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Ranks of Selmer groups in an analytic family

Author: Joël Bellaïche
Journal: Trans. Amer. Math. Soc. 364 (2012), 4735-4761
MSC (2010): Primary 11F80; Secondary 11F33
Published electronically: April 18, 2012
MathSciNet review: 2922608
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Abstract: We study the variation of the dimension of the Bloch-Kato Selmer group of a $ p$-adic Galois representation of a number field that varies in a refined family. We show that, if we restrict ourselves to representations that are, at every place dividing $ p$, crystalline, non-critically refined, and with a fixed number of non-negative Hodge-Tate weights, then the dimension of the Selmer group varies essentially lower-semi-continuously. This allows us to prove lower bounds for Selmer groups ``by continuity'', and in particular to deduce from a result of Bellaïche and Chenevier that the $ p$-adic Selmer group of a modular eigenform of weight $ 2$ of sign $ -1$ has rank at least $ 1$.

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Additional Information

Joël Bellaïche
Affiliation: Department of Mathematics, MS 050, Brandeis University, 415 South Street, Waltham, Massachusetts 02453

Received by editor(s): January 18, 2010
Received by editor(s) in revised form: October 17, 2010
Published electronically: April 18, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.