On the Algebraic Functional Equation of the Eigenspaces of Mixed Signed Selmer Groups of Elliptic Curves with Good Reduc
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
On the Algebraic Functional Equation of the Eigenspaces of Mixed Signed Selmer Groups of Elliptic Curves with Good Reduction at Primes above p Suman AHMED School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, P. R. China E-mail : [email protected]
Meng Fai LIM1) School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, P. R. China E-mail : [email protected] Abstract Let p be an odd prime number, and let E be an elliptic curve defined over a number field which has good reduction at every prime above p. Under suitable assumptions, we prove that the η-eigenspace and the η¯-eigenspace of mixed signed Selmer group of the elliptic curve are pseudoisomorphic. As a corollary, we show that the η-eigenspace is trivial if and only if the η¯-eigenspace is trivial. Our results can be thought as a reflection principle which relates an Iwasawa module in a given eigenspace with another Iwasawa module in a “reflected” eigenspace. Keywords
Algebraic functional equation, mixed signed Selmer groups
MR(2010) Subject Classification
1
11G10, 11R23
Introduction
The main conjecture of Iwasawa theory predicts a relation between a Selmer group and a conjectural p-adic L-function (see [2, 4, 8, 14, 18]). This p-adic L-function is expected to satisfy a conjectural functional equation in a certain sense. In view of the main conjecture and this conjectural functional equation, one would expect to have certain algebraic relationship between the corresponding Selmer groups. In this paper, we will examine this phenomenon for the eigenspaces of the mixed signed Selmer groups of elliptic curves with good reduction at primes above p. We shall describe our main result briefly in this introductory section. Throughout the paper, p will denote a fixed odd prime. Let F be a number field and E an elliptic curve defined over F . Let F be a finite extension of F . The following assumptions will be in full force throughout the paper. (S1) The elliptic curve E has good reduction at all primes of F above p, at least one of which is a supersingular reduction prime of E. Received December 23, 2019, accepted June 23, 2020 The second author is supported by National Natural Science Foundation of China (Grant Nos. 11550110172 and 11771164) 1) Corresponding author
Ahmed S. and Lim M. F.
2
(S2) For each prime u of F above p at which the elliptic curve E has supersingular reduction, we assume that the following statements are valid. (a) Fu = Qp . ˜u (Fp )| = 0, where E ˜u is the reduction of E at u. (b) au = 1 + p − |E (c) u is unramified in F/F . → − Denote by Fn = F (μpn+1 ) and F∞ = n Fn . We define a signed Selmer group Sel s (E/F∞ ) → − (see the body of the paper for the precise definition) and write X s (E/F∞ ) for its Pontryagin dual. We shall write Σss p for the primes of F above p at which E has good supersingular reduction. For
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