Anticyclotomic main conjecture and the non-triviality of Rankin–Selberg $L$-values in Hida families

To state our main results, fix a prime $p\geq 5$ and an integer $N$ with $p\nmid N$, and let $\mathcal{R}$ be a primitive branch of a Hida family of $p$-adic modular forms of tame conductor $N$; more precisely, $\mathcal{R}$ is a noetherian domain, finite and flat over the Iwasawa algebra $\Lambda=\mathcal{O}[[1+p\mathbb{Z}_{p}]]$, where $\mathcal{O}$ is the valuation ring of a fixed finite extension of $\mathbb{Q}_{p}$. We assume throughout that $\mathcal{R}$ is a regular ring. Let $\mathbf{f}=\sum_{n\geq 1}\mathbf{a}_{n}q^{n}\in\mathcal{R}[[q]]$ denote the Hida family of $p$-adic modular forms associated with $\mathcal{R}$. For each arithmetic prime $\kappa:\mathcal{R}\rightarrow F_{\kappa}\subseteq\bar{\mathbb{Q}}_{p}$, where $F_{\kappa}$ is a finite extension of $\mathbb{Q}_{p}$, the specialisation $f_{\kappa}=\sum_{n\geq 1}\kappa(\mathbf{a}_{n})q^{n}\in F_{\kappa}[[q]]$ of $\mathbf{f}$ at $\kappa$ is a $p$-ordinary cuspform of level $\Gamma_{1}(Np^{s_{\kappa}})$, weight $k_{\kappa}$ and character $\psi_{\kappa}$, for an integer $s_{\kappa}\geq 1$ and an integer $k_{\kappa}\geq 2$; see §3.1 for a more accurate exposition.

Let $\mathbf{T}^{\dagger}$ denote the self-dual twist of Hida’s big Galois representation attached to $\mathcal{R}$; therefore, $\mathbf{T}^{\dagger}$ is a free $\mathcal{R}$-module of rank $2$, and for each arithmetic prime $\kappa:\mathcal{R}\rightarrow F_{\kappa}$, the specialisation $V_{\kappa}^{\dagger}=\mathbf{T}^{\dagger}\otimes_{\mathcal{R},\kappa}F_{\kappa}$ at $\kappa$ is isomorphic to the base change to $F_{\kappa}$ of the self-dual twist of Deligne’s Galois representation attached to the modular form $f_{\kappa}$. Let $\mathbf{A}^{\dagger}=\mathbf{T}^{\dagger}\otimes_{\mathcal{R}}\mathcal{R}^{\vee}$. Here, $(-)^{\vee}$ means the Pontryagin dual.

Let $K$ be a quadratic imaginary field of discriminant prime to $Np$, and write the factorisation $N=N^{+}N^{-}$ where a prime divisor $\ell$ of $N$ divides $N^{+}$ if and only if it is split in $K$. Throughout the paper, we place ourselves in the definite setting; more precisely, we assume that

• •

  $N^{-}$ is a square-free product of an odd number of distinct primes; we also assume that $p$ is split in $K$.

If $f_{\kappa}$ has trivial character, the order of vanishing at $s=k/2$ of the $L$-series $L(f_{\kappa}/K,s)$ of $f_{\kappa}$ over $K$ is even by the assumption above on $N^{-}$, and therefore, in light of an analogue in this setting of Greenberg’s conjecture, one expects that these $L$-values do not generically vanish when the order of vanishing of the $L$-series $L(f_{\kappa},s)$ of $f_{\kappa}$ over $\mathbb{Q}$ is also even. As a consequence of the Tamagawa number conjecture of Bloch–Kato, one expects that the Bloch–Kato Selmer groups of $V_{\kappa}^{\dagger}$ over $K$ is generically trivial in the same setting, so one expects that Nekovář’s extended Selmer group of $\mathbf{A}^{\dagger}$ over $K$ is a cotorsion $\mathcal{R}$-module. when the order of vanishing of the $L$-series $L(f_{\kappa},s)$ of $f_{\kappa}$ over $\mathbb{Q}$ is even. See [LV11, Conjecture 9.5] for a more detailed discussion of this heuristic.

