Anticyclotomic Iwasawa theory of CM elliptic curves. II.

*(English)*Zbl 1130.11058Consider an elliptic curve \(E\) over \({\mathbb Q}\). For primes of ordinary reduction much is known about the Iwasawa theory of \(E\). This is no longer true at supersingular primes. In this case the natural Iwasawa modules considered are not torsion and the natural candidates for \(p\)-adic \(L\)-functions do not lie in the Iwasawa algebra.

However, there has been important progress in the theory of cyclotomic Iwasawa theory at primes \(p\) of supersingular reduction. S. Kobayashi [Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)] has formulated a cyclotomic main conjecture in this case by defining restricted plus and minus Selmer groups and relating them to the modified \(p\)-adic \(L\)-functions defined by R. Pollak in [Duke Math. J. 118, No. 3, 523–558 (2003; Zbl 1074.11061)]. It turns out that the conjecture is equivalent to a cyclotomic main conjecture proposed by K. Kato in [Berthelot, Pierre (ed.) et al., \(p\)-adic cohomology and arithmetic applications (III). Paris: Société Mathématique de France. Astérisque 295, 117–290 (2004; Zbl 1142.11336)] and by B. Perrin-Riou [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1061.11031); Ann. Inst. Fourier 43, No. 4, 945–995 (1993; Zbl 0840.11024)]. K. Rubin and R. Pollack [Ann. Math. (2) 159, No. 1, 447–464 (2004; Zbl 1082.11035)] proved Kobayashi’s conjecture when \(E\) has complex multiplication.

In the general case, Kobayashi himself proved one divisibility of the main conjecture using methods of Kato.

In the paper under review, the authors consider an elliptic curve \(E\) which has complex multiplication by the maximal order \({\mathcal O}\) of an imaginary quadratic field \(K\). If \(\psi\) is the Grössencharacter of \(E\) of conductor \({\mathfrak f}\) and \(p>3\) is a fixed rational prime inert in \(K\) and at which \(E\) has good reduction, then \(E\) has supersingular reduction at \(p\). Let \({\mathfrak p}\) be the unique prime of \(K\) above \(p\) and let \(D_ \infty\) be the anticyclotomic \({\mathbb Z}_ p\)-extension of \(K\). The prime \({\mathfrak p}\) is totally ramified in \(D_ \infty\).

The paper studies the Iwasawa theory of \(E\) over \(D_ \infty\). Anticyclotomic versions of Kobayashi’s plus and minus Selmer groups are defined and it turns out that one of the Selmer groups is a cotorsion Iwasawa module, while the other is not. The main result is the proof of a conjecture of R. Greenberg [Invent. Math. 72, 241–265 (1983; Zbl 0546.14015)]:

Let \(\varphi\) be the Euler totient function. Let \(\varepsilon = \pm 1\) be the sign of the functional equation of \(L(E/{\mathbb Q},s)\), \({\text{Sel}}_ {{\mathfrak p}^ n}(E/D_ \infty)\) the \({\mathfrak p}\)-primary Selmer group of \(E/D_ n\) and \({\mathcal O} _ {\mathfrak p}\) the completion of \({\mathcal O}\) at \({\mathfrak p}\). Then there exists an integer \(e\), independent of \(n\), such that \[ \text{corank}_{{\mathcal O}_ {\mathfrak p}} \text{Sel} _{{\mathfrak p}^ {\infty}}(E/D_ n) = e + \sum _ {1\leq k\leq n, (-1)^ k= \varepsilon} \varphi(p_ k) \] for all large enough \(n\), where \(D_ n\) is the subfield of \(D_ \infty\) such that \([D_ n: K] = p^ n\).

The results given here may be considered as a step towards a supersingular main conjecture of the same type as that considered in Part I of this work [Ann. Inst. Fourier 56, No. 6, 1001–1048 (2006; Zbl 1168.11023)].

However, there has been important progress in the theory of cyclotomic Iwasawa theory at primes \(p\) of supersingular reduction. S. Kobayashi [Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)] has formulated a cyclotomic main conjecture in this case by defining restricted plus and minus Selmer groups and relating them to the modified \(p\)-adic \(L\)-functions defined by R. Pollak in [Duke Math. J. 118, No. 3, 523–558 (2003; Zbl 1074.11061)]. It turns out that the conjecture is equivalent to a cyclotomic main conjecture proposed by K. Kato in [Berthelot, Pierre (ed.) et al., \(p\)-adic cohomology and arithmetic applications (III). Paris: Société Mathématique de France. Astérisque 295, 117–290 (2004; Zbl 1142.11336)] and by B. Perrin-Riou [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1061.11031); Ann. Inst. Fourier 43, No. 4, 945–995 (1993; Zbl 0840.11024)]. K. Rubin and R. Pollack [Ann. Math. (2) 159, No. 1, 447–464 (2004; Zbl 1082.11035)] proved Kobayashi’s conjecture when \(E\) has complex multiplication.

In the general case, Kobayashi himself proved one divisibility of the main conjecture using methods of Kato.

In the paper under review, the authors consider an elliptic curve \(E\) which has complex multiplication by the maximal order \({\mathcal O}\) of an imaginary quadratic field \(K\). If \(\psi\) is the Grössencharacter of \(E\) of conductor \({\mathfrak f}\) and \(p>3\) is a fixed rational prime inert in \(K\) and at which \(E\) has good reduction, then \(E\) has supersingular reduction at \(p\). Let \({\mathfrak p}\) be the unique prime of \(K\) above \(p\) and let \(D_ \infty\) be the anticyclotomic \({\mathbb Z}_ p\)-extension of \(K\). The prime \({\mathfrak p}\) is totally ramified in \(D_ \infty\).

The paper studies the Iwasawa theory of \(E\) over \(D_ \infty\). Anticyclotomic versions of Kobayashi’s plus and minus Selmer groups are defined and it turns out that one of the Selmer groups is a cotorsion Iwasawa module, while the other is not. The main result is the proof of a conjecture of R. Greenberg [Invent. Math. 72, 241–265 (1983; Zbl 0546.14015)]:

Let \(\varphi\) be the Euler totient function. Let \(\varepsilon = \pm 1\) be the sign of the functional equation of \(L(E/{\mathbb Q},s)\), \({\text{Sel}}_ {{\mathfrak p}^ n}(E/D_ \infty)\) the \({\mathfrak p}\)-primary Selmer group of \(E/D_ n\) and \({\mathcal O} _ {\mathfrak p}\) the completion of \({\mathcal O}\) at \({\mathfrak p}\). Then there exists an integer \(e\), independent of \(n\), such that \[ \text{corank}_{{\mathcal O}_ {\mathfrak p}} \text{Sel} _{{\mathfrak p}^ {\infty}}(E/D_ n) = e + \sum _ {1\leq k\leq n, (-1)^ k= \varepsilon} \varphi(p_ k) \] for all large enough \(n\), where \(D_ n\) is the subfield of \(D_ \infty\) such that \([D_ n: K] = p^ n\).

The results given here may be considered as a step towards a supersingular main conjecture of the same type as that considered in Part I of this work [Ann. Inst. Fourier 56, No. 6, 1001–1048 (2006; Zbl 1168.11023)].

Reviewer: Gabriel D. Villa-Salvador (México D.F.)