Perrin-Riou's main conjecture for elliptic curves at supersingular primes (Q6557886)

Let \(E\) be an elliptic curves over \(\mathbb{Q}\), and \(p\) an odd prime. Let \(K\) be an imaginary quadratic field such that the Heegner hypothesis holds. Assume \(p\) is a good ordinary prime for \(E\), \N\textit{B. Perrin-Riou} [Invent. Math. 115, No. 1, 81--149 (1994; Zbl 0838.11071)] formulated a Heegner point main conjecture for \((E,K,p)\), i.e., the torsion part of the Pontryagin dual of the anticyclotomic \(p^\infty\)-Selmer group has a characteristic ideal (in at least the invert \(p\) Iwasawa algebra \(\Lambda[\frac{1}{p}]\)) which is a square of the index of the \(\Lambda\)-submodule generating by the Heegner points in the Iwasawa cohomology (which is the inverse limit of the finite level \(p^\infty\)-Selmer groups). \N\NThe main part of the paper under review is a generalization of this ordinary main conjecture to the case when \(p\) is a good supersingular prime for \(E\) under generalized Heegner hypothesis. The strategy is a generalization of Kobayashi's calculation using Honda's height two formal group to the two variable setting, this then helps the authors to define the \(\pm\)-Selmer groups (which of course has an analogue in the anticyclotomic setting). They can prove this supersingular Perrin Riou main conjecture under certain conditions, the ingredients are Howard's upper bound (Euler system argument), Wan's lower bound (reduce from the Iwasawa-Greenberg main conjecture (using the BSD \(p\)-adic \(L\)-functions)) and certain reciprocity laws. As an application, they show the converse theorem for semi-stable elliptic curves for \(p>3\). They also give some interesting converse theorem only assuming the modulo \(p\) Selmer group of \(E\) (over \(\mathbb{Q}\)) has \(\mathbb{F}_p\)-dimension one.