By Gazzini M., Serra E.
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Extra info for The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues
Then one can prove the following result. 7. With the above notation, we have I corankn(Sel~(F,),) I I F,/F is the cyclotomic Zp-extension, but make no assumptions on the reduction type for E at primes lying over p. The conjecture below follows from results of Kato and Rohrlich when F is abelian over $ and E is defined over $ and modular. 8. The Zp-corank of SelE(Fn), is bounded as n varies. If this is so, then the map SelE(Fn), + s e l E ( ~ , ) ~ *must have infinite cokernel when n is sufficiently large, provided that we assume that E has potentially supersingular reduction at v for at least one prime v of F lying over p.
Where $ : GF,, -+ Z; is a continuous homoAgain we let C = ($,/Z,)($), morphism, v is any prime of F lying over p. If 77 is a prime of F, lying over v, then (F,), is the cyclotomic Z,-extension of F,. 3, the Z,corank of H1((Fn),, C) differs from [(F,),, : Fv]by at most 1. Thus, if we let rv= Gal((F,)q/F,), then it follows that as n -+ oo corankzp(HI ((F,)~, ~ ) )= ~ pn[Fv f : Q,] + O(1). Iwasawa theory for elliptic curves Ralph Greenberg 68 The structure theory of A-modules then implies that H1((F,),, C) has corank equal to [F, : $,I as a Z,[[r,]]-module.
On the other hand, GK has pcohomological dimension 1 because of the hypothesis that Gal(K/F,) contains an infinite pro-p subgroup. ) Thus if C is a divisible, pprimary GK-module, then the exact sequence 0 + C[p] + C 4 C + 0 induces the cohomology exact se, + H ~ ( K C[p]). , The last group is zero and quence H1(K, C ) 4 H ~ ( K C) hence H1(K, C ) is divisible. Applying this to C = C,, we see that Im(XK) is divisible and so Im(nK) = 1 m ( X ~ ) . If F, denotes the cyclotomic Zp-extension of F , then every prime v of F lying over p is ramified in F,/F.
The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues by Gazzini M., Serra E.