On the algebraic functional equation for the mixed signed Selmer group over multiple $\mathbb {Z}_{p}$-extensions

Higher codimension Iwasawa theory for elliptic curves with supersingular reduction

“…In this case, Theorem 1.3 can be regarded as an algebraic functional equation for the multiplysigned Selmer groups. The result is a refinement of previous work (e.g., Ahmed and Lim [1,Theorem 3.3]). Actually, the previous work mainly focused on the pseudo-isomorphism classes, and the exact sequence in Theorem 1.3 provides us more information.…”

“…A bit more precisely, for p ∈ S ss p , let g be the residue degree of Now we begin the definition of C ǫ S . For each p ∈ S p (F ), as explained in the proof of Proposition 4.8(i), we have RΓ Iw (K ∞,p , T p E) ∈ D [1,1] (R), so…”

“…Therefore, by taking the projective limit with respect to positive integers f | g, we obtain the exact sequences claimed in Proposition 4.4(1) for a supernatural number g. Note that we ignored the multiplication by α f /f ′ , which is possible since it is at any rate a unit and does not affect the module structure of the limit. We briefly check that claim (2) follows from (1). Since the R g -module R nt g,0 is torsion, the statement on the generic rank is clear.…”

“…[p] = 0, together with the self-duality of T p E and the local Tate duality, implies RΓ Iw (Q p g ,∞ , T p E) ∈ D [1,1] (R g ). Then the claim follows by combining with the Euler-Poincare characteristic formula.…”

“…p {+, −, rel} and Assumption 2.6 holds for at least one multi-sign ǫ ′ with ǫ ′ ≤ ǫ, then we also have [1,2] (R). Then by Proposition 6.1 and Lemma 6.2, we obtain C ǫ S ∈ D [1,2] (R). By (6.4), we immediately obtain H 2 (C ǫ S ) ≃ X ǫ S .…”