Algebraic functional equations and completely faithful Selmer groups

Tibor Backhausz, Backhausz and Zábrádi, Gergely (2015) Algebraic functional equations and completely faithful Selmer groups. INTERNATIONAL JOURNAL OF NUMBER THEORY, 11 (4). pp. 1233-1257. ISSN 1793-0421

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Abstract

Let E be an elliptic curve—defined over a number field K—without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses this gives an algebraic func- tional equation of the conjectured p-adic L-function. As an application we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power divi- sion points of another non-CM elliptic curve A.

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