The p-integral anticyclotomic Iwasawa main conjecture for GL(n)×GL(n+1) Rankin–Selberg motives over CM fields

For a pair of conjugate self-dual cuspidal automorphic representations $\Pi_0$ of $\mathrm{GL}_{n_0}(\mathbb{A}_F)$ and $\Pi_1$ of $\mathrm{GL}_{n_1}(\mathbb{A}_F)$ ($\{n_0,n_1\}=\{n,n+1\}$) over a CM field $F$, Y. Liu posed anticyclotomic Iwasawa main conjectures over the localized Iwasawa algebra $\mathbb{L}^\circ[[\mathrm{Gal}(\mathcal{F}/F)]]\otimes_{\mathbb{L}^\circ}\mathbb{L}$: in the root number $+1$ (coherent) case the anticyclotomic $p$-adic $L$-function should generate the characteristic ideal of the Iwasawa Bloch–Kato Selmer group, and in the root number $-1$ (incoherent) case the square of the characteristic ideal of the Heegner-type quotient should equal that of the torsion Selmer group. Liu–Tian–Xiao proved the one-sided divisibility in both cases over that algebra (Theorem 5.1.3 of arXiv:2406.00624), and in Remark 1.2.10 they explicitly pose the $p$-integral strengthening — the same divisibilities over the $p$-integral Iwasawa algebra $\mathcal{O}_{E_\wp}[[\mathrm{Gal}(\mathcal{F}/F)]]$ — as an open problem, feasible in the authors' view under conditions on $(\Pi_0,\Pi_1)$ ensuring a good local newform theory. The problem is to prove that strengthening: it requires an integral normalization of the analytic side and yields control of $\mu$-invariants and integral torsion. No resolution in the higher-rank setting is known in the literature to date.

Background

The Iwasawa main conjecture predicts that a $p$-adic $L$-function (the analytic side of a motive over a $p$-adic tower of number fields) generates the characteristic ideal of a compactly realized Selmer group (the algebraic side). This record concerns the higher-rank anticyclotomic instance of that philosophy and, specifically, its refinement with $p$-integral coefficients.

Let $F/F^+$ be a CM field, i.e. a totally imaginary quadratic extension of a totally real field, with complex conjugation $c$. An anticyclotomic extension $\mathcal{F}/F$ is a Galois extension contained in the maximal abelian extension of $F$ such that the nontrivial element of $\mathrm{Gal}(F/F^+)$ acts on $\mathrm{Gal}(\mathcal{F}/F)$ by $-1$; it is called free when $\mathrm{Gal}(\mathcal{F}/F)$ is torsion-free. Fix an integer $n\geq 1$ and two cuspidal automorphic representations $\Pi_0$, $\Pi_1$ of $\mathrm{GL}_{n_0}(\mathbb{A}_F)$ and $\mathrm{GL}_{n_1}(\mathbb{A}_F)$ — irreducible representations of these adelic matrix groups arising from automorphic forms, the higher-rank generalization of modular-form representations — where $n_0$ and $n_1$ are the even and odd elements of $\{n,n+1\}$, satisfying: $\Pi_i\circ c\simeq\Pi_i^\vee$ (conjugate self-duality) and prescribed archimedean principal-series type (each archimedean component is induced from an explicit list of powers of the argument character $z\mapsto z/|z|$) — the "relevant" representations of Liu–Tian–Xiao. To such $\Pi_i$ one associates, for a $p$-adic place $\wp$ of a sufficiently large coefficient field $E$, a continuous Galois representation $\rho_{\Pi_i,\wp}$ of $F$, so that $\rho_{\Pi_0,\wp}\otimes\rho_{\Pi_1,\wp}(n)$ is the Rankin–Selberg motive, polarizable (self-dual up to a twist). An anticyclotomic extension is $\wp$-ordinary (with respect to $\Pi_0,\Pi_1$) if every prime of $F$ ramified in it divides $p$, splits over $F^+$, and both $\Pi_i$ are unramified and ordinary there — ordinary meaning that the local $p$-adic Galois representation admits a filtration whose graded pieces have Hodge–Tate weights ordered so that the top quotient is unramified with unit Frobenius eigenvalue.

For such a tower one studies Iwasawa-theoretic Bloch–Kato Selmer groups: the compact module $\mathcal{S}(\mathcal{F},\rho_{\Pi_0,\wp}\otimes\rho_{\Pi_1,\wp}(n))$, an inverse limit of the Selmer groups (cohomology classes satisfying the Bloch–Kato local conditions) of Galois-stable lattices over the finite layers of $\mathcal{F}/F$, and its Pontryagin dual $\mathcal{X}$ (the continuous dual with values in $\mathbb{Q}_p/\mathbb{Z}_p$); both are finitely generated over the Iwasawa algebra $\Lambda=\mathbb{Z}_p[[\mathrm{Gal}(\mathcal{F}/F)]]\otimes_{\mathbb{Z}_p}E_\wp$. When $\mathcal{F}/F$ is free, $\Lambda$ is a unique factorization domain and the characteristic ideal $\mathrm{char}(M)$ of a torsion $\Lambda$-module $M$ is defined. The global Rankin–Selberg root number $\epsilon(\Pi_0\times\Pi_1)=\pm 1$ — the sign in the functional equation of the Rankin–Selberg $L$-function — splits the theory into a coherent case ($+1$) and an incoherent case ($-1$).

