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Extra info for The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues

Example text

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.

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The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues by Gazzini M., Serra E.


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