Two-variable anticyclotomic Iwasawa main conjecture for Hida families of Hilbert modular forms in the totally definite quaternionic setting over totally real fields

For an ordinary Hida family of Hilbert modular forms over a totally real field $F$, Jiménez (arXiv:2510.26332, 2025) constructed, in the totally definite quaternionic setting under the weak Heegner hypothesis, a two-variable $p$-adic $L$-function $\mathcal{L}_p(f/K)=\theta_\infty\cdot\theta_\infty^*$ from big theta elements on towers of Gross curves, indexed by the Hida family and by the $p$-anticyclotomic $\mathbb{Z}_p^{[F:\mathbb{Q}]}$-extension $K_\infty$ of a CM extension $K/F$. He formulated (Conjecture 1.2/7.2) but did not prove the two-variable anticyclotomic Iwasawa main conjecture: the equality of nontrivial ideals $(\mathcal{L}_p(f/K))=\mathrm{Char}_{\mathcal{R}_\infty}\big((\widetilde{H}^1_{f,\mathrm{Iw}}(K_\infty,\mathbf{A}^\dagger))^\vee\big)$ between the ideal generated by the $p$-adic $L$-function and the characteristic ideal of the Pontryagin dual of the extended Selmer group. In the literature, over $\mathbb{Q}$ the fixed-weight definite analogue is proved (Chida–Hsieh) and the Hida-family quaternionic version is known only up to one divisibility (Zerman), while over totally real fields only one-sided divisibility results in adjacent (indefinite or parallel-weight) settings exist (Longo, Van Order, Wang). The conjecture as stated is open, with the expected interpolation property of $\mathcal{L}_p$ — stated in the source as unnumbered prose and summarized there by the weaker non-vanishing equivalence that is its Conjecture 7.1 — as its expected analytic prerequisite.

Background

Let $F$ be a totally real field and $K/F$ a CM extension (a totally imaginary quadratic extension of $F$). An ideal $\mathfrak{n}$ of $F$ factors as $\mathfrak{n}=\mathfrak{n}^+\mathfrak{n}^-$ where the primes dividing $\mathfrak{n}^+$ split in $K$ and those dividing $\mathfrak{n}^-$ are inert; the weak Heegner hypothesis requires $\mathfrak{n}^-$ to be squarefree. A quaternion algebra $B/F$ is definite at an archimedean place when it ramifies there (is isomorphic to Hamilton's quaternions) and indefinite when it splits; $B$ is totally definite when it ramifies at every archimedean place. The number of finite ramified primes plus $[F:\mathbb{Q}]$ governs the parity of the root number of the associated Rankin–Selberg $L$-function: even parity (root number $+1$) places one in the definite setting, where central $L$-values are expected to be nonzero and the arithmetic objects are theta elements on Gross curves (zero-dimensional Shimura sets of optimal embeddings); odd parity places one in the indefinite setting of Shimura curves and Heegner points.

Bertolini and Darmon (1996) introduced theta elements over $\mathbb{Q}$ from Gross points and used their Kolyvagin derivative classes to study Selmer groups of modular forms in the anticyclotomic direction. Howard (2006) enlarged this to Hida families: a Hida family is a $p$-adic analytic family of ordinary modular forms of varying weight passing through a given form, governed by Hida's ordinary Hecke algebra $\mathcal{R}$ (a complete noetherian local $\mathbb{Z}_p$-algebra whose arithmetic prime ideals $\mathfrak{p}$ correspond to classical members of the family of weight $k_\mathfrak{p}$). Howard's big Heegner points interpolate classical Heegner points across the family and form an Euler system. Longo and Vigni (2011) transported this to the quaternionic setting over $\mathbb{Q}$, constructed a big theta element $\theta_\infty$ and the two-variable $p$-adic $L$-function $\mathcal{L}_p=\theta_\infty\cdot\theta_\infty^*$ (with $*$ the involution inverting group elements), and posed the corresponding two-variable anticyclotomic Iwasawa main conjecture (their Conjecture 9.12), which remains unproved in full.

