Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two (Q2888925)

The goal of this paper is to prove a rather general results on anticyclotomic Selmer groups of weight 2 Hilbert modular forms of parallel weight using the theory of congruences. The version of the congruence method presented in this paper is probably optimal, in the sense that it seems to the reviewer that all unnecessary assumptions have been removed.NEWLINENEWLINEThe congruence method to control Selmer groups in anticyclotomic extensions of elliptic curves has been successfully adopted by \textit{M. Bertolini} and \textit{H. Darmon} in the paper [Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)]. This method has been further developed in a series of papers by Pollack-Weston, B. Howard, M. Chida, S. Vigni and the reviewer, including cases of Hilbert modular forms of parallel weight 2. However, many technical assumptions have been imposed in these papers, and one of the goals of the paper under review consists in removing all unnecessary ones, and refining as much as possible the method. Notably, the method also applies to elliptic curves without CM defined over totally real number fields with everywhere good reduction, a case which is outside the scope of usual Kolyvagin system arguments because these elliptic curves do not admit a Shimura curve parametrization (similar results were also obtained by the reviewer).NEWLINENEWLINEThe main result of this paper can be formulated as follows. Fix an Hilbert modular form \(f\) over the totally real number field \(F\) of parallel weight 2, level \(\mathfrak n\) and character \(\omega\), which is an eigenform for the Hecke operators, a totally and an anticyclotomic character \(\chi:\mathbb A_K^\times/K^\times\rightarrow \mathbb C^\times\) of finite order such that \(\chi_{|\mathbb A_F^\times}=\omega^{-1}\). The above assumption one \(\chi\) and \(\omega\) implies that the Rankin-Selberg \(L\)-function \(L(\pi(f)\times\theta_\chi,s)\) admits a self-dual functional equation; here \(\pi(f)\) is the automorphic representation of \(\mathrm{GL}_2\) attached to \(f\), and \(\theta_\chi\) is that generated by the \(\theta\)-series of \(\chi\). Let \(L_f\) denote the extension of \(f\) generated by the Hecke eigenvalues on \(f\). Fix a field \(L\) containing \(L_f\) and the values of \(\chi\). If \(f\) has CM by a totally imaginary field \(K(f)\), assume that \(K(f)\) is not contained in the field \(K_\chi F_\omega\), where \(K_\chi=\bar K^{\ker(\chi)}\) and \(F_\omega=\bar F^{\ker(\omega)}\). Let \(V\) denote the self-dual twist of the \(p\)-adic representation attached to \(f\) and a prime ideal \(\mathfrak p| p\) of \(F\). Assume there exists an element \(g_{\mathfrak p}\in G_F=\mathrm{Gal}(\bar F/F)\) such that the following three conditions are satisfied: (1) \(\omega(g_{\mathfrak p})=1\); (2) \(\det(1-g_{\mathfrak p} X|V)=(1-\lambda_1X)(1-\lambda_2X)\) with \(\lambda_1^2=1\) and \(\lambda_2^2\neq 1\), and, if \(f\) has CM, \(\lambda_2\) is not a root of unity; (3) \(g_{\mathfrak p}\) does not act trivially on \(K\). Then the following implication holds: NEWLINE\[NEWLINEL(f_K,\chi,1)\neq 0\Longrightarrow H^1_f(K,V\otimes \chi)=0.NEWLINE\]NEWLINE The right-hand side denotes the Bloch-Kato Selmer group of \(V\otimes\chi\), while the left-hand side is related to \(L(\pi(f)\times\theta_\chi,s)\) by the equation \(L(\pi(f)\times\theta_\chi,s-1/2)=\Gamma_\mathbb C(s)^{[K:\mathbb Q]}L(f_K,\chi,s)\). Similar results hold for the representation \(V/T\), where \(T\) si a Galois-stable lattice in \(V\). Also, the set of primes \(\mathfrak p\) satisfying the three above assumptions has positive density.NEWLINENEWLINESeveral consequences of this result are then deduced, concerning finiteness of the Tate-Shafarevich group attached to \(f\), and modular abelian varieties. In particular, if \(A\) is a modular abelian variety attached to a Hilbert modular form with trivial character \(\omega\), then, if \(A\) does not have CM or the degree of \(F\) over \(\mathbb Q\) is not even, the implication NEWLINE\[NEWLINE\text{\(L(A/F,1)\neq 0\Longrightarrow A(F)\) is finite and Ш\((A/F)(\mathfrak p^\infty)=0\)}NEWLINE\]NEWLINE holds, for all but finitely many primes \(\mathfrak p\) of \(F\). Here \(L(A/F,s)\) is the \(L\)-function of \(A\) over \(F\) (with respect to a fixed inclusion \(L_f\subseteq \mathbb C\)) and Ш\((A/F)(\mathfrak p^\infty)=0\) is the \(\mathfrak p\)-primary part of the Tate-Shafarevich group of \(A\).NEWLINENEWLINEThe main ingredients to overcome the technical difficulties alluded above are: a use of results by \textit{J. B. Tunnell} [Am. J. Math. 105, 1277--1307 (1983; Zbl 0532.12015)] and \textit{H. Saito} [Compos. Math. 85, No. 1, 99--108 (1993; Zbl 0795.22009)] on local toric linear forms, a careful study of the image of the Galois representation \(V\), and the use of flat cohomology to compare finiteness conditions on Selmer groups. The arguments are explained carefully, and the article is very well written and very clear.