Denote by $K_{\infty}$ the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$, and denote $\Gamma_{\infty}=\operatorname{Gal}(K_{\infty}/K)\simeq\mathbb{Z}_{p}$. In [LV11] a big theta element $\Theta_{\infty}(\mathbf{f})$ in $\mathcal{R}[[\Gamma_{\infty}]]$ is constructed by means of compatible families of Gross points on towers of Shimura curves associated with the definite quaternion algebra $B$ ramified at all primes dividing $N^{-}$ and Eichler orders of increasing $p$-power level; we also define the two-variable anticyclotomic $p$-adic $L$-function

where $x\mapsto x^{*}$ is the involution of $\mathcal{R}[[\Gamma_{\infty}]]$ defined by $\gamma\mapsto\gamma^{-1}$ on group-like elements. The construction $\Theta_{\infty}(\mathbf{f})$, $L_{p}(\mathbf{f}/K)$, and the notion of compatibility of Gross points on towers of Shimura curves, are reviewed in §3.3.

The element $L_{p}(\mathbf{f}/K)$ is the analogue of the $p$-adic $L$-function $L_{p}(E/K)$ introduced by Bertolini–Darmon [BD96a], [BD96b] and [BD05] using Gross points on definite Shimura curves to study the arithmetic of elliptic curves over $K$ in a similar definite setting. In particular, if the conductor $N$ of $E$ admits the same factorisation $N=N^{+}N^{-}$ as above and $p$ is a prime of good ordinary reduction for $E$ which splits in $K$, then, under mild technical hypotheses, it is known that the $p$-adic $L$-function $L_{p}(E/K)$ is a non-trivial element of $\mathbb{Z}_{p}[[\Gamma_{\infty}]]$, the Pontryagin dual of the Selmer group of the $p$-power torsions of $E$ over $K_{\infty}$ is a torsion $\mathbb{Z}_{p}[[\Gamma_{\infty}]]$-module and its characteristic ideal of equal to the ideal generated by the $p$-adic $L$-function (see [BD05, Theorem 1] and [SU14, Theorem 3.37]). Therefore, it is natural to expect a similar main conjecture holds for families. In other words, if $\mathcal{R}[[\Gamma_{\infty}]]$ is the Iwasawa algebra of $\Gamma_{\infty}$ with coefficients in $\mathcal{R}$, then one expects that the two-variable $p$-adic $L$-function $L_{p}(\mathbf{f}/K)$ is a non-zero element of $\mathcal{R}[[\Gamma_{\infty}]]$, the Selmer group of $\mathbf{A}^{\dagger}$ over $K_{\infty}$ is a cotorsion $\mathcal{R}[[\Gamma_{\infty}]]$-module, and its characteristic ideal is equal to the ideal generated by the $p$-adic $L$-function. See [LV11, §9.3] for a more detailed discussion of this topic. The proof of these assertions is one of the main result of this paper, from which we also deduce some results on the Selmer group of $\mathbf{T}^{\dagger}$ over $K$.

Let $\tilde{H}^{1}_{f}(K_{\infty},\mathbf{A}^{\dagger})$ be Nekovář’s extended Selmer group of $\mathbf{A}^{\dagger}$ over $K_{\infty}$ and $\tilde{H}^{1}_{f}(K_{\infty},\mathbf{A}^{\dagger})^{\vee}$ its Pontryagin dual, which are discrete and compact $\mathcal{R}[[\Gamma_{\infty}]]$-modules, respectively. Before stating our main results, we fix our assumptions. We suppose that there exists an arithmetic prime $\kappa_{0}$ such that $f_{0}=f_{\kappa_{0}}=\sum_{n\geq 1}a_{n}q^{n}\in S_{k_{0}}(\Gamma_{0}(Np))$ is an ordinary $p$-stabilised newform of weight $k_{0}\geq 2$ with $k_{0}\equiv 2\pmod{p-1}$ and trivial nebentypus character. We denote $\bar{\rho}_{f_{0}}$ the residual representation attached to $f_{0}$.