The anticyclotomic theory in rank one goes back to Bertolini–Darmon (2005), who proved one divisibility of the main conjecture for rational elliptic curves over the anticyclotomic tower of an imaginary quadratic field using an Euler system of Heegner points, later streamlined by Howard as the bipartite Euler system — a combinatorial system of cohomology classes indexed by sets of level-raising primes, tied together by two explicit reciprocity laws. Y. Liu (arXiv:2306.07039) generalized the analytic and algebraic objects to arbitrary rank: when $\epsilon(\Pi_0\times\Pi_1)=1$ he constructed an anticyclotomic $p$-adic $L$-function $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)$ interpolating central $L$-values of twists, and when $\epsilon(\Pi_0\times\Pi_1)=-1$ a module $\mathcal{K}(\mathcal{F},\rho_{\Pi_0,\wp}\otimes\rho_{\Pi_1,\wp}(n))$ of Heegner-type classes generated by Abel–Jacobi images of Gan–Gross–Prasad diagonal cycles — special algebraic cycles on unitary Shimura varieties cut out by a diagonal subgroup — in the anticyclotomic tower, with quotient $\mathcal{T}=\mathcal{S}/\mathcal{K}$. He posed the main conjectures in both cases over the localized Iwasawa algebra $\mathbb{L}^\circ[[\mathrm{Gal}(\mathcal{F}/F)]]\otimes_{\mathbb{L}^\circ}\mathbb{L}$ (his notation $\mathbb{L}[[\mathrm{Gal}]]^\circ$, where $\mathbb{L}^\circ$ is the ring of integers of the coefficient field and the tensor with $\mathbb{L}$ inverts $p$): Conjecture 6.4, that $\mathrm{char}(\mathcal{X})$ is generated by $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)$; and Conjecture 7.7, a higher-rank analogue of the Heegner point main conjecture, that $\mathrm{char}(\mathcal{X}^{\mathrm{tor}})=\mathrm{char}(\mathcal{T})^2$.

Liu–Tian–Xiao (arXiv:2406.00624) proved one side of both conjectures over that same localized algebra — the divisibilities $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)\in\mathrm{char}(\mathcal{X})$ and $\mathrm{char}(\mathcal{T})^2\subseteq\mathrm{char}(\mathcal{X}^{\mathrm{tor}})$ (Theorem 5.1.3, under an admissibility hypothesis on $\wp$ and the parity restriction $n\leq 2$ when $F^+=\mathbb{Q}$, which is due to a hypothesis on automorphic forms on unitary Shimura varieties); the paper itself notes that the two cases of the theorem "confirm one direction of Conjecture 6.4 and Conjecture 7.7 in [Liu]". Their Remark 1.2.10 states that the main conjectures "we study in this article are the ones inverting $p$", that it is natural to ask for the analogous results in the $p$-integral Iwasawa algebra $\Lambda^\circ=\mathcal{O}_{E_\wp}[[\mathrm{Gal}(\mathcal{F}/F)]]$, that under conditions on $(\Pi_0,\Pi_1)$ ensuring a good local newform theory "it is possible to do this", and that they plan to pursue it. This $p$-integral strengthening — not the conjectures as posed, which already invert $p$ — is the open target of the present problem. The $p$-inverted statement controls only the $\lambda$-part of the modules (the codimension-one data); the integral statement additionally governs the $\mu$-part — the $p$-power torsion — and requires an integral normalization of the analytic side, since $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)$ is a priori only well-defined up to a scalar in $E_\wp^\times$. It also requires fixing Galois-stable lattices in the Galois representations: over $\Lambda^\circ$ the Selmer modules genuinely depend on that choice, a dependence invisible after inverting $p$.

Problem Statement

Prove, without inverting $p$, the $p$-integral strengthening of the one-sided divisibilities of the anticyclotomic Iwasawa main conjecture for Rankin–Selberg motives established by Liu–Tian–Xiao over the $p$-inverted (localized) Iwasawa algebra — that is, prove the same divisibilities over the $p$-integral Iwasawa algebra, the open problem posed in Remark 1.2.10 of arXiv:2406.00624, which frames it conditionally: the integral results are asserted to be possible "under certain conditions on $(\Pi_0,\Pi_1)$ so that we have a good new form theory locally". Such a proof implies, after inverting $p$, one divisibility in Conjectures 6.4 and 7.7 of Y. Liu (arXiv:2306.07039) as those conjectures are posed (over the localized algebra); the integral statement is strictly stronger and is the target here.

