On Iwasawa main conjectures for elliptic curves at supersingular primes: beyond the case \(a_p = 0\) (Q6553207)

This interesting paper studies Iwasawa theoretic questions for elliptic curves \(E/\mathbb{Q}\) at supersingular odd primes \(p\), especially in the case when \(p=3\) and \(a_3\neq 0\). The main application, and maybe also one of the main motivations of this paper, is to prove the \(3\)-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves when \(a_p\neq 0\) (the ordinary cases are known by works of many people, including, among others, Skinner-Urban, W. Zhang, Skinner-Zhang, Castella, Berti-Bertolini-Venerucci, while the supersingular case when \(a_p=0\) is due to Wan, Castella-Wan and Fouquet-Wan).\N\NThe results of this paper on Iwasawa main conjectures are proved under the following assumption (which is stated in this form in the introduction, and made precise in Conjecture 3.33 in the paper): There exist two-variable Beilinson-Flach classes attached to an appropriate quadratic imaginary field \(K\) in which \(p\) splits that satisfy certain reciprocity laws at the two primes above \(p\). The main conjectures here are both the cyclotomic one and the two variable one (relative to the \(\mathbb{Z}_p^2\)-extension of \(K\)). The existence of these Beilinson-Flach elements is expected, but not fully known (at least not in the case of interest of this paper, when \(3\mid a_3\) and \(a_3\neq 0\)).\N\NThe application to the BSD conjecture and the results regarding the Iwasawa main conjectures are of course very important; however, the methods and ideas introduced in the paper to deal with the general supersingular case are also important and of independent interest; especially, the idea of \emph{double signed} or \emph{cromatic} Selmer groups, \(p\)-adic \(L\)-functions and Coleman maps are very flexible and have already been used and generalized by several authors, including Büyükboduk, Lei, Loeffler and Venkat.\N\NFinally, the paper contains in the introduction a rich overview of the results and methods in this field of research, including some of the most recent works, which makes it readable (and useful) even to researchers not primarily interested in the case \(p=3\) and \(a_3\neq 0\).