Number Theory

Iwasawa theory of elliptic curves with complex by Ehud De Shalit

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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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Additional resources for Iwasawa theory of elliptic curves with complex multiplication: p-adic L functions

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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.

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