Euler systems and modular elliptic curves.(English)Zbl 0952.11016

Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 351-367 (1998).
The paper consists of two (small) parts (almost without proofs): (I) Generalities; (II) Elliptic Curves.
Let $$G_{\mathbb Q}=\text{ Gal}(\bar{\mathbb Q}/{\mathbb Q})$$ and define a $$p$$-adic representation of $$G_{\mathbb Q}$$ as a free $${\mathbb Z}_p$$-module $$T$$ of finite rank with a continuous $${\mathbb Z}_p$$-linear action of $$G_{\mathbb Q}$$. For such $$T$$ fix a positive integer $$N$$ divisible by $$p$$ and by all the primes where $$T$$ is ramified. Let $$\mathcal R=\mathcal R(N)$$ denote the set of all squarefree integers $$r$$ such that $$(r,N)=1$$. Let $$P_q(x)=\det(1-\text{ Fr}_qx|T) \in{\mathbb Z}_p[x]$$ denote the characteristic polynomial of Frobenius $$\text{ Fr}_q$$ at a prime $$q$$ where $$T$$ is unramified. Then an Euler system for $$T$$ is a collection $$c_{{\mathbb Q}_n(\mu_r)}\in H^1({\mathbb Q}_n(\mu_r),T)$$ for every $$r\in\mathcal R$$ and every $$n\geq 0$$ such that if $$m\geq n,q$$ is prime, and $$rq\in \mathcal R$$, then
(i) $$\text{Cor}_{{\mathbb Q}_n(\mu_{rq})/{\mathbb Q}_n (\mu_r)}c_{{\mathbb Q}_n(\mu_{rq})}=P_q(q^{-1}\text{Fr}_q^{-1})c_{{\mathbb Q}_n (\mu_r)}$$;
(ii) $$\text{Cor}_{{\mathbb Q}_m(\mu_r)/{\mathbb Q}_n(\mu_r)}c_{ {\mathbb Q}_m(\mu_r)}=c_{{\mathbb Q}_n(\mu_r)}$$.
Here $${\mathbb Q}_n\subset{\mathbb Q}( \mu_{p^{n+1}})$$ denotes the extension of $${\mathbb Q}$$ of degree $$p^n$$ in the cyclotomic $${\mathbb Z}_p$$-extension $${\mathbb Q}_{\infty}\subset{\mathbb Q}(\mu _{p^{\infty}})$$ of $${\mathbb Q}$$. It is recalled that under suitable conditions on $$T$$ the existence of an Euler system implies the finiteness (and even an upper bound) of the Selmer groups related to $$T$$. Also, by passing to the limit one obtains Iwasawa-like results.
For elliptic curves the dual exponential map $$\exp^*$$ is defined à la Kato. The basic facts on $$p$$-adic $$L$$-functions are recalled. Then Kato’s Euler system for $$T_p(E)$$ and its relation to the $$L$$-function value at $$s=1$$ are reminded. As consequences Kato’s results follow: For a modular elliptic curve $$E$$ without complex multiplication one has
(i) If $$L(E,1)\neq 0$$ then $$E({\mathbb Q})$$ and $$\text{ Ш}(E)$$ are finite;
(ii) If $$L$$ is a finite abelian extension of $${\mathbb Q}$$, $$\chi$$ a character of $$\text{ Gal}(L/{\mathbb Q})$$, and $$L(E,\chi,1)\neq 0$$ then $$E(L)^{\chi}$$ and $$\text{ Ш}(E)^{\chi}$$ are finite.
As a corollary one finds that $$E({\mathbb Q}_{\infty})$$ is finitely generated. These results are proved in some detail (at least in the case of good reduction). The paper closes with an appendix with an explicit description of the so-called Coleman map.
For the entire collection see [Zbl 0905.00052].

MSC:

 11G05 Elliptic curves over global fields 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14G05 Rational points 14H52 Elliptic curves