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].


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