Arithmetic Theory of Elliptic Curves by J. Coates, R. Greenberg, K.A. Ribet, K. Rubin, C. Viola

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

This quantity includes the accelerated types of the lectures given through the authors on the C. I. M. E. educational convention held in Cetraro, Italy, from July 12 to 19, 1997. The papers accrued listed below are huge surveys of the present study within the mathematics of elliptic curves, and in addition comprise a number of new effects which can't be came across somewhere else within the literature. as a result of readability and magnificence of exposition, and to the historical past fabric explicitly integrated within the textual content or quoted within the references, the quantity is easily suited for examine scholars in addition to to senior mathematicians.

Show description

Read or Download Arithmetic Theory of Elliptic Curves PDF

Best popular & elementary books

Thirteen Books of Euclid's Elements

Quantity 1 of three-volume set containing whole English textual content of all thirteen books of the weather plus severe equipment studying each one definition, postulate and proposition in nice element. Covers textual and linguistic concerns; mathematical analyses of Euclid's principles; classical, medieval, Renaissance and smooth commentators; refutations, helps, extrapolations, reinterpretations and ancient notes.

Pre-Calculus Workbook For Dummies

Get the boldness and math talents you want to start with calculusAre you getting ready for calculus? This hands-on workbook is helping you grasp uncomplicated pre-calculus recommendations and perform the kinds of difficulties you are going to stumble upon within the direction. you will get hundreds of thousands of invaluable workouts, problem-solving shortcuts, lots of workspace, and step by step recommendations to each challenge.

The Thirteen Books of The Elements Vol 2(Books 3-9)

Quantity 2 of 3-volume set containing entire English textual content of all thirteen books of the weather plus severe research of every definition, postulate, and proposition. Covers textual and linguistic issues; mathematical analyses of Euclid's rules; classical, medieval, Renaissance and sleek commentators; refutations, helps, extrapolations, reinterpretations and old notes.

Functional Analysis for Physics and Engineering

This e-book presents an creation to practical research for non-experts in arithmetic. As such, it's particular from so much different books at the topic which are meant for mathematicians. techniques are defined concisely with visible fabrics, making it available for these unusual with graduate-level arithmetic.

Additional info for Arithmetic Theory of Elliptic Curves

Example text

That is, At/X r W, where W = M$ n pp-. On the other hand, if ppm M,, then XA-torsZ Zp(l), the Tate module for ppm. In this case, X/XA-torsis free and hence X Z At x Z p ( l ) . In the preceding discussion, the A-module At is in fact canonical. It is the reflexive hull of X/XA-t,,s. Thus, the action of A on X gives an action on At. Examining the above arguments more carefully, one finds that, for p odd, (One just studies the A-module X @for each At is isomorphic to A[A][~:QPI. character 4 of A. ) For p = 2, we can at least make such an identification up to a group of exponent 2.

Now if one considers the A-module Y = A/( fi (T)ai),where f i (T) is irreducible in A, then Y/TY is infinite if and only if fi(T) is an associate of T. Therefore, if F is an imaginary quadratic field in which p splits and if F, is the cyclotomic Bpextension of F, then TI f (T), where f (T) is a generator of the characteristic ideal of X . One can prove that T2 I( f (T). (This is an interesting exercise. It is easy to show that X/TX has Zp-rank 1. One must then show that X/T2X also has Zp-rank 1.

We refer the reader to [Be], [BeDal, 21, and [Maz4] for a discussion of this topic. 1. If q i p , then Im(n,) = 0. If qlp, then The first assertion can also be explained by using the fact that, for q p, H1(M,, E[pw]) is a finite group. But E(M,) 8 (Qp/Zp), and hence Im(tc,) are divisible groups. Even if M, is an infinite extension of Fv,it is clear from the above that Im(n,) = 0 if q i p. Assume that E has good, ordinary reduction at v, where v is a prime of F lying over p. Then, considering Eb*] as a subgroup of E ( F v ) , we have the reduction map E [ y ] t E[pm], where E is the reduction of E modulo v.

Download PDF sample

Rated 4.69 of 5 – based on 40 votes