By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola
This quantity includes the elevated models of the lectures given by means of the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers amassed listed below are wide surveys of the present learn within the mathematics of elliptic curves, and in addition include a number of new effects which can't be chanced on in other places within the literature. because of readability and magnificence of exposition, and to the heritage fabric explicitly incorporated within the textual content or quoted within the references, the quantity is definitely fitted to learn scholars in addition to to senior mathematicians.
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For F = $, this is a theorem. In this case, Manin showed that it would suffice to prove that logp(qE) # 0, where q~ denotes the Tate period for E, assuming that E has multiplicative reduction at p. But a recent theorem of Barre-Sirieix, Diaz, Gramain, and Philibert [B-D-G-P] shows that q~ is transcendental when the j-invariant jE is algebraic. Since jE E $, it follows that U--E -~ - O ' ~ ( ~ E ) is not a root of unity and so logp(qE) # 0. For arbitrary F , one would need to prove that l ~ g ~ ( N ~ ~# ~0 for ~ ~all( primes ~ ~ )v ) ) of F lying over p where E has multiplicative reduction.
Let run Gal(K/(F,),,,), which is isomorphic to H,, topologically generated by y,,,, say. Then + ker(g,) + coker(s,) + coker(h,). Therefore, we must study ker(h,), coker(h,), and ker(g,), which we do in a sequence of lemmas. 1. The kernel of h, is finite and has bounded order as n varies. 1) and hence 74 Ralph Greenberg Iwasawa theory for elliptic curves where we regard Elpw] as a subProof. Let C, = ker(E[pw] -, &PI), group of E(Q. 2. 4, we have Im(n,) = Im(X,), since the inertia subgroup of Gal(F,/F) for v has finite index.
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.