The algebraic functional equation of an elliptic curve at supersingular primes
The author proves an algebraic functional equation for the \(\pm\)-Selmer groups of an elliptic curve at supersingular primes. Let \(E\) be an elliptic curve over \(\mathbb Q\) and \(p > 3\) be a supersingular prime of \(E\). Let \(K\) be an abelian extension of \(\mathbb Q\) such that \([K:\mathbb Q] = g\) is prime to \(p\) and \(p\) is unramified over \( K / \mathbb Q\). Denote \(O = \mathbb Z_{p}[\mu _{g}]\). Let \(K_{\infty}\) be the cyclotomic \( \mathbb Z_{p}\)-extension of \(K\) and \(\text{Sel}^{\pm}_{p} (E/K_{\infty})\) be the \(\pm\)-Selmer groups. Corresponding to Pollack's analytic functional equation for the \(p\)-adic \(L\)-function [see \textit{R. Pollack}, Duke Math. J. 118, No. 3, 523--558 (2003; Zbl 1074.11061)], the author of this paper shows the algebraic counterpart of this result. More precisely, he proves firstly that \(\text{Sel}^{-}_{p} (E/K_{\infty})\) is \( \Lambda\)-cotorsion by using Kato's and Rohrlich's work, where \( \Lambda = \mathbb Z_{p} \Gamma\) is the Iwasawa algebra of \(\Gamma = \text{Gal}(K_{\infty}/K)\). Then following \textit{R. Greenberg}'s ideas [see Adv. Stud. Pure Math. 17, 97--137 (1989; Zbl 0739.11045)] and his own previous work [Compos. Math. 143, No. 1, 47--72 (2007; Zbl 1169.11022)], he shows \[ (\text{Sel}^{-}_{p} (E/K_{\infty}) \otimes O ) ^{\vee } \thicksim (\text{Sel}^{-}_{p} (E / K_{\infty})^{\iota}\otimes O ) ^{\vee } \] where \(\thicksim\) is a \(O[[\text{Gal}(K_{\infty}/\mathbb Q)]]\)-pseudo isomorphism and \(\iota\) is the standard involution. This implies that the characteristic ideal \((a) \subset \Lambda \) of the Pontryagin dual of \(\text{Sel}^{-}_{p} (E / K_{\infty})\) is nonzero and satisfies the algebraic functional equation \((a)= (a^{\iota})\). The author also obtain similar results for the plus-Selmer groups in some special cases.
scientific article; zbMATH DE number 5302016