Setting: $F/F^+$ a CM field; $n\geq 1$ (at most $2$ when $F^+=\mathbb{Q}$); $\Pi_0$ and $\Pi_1$ relevant (conjugate self-dual, prescribed archimedean principal-series type) cuspidal automorphic representations of $\mathrm{GL}_{n_0}(\mathbb{A}_F)$ and $\mathrm{GL}_{n_1}(\mathbb{A}_F)$ with $\{n_0,n_1\}=\{n,n+1\}$ ($n_0$ even, $n_1$ odd); $E$ a strong coefficient field of both $\Pi_0$ and $\Pi_1$ (a sufficiently large $p$-adic coefficient field over which the attached Galois representations are defined, in the sense of the source papers); $\wp$ an admissible prime of $E$ in the sense of Definition 5.1.2 of arXiv:2406.00624; $\mathcal{F}/F$ a free $\wp$-ordinary anticyclotomic extension. Let $\Lambda^\circ=\mathcal{O}_{E_\wp}[[\mathrm{Gal}(\mathcal{F}/F)]]$ be the $p$-integral Iwasawa algebra. Fix Galois-stable $\mathcal{O}_{E_\wp}$-lattices in $\rho_{\Pi_0,\wp}\otimes\rho_{\Pi_1,\wp}(n)$ — their choice is part of the data, since over $\Lambda^\circ$ the Selmer modules depend on the choice — and let the Iwasawa Bloch–Kato Selmer modules $\mathcal{X}$ (Pontryagin dual), $\mathcal{S}$, the torsion submodule $\mathcal{X}^{\mathrm{tor}}$, and the Heegner-type submodule $\mathcal{K}\subseteq\mathcal{S}$ with quotient $\mathcal{T}=\mathcal{S}/\mathcal{K}$ be constructed over $\Lambda^\circ$ from those lattices (not merely after tensoring with $E_\wp$). The two requested branches, which are the two root-number cases of the single overarching objective of replacing the coefficient ring $\Lambda^\circ[1/p]$ by $\Lambda^\circ$ throughout Theorem 5.1.3 of arXiv:2406.00624, are:

(1) Coherent case, $\epsilon(\Pi_0\times\Pi_1)=1$: construct the anticyclotomic $p$-adic $L$-function $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)$ as an element of $\Lambda^\circ$ with a specified integral normalization — a measure or element defined up to a unit of $\mathcal{O}_{E_\wp}$, whose image in $\Lambda^\circ[1/p]$ coincides, up to a scalar in $E_\wp^\times$, with the $p$-adic $L$-function constructed from Bessel periods (period integrals of automorphic forms over torus orbits on unitary groups) and diagonal cycles in arXiv:2306.07039 and used in Theorem 5.1.3 — and prove that $\mathcal{L}_\mathcal{F}(\Pi_0\times\Pi_1)$ belongs to the characteristic ideal of $\mathcal{X}$ as a $\Lambda^\circ$-module.

(2) Incoherent case, $\epsilon(\Pi_0\times\Pi_1)=-1$: prove that the square of the characteristic ideal of $\mathcal{T}$ is contained in the characteristic ideal of $\mathcal{X}^{\mathrm{tor}}$, both taken over $\Lambda^\circ$.

The statement is considered established when both inclusions of characteristic ideals hold over $\Lambda^\circ$ under the hypotheses of Theorem 5.1.3 of arXiv:2406.00624 (including its nonvanishing hypotheses where stated), possibly strengthened by explicit conditions on $(\Pi_0,\Pi_1)$ ensuring a good local newform theory in the sense of Remark 1.2.10 of arXiv:2406.00624 — that remark poses the $p$-integral problem under such conditions; a proof of the full equalities — the integral analogue of the conjectures of Y. Liu — or of the accompanying torsion/rank assertions over $\Lambda^\circ$ exceeds and hence satisfies this target. Beyond the source's own hypotheses and such explicit good-newform conditions on $(\Pi_0,\Pi_1)$, which must be stated precisely and must not empty the admissible class, no restriction on $n$, on the ground field, on the number of ramified $p$-adic places, or on the answer's method is admitted; an unconditional proof for all relevant pairs likewise satisfies the target.

References

1. Y. Liu, Y. Tian, L. Xiao, "Iwasawa's main conjecture for Rankin–Selberg motives in the anticyclotomic case", 2024, arXiv:2406.00624, DOI 10.48550/arxiv.2406.00624, https://arxiv.org/abs/2406.00624 (v3, 25 Dec 2024)
2. Y. Liu, "Anticyclotomic p-adic L-functions for Rankin–Selberg product", 2023, arXiv:2306.07039, DOI 10.48550/arxiv.2306.07039, https://arxiv.org/abs/2306.07039 (poses Conjectures 6.4 and 7.7 over the localized, $p$-inverted Iwasawa algebra)
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