Our first main result is the two-variable anticyclotomic Iwasawa main conjecture for Hida families, which proves [LV11, Conjecture 9.12].

We also deduce a result on the arithmetic of $\mathbf{f}$ over $K$, which is an definite analogue of the horizontal non-vanishing conjecture [LV11, Conjecture 9.5]. Define $\mathcal{J}_{0}=\chi_{\mathrm{triv}}\left(\Theta_{\infty}(\mathbf{f})\right)$, where $\chi_{\mathrm{triv}}$ is the trivial character of $\Gamma_{\infty}$ and let $\tilde{H}^{1}_{f}(K,\mathbf{T}^{\dagger})$ denote Nekovář’s extended Selmer group of $\mathbf{T}^{\dagger}$ over $K_{\infty}$.

The proofs of these results are the combination of the following ingredients.

• •

  A control theorem for Selmer groups of Hida’s big Galois representations over the antiyclotomic $\mathbb{Z}_{p}$-extension, similar to analogous results for the cyclotomic $\mathbb{Z}_{p}$-extension by Ochiai [Och06], which we prove in §2.6 of this paper;

• •

  The results from [CL16], [CKL17] and [KL22] proving a close relation between $L_{p}(\mathbf{f}/K)$ and the self-dual twist of the specialisation to the anticyclotomic line of the three-variable $p$-adic $L$-functions of Skinner–Urban [SU14];

• •

  The three-variable Iwasawa main conjecture proved by Skinner–Urban [SU14].

As hinted at in the lines above, the proof of the three-variable main conjecture in [SU14] has a prominent role in our argument; however, the careful comparison of the two setting is required, for which we use the results from [CL16], [CKL17] and [KL22].

In this section, we first review Iwasawa algebras over complete noetherian regular local rings of Krull dimension $\geq 1$ and Selmer groups of ordinary Galois representations over such rings. Then we prove a general control theorem for these Selmer groups and relate them with classical Selmer groups via Shapiro’s lemma. This generality certainly includes the case of Hida deformations. The notation of this section is independent of the notation of the other sections of the paper. Some of the arguments are similar to those in [Och00], [Och01], and [Och06].

We first set some general conventions. Let $R$ be a complete noetherian regular local ring with maximal ideal $\mathfrak{m}_{R}$, of Krull dimension $d\geq 1$, with finite residue field $k=R/\mathfrak{m}_{R}R$ of characteristic $p$, a prime number. For any ideal $I\subseteq R$, and any $R$-module $M$, denote $M[I]$ the $I$-torsion $R$-submodule of $M$ and $M_{I}$ the localization of $M$ at $I$. Denote

the $R$-linear dual of $M$ (where $\operatorname{Hom}_{R}$ denotes $R$-linear homomorphisms) and

the Pontryagin dual of $M$ (where $\operatorname{Hom}_{\mathrm{cont}}$ denotes continuous group homomorphisms). By [Nek06, §2.9.1, §2.9.2],

under our assumptions for any $R$-module $M$ of finite type, hence compact, or any $R$-module $M$ of cofinite type equipped with the discrete topology. Following [Nek06, §0.4], define

In particular, $(M^{*})^{\vee}\simeq\Phi(M)$ and $(\Phi(M))^{\vee}\simeq M^{*}$ for any $R$-module $M$ of finite type ([Nek06, (0.4.4)]). Further, by basic properties of Pontryagin duality, $(M[\mathfrak{p}])^{\vee}\simeq M^{\vee}/\mathfrak{p}M^{\vee}$ and, if $M$ is a $G$-module for some profinite group $G$, we have $(M^{G})^{\vee}\simeq(M^{\vee})_{G}$.