## Iwasawa theory of elliptic curves with complex by Ehud De Shalit

By Ehud De Shalit

Within the final fifteen years the Iwasawa concept has been utilized with awesome good fortune to elliptic curves with complicated multiplication. a transparent but normal exposition of this idea is gifted during this book.

Following a bankruptcy on formal teams and native devices, the p-adic L services of Manin-Vishik and Katz are built and studied. within the 3rd bankruptcy their relation to category box concept is mentioned, and the functions to the conjecture of Birch and Swinnerton-Dyer are handled in bankruptcy four. complete proofs of 2 theorems of Coates-Wiles and of Greenberg also are offered during this bankruptcy that may, moreover, be used as an creation to the more moderen paintings of Rubin.

The booklet is basically self-contained and assumes familiarity purely with basic fabric from algebraic quantity idea and the speculation of elliptic curves. a few effects are new and others are awarded with new proofs.

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**Example text**

Let us explain the answer without going into every detail. Congruences mod p behave, in many ways, just like ordinary equations. Now we know that an equation of the form coXk + CIX k - 1 + ... + Ck-lX + Ck = 0, of degree k, cannot have more than k roots. In the same way, the congruence coXk + CIXk- 1 + ... + Ck-lX + Ck == 0 mod p cannot have more than k solutions. ::.!. 2 - 1 == o mod p cannot have more than P;l solutions. But we know by our previous arguments that all quadratic residues satisfy this congruence, and there are P;l quadratic residues.

You probably also managed to show that the same thing happened for any polar curve of the form r = me + c. (2) The argument is the same. At the ray e at first. The next time past this ray and r2 = = e], we get r] m(e] = me] +c + 2n) + c. So the difference between the two values of r is r2 - r] = (me] + m2n + c) - (me] + c) = 2nm. Again, a constant increase. Again, the same increase occurs for every ray. Such curves are known as Archimedean spirals (see [2] and [3], for example). • •• BREAK Can you think where you might have seen Archimedean spirals?

CP - (16) l)a. We claim that no two of the integers on the list (16) are congruent mod p. For suppose we were wrong; then we would have numbers k, £, with 1 ::: k < l ::: p - 1, such that ka == la mod p. This means that p I a(l - k). But p f a, by hypothesis; and p f l - k since 1 ::: l - k < p. Thus, by Theorem 4,p f k). This contradiction shows that, after all, we were right-no two of the integers in (16) are congruent mod p. A very similar argument shows that no integer in the list (16) is divisible by p.