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**Extra resources for A course of applied functional analysis**

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Assume now that p is odd and that E has nonsplit, multiplicative reduction. We then show that ker(rvn) = 0. We have an exact sequence I where q5 is the unramified character of GFy of order 2. As discussed in section 2, we have Im(rcv,) = Im(X,,). Also finis surjective. We can identify ker(rUn) with ker(d,, ), where dun is the map I In the above result, q(l ) denotes the Tate period for E over F,. ( Thus, N ~(q;)) / = (q;))IFvi~pl ~ ~ is transcendental according to the theorem of Barre-Sirieix, Diaz, Gramain, and Philibert.

We first assume that E has split, multiplicative reduction. Then C, S ppm and we have an exact sequence r of GF,-modules, where the action on QP/Zp is trivial. Then H1((Fn),, ,pp-) and hence Im(X,,) are divisible. We have I ~ ( K , , ) = Im(X,,) as well as Im(rc,) = Im(X,). Thus, ker(r,, ) = ker(b,, ), where bun is the map Iwasawa theory for elliptic curves + 77 +- 9-I -t- 744 1968849 - for ~ql,< 1 and jE is the j-invariant for E. Since j~ E F is algebraic, the theorem of [B-D-G-P] referred to in section 1 implies that q~ is transcendental.

11. Let E be an elliptic curve defined over $. Assume that SelE($,), is A-cotorsion. Then there exists a $-isogenous elliptic curve E' such that p ~ = t 0. In particular, if Eb] is irreducible as a (ZIP+)representation of GQ,then p~ = 0. I I I Here E b ] = k e r ( ~ ( $ ) 3 E($)). P. Schneider has given a simple formula for the effect of an isogeny on the p-invariant of SelE(F,), for arbitrary F and for odd p. ) Thus, the above conjecture effectively predicts the value of p~ for F = $. Suppose that SelE(F,), is A-cotorsion.