant.

References