Anticyclotomic Iwasawa's main conjecture for Hilbert modular forms
Let \(F\) be a totally real number field and \(f\) a Hilbert modular cusp form of level \(\mathfrak{n}\), with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime \(\wp\) of \(F\) dividing \(\mathfrak{n}\), of residual characteristic \(p \leq 5\). Let \(K\) be a quadratic imaginary extension of \(F\) and \(K_{\wp^\infty}\) be the \(\wp\)-anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\). Suppose for simplicity that \(p\) is unramified in \(F\) and in \(K_f\), where \(K_f\) is the field generated over \(\mathbb Q\) by the eigenvalues of the Hecke algebra acting on \(f\). Fix an embedding of \(\overline{\mathbb Q}\) into \(\overline{\mathbb Q}_{p}\) and denote by \(\pi\) the prime ideal of the ring of integers \(O_f\) of \(K_f\). Assuming certain technical conditions on \(\wp\) and \(\mathfrak{n}\), it is possible to associate to f a \(p\)-adic \(L\)-function \(L_{\wp,\pi} (f/K)\) which is an element of the Iwasawa algebra \(\Lambda_{\wp,\pi}\) of the Galois group of \(K_{\wp^\infty }/K\) over the completion \(O_{f,\pi}\) of \(O_f\) at \(\pi\). The so-called anticyclotomic Iwasawa Main Conjecture (AMC for short) states that the ideals of \(\Lambda_{\wp,\pi}\) generated by \(L_{\wp,\pi} (f/K)\) and by the characteristic series \([[\text{Char}]]_{\wp,\pi} (f/K)\) of the Pontryagin dual of the Selmer group attached to \((f, K_{\wp^\infty})\) must coincide. The central result of this paper is that, under suitable arithmetic assumptions and some technical conditions (among them, a condition which consists in a version of Ihara's lemma for Shimura curves over totally real fields), \([[\text{Char}]]_{\wp,\pi} (f/K)\) divides \(L_{\wp,\pi} (f/K)\). This generalizes the analogous result proved by \textit{M. Bertolini} and \textit{H. Darmon} when \(F = \mathbb Q\) and \(O_{f,\pi} = \mathbb Z_p\) [Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)]. The author hopes that, using the techniques announced by Skinner and Urban, it should be possible to prove the opposite divisibility property, hence the AMC. Several structural properties on Selmer and Shafarevich groups above \(K_{\wp^\infty}\) can be derived from the central theorem.
scientific article; zbMATH DE number 6043061