The Iwasawa-theoretic framework is as follows. The $p$-anticyclotomic extension $K_\infty/K$ is the $\mathbb{Z}_p^{[F:\mathbb{Q}]}$-extension on which the nontrivial automorphism of $K/F$ acts by inversion; write $\mathcal{R}_\infty=\mathcal{R}[[\mathrm{Gal}(K_\infty/K)]]$ for the two-variable Iwasawa algebra. The algebraic side is the extended Selmer group $\widetilde{H}^1_{f,\mathrm{Iw}}(K_\infty,\mathbf{A}^\dagger)=\varinjlim_n\widetilde{H}^1_f(K_n,\mathbf{A}^\dagger)$, a direct limit of discrete Selmer-type cohomology groups over the finite layers $K_n$, with coefficients in the Pontryagin dual representation $\mathbf{A}^\dagger$ of the big Galois representation attached to the family. For a finitely generated torsion $\mathcal{R}_\infty$-module $M$, the characteristic ideal $\mathrm{Char}_{\mathcal{R}_\infty}(M)$ is the product over height-one prime ideals of powers given by local lengths (set to $0$ if $M$ is not torsion). The main conjecture predicts that the ideal generated by $\mathcal{L}_p(f/K)$ in $\mathcal{R}_\infty$ equals this characteristic ideal of the Pontryagin dual $(\widetilde{H}^1_{f,\mathrm{Iw}}(K_\infty,\mathbf{A}^\dagger))^\vee$, tying the analytic side ($p$-adic $L$-function) to the arithmetic side (Selmer groups).

Jiménez (2025) extended the Longo–Vigni construction to totally real fields: he builds big Heegner points from towers of Gross curves and Shimura curves via optimal embeddings (embeddings of orders of $K$ into orders of the quaternion algebra whose image is its own normalizer stabilizer), treats definite and indefinite quaternion algebras uniformly, and in the definite case constructs $\theta_\infty\in\mathcal{R}_\infty$ and $\mathcal{L}_p(f/K)=\theta_\infty\cdot\theta_\infty^*$ for a Hida family of Hilbert modular forms of parallel weight. Two standing hypotheses are imposed: Assumption 7.1 (multiplicity one: the Picard module $\mathbf{J}$ of the tower of Gross (or Shimura) curves $X_m$ — the inverse limit of the ordinary parts of $\mathrm{Pic}(X_m)\otimes_{\mathbb{Z}}\mathcal{O}$, tensored with $\mathcal{R}$ — satisfies $\dim_{\mathcal{R}/\mathfrak{m}}\mathbf{J}/\mathfrak{m}\mathbf{J}=1$, ensuring $\mathbf{J}\cong\mathcal{R}$) and Assumption 7.2 (the ring $\mathcal{R}$ is regular), together with common root number $\omega=1$. The $p$-adic $L$-function is only conjectured to interpolate the algebraic parts of the central values of the Rankin–Selberg $L$-functions $L(f^\dagger_\mathfrak{p}/K,\chi,k_\mathfrak{p}/2)$ of the specializations of the family, twisted by finite-order anticyclotomic characters $\chi$ and by the critical character $\theta_p^{-1}$ (stated in the source as an unnumbered expected property; its numbered Conjecture 7.1 is the weaker non-vanishing equivalence $L(f^\dagger_\mathfrak{p}/K,\chi,k_\mathfrak{p}/2)\neq 0\Longleftrightarrow\mathcal{L}_p(f/K;\chi,\mathfrak{p})\neq 0$); over $\mathbb{Q}$ this interpolation was proved by Castella–Longo (2016). Over $\mathbb{Q}$, Chida–Hsieh proved the anticyclotomic main conjecture for modular forms of fixed higher weight in the definite setting (2015) and the full interpolation and non-vanishing theory for such theta-element $L$-functions (2018), while Zerman (2025) proved one divisibility of the Hida-family version using Kolyvagin systems. Over totally real fields, Longo (2012), Van Order (2013), and Wang (2023) proved the main conjecture or one of its divisibilities for Hilbert modular forms in the Heegner-point/indefinite settings, under hypotheses; the totally definite, Hida-family, two-variable case has no published partial result.

