Control theorem and functional equation of Selmer groups over p -adic Lie extensions
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Control theorem and functional equation of Selmer groups over p-adic Lie extensions Somnath Jha1 · Tadashi Ochiai2 Accepted: 20 October 2020 / Published online: 19 November 2020 © Springer Nature Switzerland AG 2020
Abstract Let K ∞ be a p-adic Lie extension of a number field K which fits into the setting of non-commutative Iwasawa theory formulated by Coates–Fukaya–Kato–Sujatha– Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. As an application of this control theorem, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, PerrinRiou and Zábrádi. Especially, we generalize the result of Zábrádi for elliptic curves to general motives. Note that our proof is different from the proof of Zábrádi even in the case of elliptic curves. We also discuss the functional equation for the analytic p-adic L-functions and check the compatibility with the functional equation of the dual Selmer groups. Mathematics Subject Classification Primary 11R23; Secondary 19A31 · 16E20 · 11G40 · 20C07 · 11G05 · 11F67
The first author acknowledges the support of SERB ECR Grant and SERB MATRICS Grant. The second author is partially supported for this work by KAKENHI (Grant-in-Aid for Exploratory Research: Grant Number 24654004, Grant-in-Aid for Scientific Research (Red): Grant Number 26287005).
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Somnath Jha [email protected] Tadashi Ochiai [email protected]
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Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India
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Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka 5600043, Japan
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S. Jha, T. Ochiai
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2 Proof of the control theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3 Examples of the error term of the algebraic functional equation . . . . . . . . . . . . . . . . . .
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4 Higher extension groups of Selmer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5 Proof of the algebraic functional equation (Theorem 0.3) . . . . . . . . . . . . . . . . . . . . . .
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6 Compatibility of the algebraic and the conjectural analytic functional equation . . . . . . . . . .
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Appendix A: Conjectural existence of analytic p-adic L-functions . . . . . . . . . . . . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction Let us fix a prime number p throughout the paper. Let K be a number field and let K be a finite extension of Q p whose ring of integers is denoted by O. For any p-adic Lie group G, we define (G) := lim Z p [G/U ]
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