Euler Systems and Modular Elliptic Curves
نویسنده
چکیده
This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group associated to the dual module Hom(T, μp∞). These theorems, which generalize work of Kolyvagin [Ko], have been obtained independently by Kato [Ka1], Perrin-Riou [PR2], and the author [Ru3]. We will not prove these theorems here, or even attempt to state them in the greatest possible generality. In the second part of the paper we show how to apply the results of Part I and an Euler system recently constructed by Kato [Ka2] (see the article of Scholl [Scho] in this volume) to obtain Kato’s theorem in the direction of the Birch and Swinnerton-Dyer conjecture for modular elliptic curves (Theorem 8.1).
منابع مشابه
Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer
1. Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6. Descent 22 7. Elliptic Units 27 8. Euler Systems 37 9. Bounding Ideal Class Groups 43 10. The Theorem of Coates and Wiles 47 11. Iwasawa Theory and the “Main Conjecture” 50 12. Computing the Selmer G...
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