Problem Statement

In the setting of Jiménez (arXiv:2510.26332): let $F$ be a totally real field of degree $d$, $p>3$ a rational prime unramified in $F$, $K/F$ a CM extension with $\mathfrak{n}=\mathfrak{n}^+\mathfrak{n}^-$ the split/inert factorization of an ideal of $F$ coprime to $p$ and to the relative discriminant, with $\mathfrak{n}^-$ squarefree (weak Heegner hypothesis) and $r+d\equiv 0\pmod{2}$ where $r$ is the number of prime divisors of $\mathfrak{n}^-$ (totally definite case), let $B$ be the quaternion algebra over $F$ of finite discriminant $\mathfrak{n}^-$ ramified at every archimedean place (uniquely determined by this discriminant and the parity condition), and let $\mathcal{R}$ be Hida's ordinary Hecke algebra for the Hida family through an ordinary Hilbert modular form $f$ of parallel weight. Assume multiplicity one for the Picard module $\mathbf{J}$ of the tower of Gross curves $X_m$ of $B$ — the inverse limit of the ordinary parts of $\mathrm{Pic}(X_m)\otimes_{\mathbb{Z}}\mathcal{O}$, tensored with $\mathcal{R}$ — that is, $\dim_{\mathcal{R}/\mathfrak{m}}\mathbf{J}/\mathfrak{m}\mathbf{J}=1$ (Assumption 7.1), regularity of $\mathcal{R}$ (Assumption 7.2), and common root number $\omega=1$. Let $K_\infty$ be the $p$-anticyclotomic $\mathbb{Z}_p^d$-extension of $K$, $\mathcal{R}_\infty=\mathcal{R}[[\mathrm{Gal}(K_\infty/K)]]$, let $\theta_\infty\in\mathcal{R}_\infty$ be the big theta element constructed from the big Heegner points on the tower of Gross curves of $B$, $\mathcal{L}_p(f/K)=\theta_\infty\cdot\theta_\infty^*$ the two-variable $p$-adic $L$-function constructed by Jiménez as a totally real analogue of Longo–Vigni's construction over $\mathbb{Q}$, and $\widetilde{H}^1_{f,\mathrm{Iw}}(K_\infty,\mathbf{A}^\dagger)=\varinjlim_n\widetilde{H}^1_f(K_n,\mathbf{A}^\dagger)$ the Iwasawa extended Selmer group with coefficients in the Pontryagin dual $\mathbf{A}^\dagger$ of the big Galois representation. Prove or refute Conjecture 7.2 of the source: the equality of nontrivial ideals

$$\big(\mathcal{L}_p(f/K)\big)=\mathrm{Char}_{\mathcal{R}_\infty}\Big(\big(\widetilde{H}^1_{f,\mathrm{Iw}}(K_\infty,\mathbf{A}^\dagger)\big)^\vee\Big)$$

in $\mathcal{R}_\infty$, where $\vee$ denotes the Pontryagin dual and $\mathrm{Char}_{\mathcal{R}_\infty}$ the characteristic ideal. Since $\mathcal{L}_p(f/K)$ is itself only conjectured to interpolate the algebraic parts of the central values $L(f^\dagger_{\mathfrak{p}}/K,\chi,k_{\mathfrak{p}}/2)$ at arithmetic primes $\mathfrak{p}$ and finite-order anticyclotomic characters $\chi$ (an unnumbered expected property in the source, with the critical-character twist made precise; the source's numbered Conjecture 7.1 records the weaker non-vanishing equivalence), the target is the equality of nontrivial ideals of Conjecture 7.2 itself; the expected interpolation property is an analytic prerequisite that a proof relying on it must establish or cite, not an independent deliverable.

References

1. Ignacio M. Jiménez, "Quaternionic big Heegner points over totally real fields", preprint (2025), arXiv:2510.26332, https://arxiv.org/abs/2510.26332 (source of the conjecture)
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