By J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola
This quantity includes the multiplied models of the lectures given by means of the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers gathered listed below are vast surveys of the present examine within the mathematics of elliptic curves, and in addition include a number of new effects which can't be stumbled on in other places within the literature. because of readability and magnificence of exposition, and to the history fabric explicitly integrated within the textual content or quoted within the references, the amount is easily fitted to examine scholars in addition to to senior mathematicians.
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Additional resources for Arithmetic Theory of Elliptic Curves
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.