The full Iwasawa Main Conjecture for the $\mathrm{GL}_2 \times \mathrm{GU}(1)$ $p$-adic $L$-function over totally real fields in the non-split case

For a totally real field $F$, a CM quadratic extension $E/F$, and a prime $p$, Disegni constructed a $p$-adic $L$-function $\mathscr{L}_p(\mathscr{V})$ on Hida families for $\mathrm{GL}_{2/F} \times \mathrm{Res}_{E/F}\mathrm{GL}_1$, characterized by exact interpolation of critical Rankin–Selberg $L$-values $L(1/2, \pi_E \otimes \chi)/L(1, \pi, \mathrm{ad})$. The author identifies the full Iwasawa Main Conjecture for this function — equality, up to the standard factors, of the characteristic ideal of the Pontryagin dual of the associated Selmer group with the ideal generated by $\mathscr{L}_p(\mathscr{V})$ — as the main remaining goal. The conjecture is known in the case where $E/F$ splits above $p$ (Skinner–Urban, Wan), and only one-divisibility partial results exist for $F = \mathbb{Q}$ at inert primes (Büyükboduk–Lei); a derivative-level variant has one divisibility from Disegni's universal $p$-adic Gross–Zagier formula. The general case — arbitrary totally real $F$, $E/F$ not necessarily split above $p$, both divisibilities — remains open.

Background

A CM quadratic extension is a degree-2 extension $E/F$ of a totally real field $F$ (all embeddings of $F$ into $\mathbb{R}$ are real) such that $E$ is totally imaginary; Hecke characters of $E$ then have infinity types governed by the embeddings $E \hookrightarrow \mathbb{C}$. To a $p$-ordinary (nearly ordinary at the places above $p$) Hilbert modular representation $\pi$ of $\mathrm{GL}_2(F)$ — 'ordinary' means the $U_p$-eigenvalues are $p$-adic units — and a Hecke character $\chi$ of $E$ whose weights are smaller than those of $\pi$, one attaches the Rankin–Selberg $L$-function $L(1/2, \pi_E \otimes \chi)$, where $\pi_E$ is the base change of $\pi$ to $\mathrm{GL}_2(E)$; Waldspurger's theory relates its central values to periods and theta lifts. A Hida family is a $p$-adic analytic family of such ordinary representations varying over a rigid-geometric 'weight space'; the product group $\mathrm{GL}_{2/F} \times \mathrm{Res}_{E/F}\mathrm{GL}_1$ (the second factor, often denoted $\mathrm{GU}(1)$, is the multiplicative group of $E$ viewed as an $F$-group) carries a universal Galois representation $\mathscr{V}$ over such a family. Disegni (2021, published 2022) constructed a meromorphic function $\mathscr{L}_p(\mathscr{V})$ on these families, characterized by an exact interpolation property for the critical values $L(1/2, \pi_E \otimes \chi)/L(1, \pi, \mathrm{ad})$ at all classical points outside the polar locus and subject to a weight condition. The construction is 'soft': $\mathscr{L}_p(\mathscr{V})$ is glued as a ratio of a family of global Waldspurger zeta integrals to a family of local zeta integrals, the latter built via the local Langlands correspondence in families; as a quotient it is a priori only meromorphic, and the polar locus is controlled only on the self-dual locus (the closed subspace where the central character $\omega$ is trivial). The Iwasawa Main Conjecture (IMC) for such a $p$-adic $L$-function asserts an arithmetic counterpart: the Pontryagin dual of the Bloch–Kato Selmer group of $\mathscr{V}$ (Galois cohomology classes satisfying local conditions determined by the $p$-adic local Langlands data) over the family is a torsion module over the relevant Iwasawa algebra (the completed group ring of the Galois group of the relevant cyclotomic $\mathbb{Z}_p$-extension, extended by the weight variables parametrizing the Hida family), and its characteristic ideal equals the ideal generated by $\mathscr{L}_p(\mathscr{V})$ up to explicit interpolation factors. Prior status: for $\mathrm{GL}_2$ itself the IMC was proved by Skinner–Urban (2014) over $\mathbb{Q}$ and by Wan (2015) for Hilbert modular forms; Disegni's account states the IMC for $\mathscr{L}_p(\mathscr{V})$ was proved 'in the split case' ($E/F$ split above $p$) by these works, while in the non-split case only partial results toward it exist for $F = \mathbb{Q}$ (Büyükboduk–Lei, one divisibility in a three-variable semi-ordinary setting at inert primes). A companion 'derivative' conjecture along the self-dual locus — where $L(1/2, \pi_E \otimes \chi)$ vanishes and the $p$-adic Gross–Zagier formula of Disegni's universal-Heegner-class paper governs the cyclotomic derivative $d^{\sharp}\mathscr{L}_p(\mathscr{V})$ — has one divisibility (that paper's Theorem E); the reverse divisibility is equivalent to a generalization of Perrin-Riou's Heegner point main conjecture, whose original form was proved by Burungale–Castella–Kim (2021).

Problem Statement

Prove the full Iwasawa Main Conjecture for Disegni's $p$-adic $L$-function $\mathscr{L}_p(\mathscr{V})$: for every Hida family $\mathscr{X}$ for $\mathrm{GL}_{2/F} \times \mathrm{Res}_{E/F}\mathrm{GL}_1$ satisfying the standing hypotheses of the construction ($F$ totally real, $E/F$ a CM quadratic extension, $p$-ordinary forms, nonempty self-dual locus), establish both divisibilities — hence equality up to the explicit interpolation factors — between the characteristic ideal of the Pontryagin dual of the Selmer group of the universal Galois representation $\mathscr{V}$ with its standard local conditions, and the ideal generated by $\mathscr{L}_p(\mathscr{V})$ in the fraction field of the cyclotomic and weight-variable Iwasawa algebra. The open content is the general case in which $F$ is an arbitrary totally real field and $E/F$ is not assumed to split above $p$ (the split case being known from the work of Skinner–Urban and Wan, and only partial divisibilities being known for $F = \mathbb{Q}$ at inert primes). The source treats the following as related but separate companion problems, distinct from the objective above: the reverse divisibility for the cyclotomic-derivative main conjecture (equivalent to a generalized Perrin-Riou Heegner point main conjecture), the finite-slope (non-ordinary) analogue, control of the polar locus of $\mathscr{L}_p(\mathscr{V})$ away from the self-dual locus, and finer integrality properties.

References

1. Daniel Disegni, '$p$-adic L-functions via local-global interpolation: the case of ${\rm GL}_2 \times {\rm GU}(1)$', arXiv:2102.02591, DOI 10.48550/arXiv.2102.02591; published in the Canadian Journal of Mathematics 75 (2022), no. 3, 965-1017, DOI 10.4153/S0008414X22000256. https://arxiv.org/abs/2102.02591
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