By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

This quantity comprises the improved types of the lectures given through the authors on the C. I. M. E. tutorial convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accrued listed here are extensive surveys of the present learn within the mathematics of elliptic curves, and in addition comprise a number of new effects which can't be discovered somewhere else within the literature. due to readability and style of exposition, and to the historical past fabric explicitly integrated within the textual content or quoted within the references, the amount is definitely suited for learn scholars in addition to to senior mathematicians.

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Conjecturally, this power series should be in A. This is proven in [St] if E[p] is irreducible as a GQ-module. In general, it is only known to be A @zp$,. That is, p t f y ' ( ~ )E A for some t 2 0. Kato then proves that the characteristic ideal at least contains pm f y l (T) for some m 0. Rohrlich proves that L(E/$, $,I) # 0 for all but finitely many characters $ of r, which is equivalent to the statement f y ' ( ~ )# 0 as an element of A @zp Qp. 10 in these notes). This will be discussed in [GrVa].

Then by (2) we see that H1(F,, C) is divisible. The Zp-corank of H1(F,, C) is [F, : $,I + 6, where 6 = 0 if $ is nontrivial, 6 = 1 if $ is trivial. By the inflation-restriction sequence we see that H' ((F,),, C)rv E ( $ , / Z , ) [ ~ ~ : ~ PIt~follows . ,[[r,]]-cofree of corank [F, : $,I, under the hypotheses that $ is unrapified and e, p - 2. These remarks are a special case of results proved in [Gr2]. Now we return to the case where C, = ker(E[pw] + &PI). The action of GF, on C, is by a character $, the action on is by a character 4, and we have $$ = x since the Weil pairing T,(E) A T,(E) E Z p ( l ) means that x is the determinant of the representation of GFv on T,(E).

Omitting the details, one obtains a functional equation for Lp(E/$, s): for all s E Z,. Here WE is the sign which occurs in the functional equation for the Hasse-Weil L-series L(E/$, s), NE is the conductor of E, and (NE) is the projection of NE to 1 2pZp as above. 13 and the above functional equation for the padic L-function Lp(E/$, s). The functional equation is in fact equivalent to the relation between f g n a l ( ~and ) T) f r l ( ( l +T) -' - 1) mentioned above. In particular, f ~ ~ l ( ~ ~ ) / f ?