Iwasawa’s main conjecture for elliptic curves over anticyclotonic \(\mathbb Z_p\)-extensions.

*(English)*Zbl 1093.11037Let \(E\) be a non-CM elliptic curve over \(\mathbb Q\) of conductor \(N_0\). Let \(p\) be a prime of good ordinary reduction (in which case, set \(N=pN_0\)) or multiplicative reduction (in which case, set \(N=N_0\)). Let \(K\) be an imaginary quadratic field of discriminant prime to \(N\). Let \(K_{\infty}/K\) be the anticyclotomic \(\mathbb Z_p\)-extension of \(K\) and \(G_{\infty}=\text{Gal}(K_{\infty}/K)\). Previous work of the authors [Invent. Math. 126, 413–456 (1996; Zbl 1029.11027)] constructed a \(p\)-adic \(L\)-function \(L_p(E,K)\in \mathbb Z_p[[G_{\infty}]]\).

Let \(\text{Sel}(K_{\infty},E_{p^{\infty}})^{\vee}\) be the Pontryagin dual of the \(p\)-primary Selmer group attached to \(E\) over \(K_{\infty}\), and let \({\mathcal C}\in \mathbb Z_p[[G_{\infty}]]\) be its characteristic power series.

Write \(N=pN^+N^-\), where \(N^+\) is a product of primes splitting in \(K\) and \(N^-\) is a product of primes inert in \(K\). Assume that \(N^-\) is the squarefree product of an odd number of primes. The main result of the paper is that, under certain technical additional assumptions that exclude a finite set of values of \(p\), the characteristic power series \({\mathcal C}\) divides the \(p\)-adic \(L\)-function \(L_p(E,K)\). This has several corollaries that are closely related to the conjecture of Birch and Swinnerton-Dyer. For example, if \(L_p(E,K,s)\) is the \(p\)-adic Mellin transform of the measure associated to \(L_p(E,K)\), then the order of vanishing of \(L_p(E,K,s)\) at \(s=1\) is greater than or equal to the rank of the Mordell-Weil group \(E(K)\). Moreover, if \(\chi\) is a character of \(G_{\infty}\) of finite order, and if the classical \(L\)-function value \(L(E/K,\chi,1)\neq 0\), then the \(\chi\)-parts of \(E(K_{\infty})\) and of the \(p\)-primary subgroup of the Shafarevich-Tate group of \(E\) over \(K_{\infty}\) are finite.

The assumption that \(N^-\) has an odd number of prime factors implies that the signs of the functional equations of the relevant \(L\)-functions are \(+1\). This means that the theory of Heegner points does not apply. Instead, the authors use Shimura curves and congruences between modular forms to construct compatible collections of cohomology classes. In the terminology of B. Mazur and K. Rubin [Kolyvagin Systems, Mem. Am. Math. Soc. 168, AMS, Providence, RI (2004; Zbl 1055.11041)], the construction produces a “Kolyvagin system” without using an Euler system.

Let \(\text{Sel}(K_{\infty},E_{p^{\infty}})^{\vee}\) be the Pontryagin dual of the \(p\)-primary Selmer group attached to \(E\) over \(K_{\infty}\), and let \({\mathcal C}\in \mathbb Z_p[[G_{\infty}]]\) be its characteristic power series.

Write \(N=pN^+N^-\), where \(N^+\) is a product of primes splitting in \(K\) and \(N^-\) is a product of primes inert in \(K\). Assume that \(N^-\) is the squarefree product of an odd number of primes. The main result of the paper is that, under certain technical additional assumptions that exclude a finite set of values of \(p\), the characteristic power series \({\mathcal C}\) divides the \(p\)-adic \(L\)-function \(L_p(E,K)\). This has several corollaries that are closely related to the conjecture of Birch and Swinnerton-Dyer. For example, if \(L_p(E,K,s)\) is the \(p\)-adic Mellin transform of the measure associated to \(L_p(E,K)\), then the order of vanishing of \(L_p(E,K,s)\) at \(s=1\) is greater than or equal to the rank of the Mordell-Weil group \(E(K)\). Moreover, if \(\chi\) is a character of \(G_{\infty}\) of finite order, and if the classical \(L\)-function value \(L(E/K,\chi,1)\neq 0\), then the \(\chi\)-parts of \(E(K_{\infty})\) and of the \(p\)-primary subgroup of the Shafarevich-Tate group of \(E\) over \(K_{\infty}\) are finite.

The assumption that \(N^-\) has an odd number of prime factors implies that the signs of the functional equations of the relevant \(L\)-functions are \(+1\). This means that the theory of Heegner points does not apply. Instead, the authors use Shimura curves and congruences between modular forms to construct compatible collections of cohomology classes. In the terminology of B. Mazur and K. Rubin [Kolyvagin Systems, Mem. Am. Math. Soc. 168, AMS, Providence, RI (2004; Zbl 1055.11041)], the construction produces a “Kolyvagin system” without using an Euler system.

Reviewer: Lawrence C. Washington (College Park)