The unconditional Langlands-Kottwitz description of the cohomology of compact PEL Shimura varieties of type A at ramified split places at full $G(\mathbb A_f)$-level: general stable character identities for pseudo-stabilized test functions, the stable Bernstein center, the monodromy operator, and Hecke isolation of local $A$-packet representations

Scholze and Shin (JAMS 2013) described the cohomology of compact unitary group Shimura varieties at a ramified split place as a virtual $G(\mathbb A_f^p)\times G(\mathbb Z_p)\times W_{E_p}$-representation, and reduced the unconditional description for general ramified PEL data of type A at the full $G(\mathbb A_f)$-level to local inputs: their Conjectures 7.1/7.2 (stable character identities for the endoscopic transfers of the pseudo-stabilized test functions $\varphi_{\tau,h}$), their Conjecture 6.3 (a finite map from the Bernstein variety of supercuspidal supports to the variety of semisimple $L$-parameters, such that the induced action of the stable Bernstein center preserves vanishing stable orbital integrals), the determination of the monodromy operator $N$ of the Galois representations attached to the cohomology, and their Question 7.5 (linear independence of the restrictions of local $A$-packet members to a special maximal compact subgroup). The status of this interlinked cluster as of September 2026 is as follows. The geometry of the moduli of Langlands parameters has been established (Dat-Helm-Kurinczuk-Moss), Hansen has proved that the Fargues-Scholze excursion-operator construction of central distributions is very stable for every connected reductive group (which he describes as essentially confirming Conjecture 6.3), and Chi-Haines extended the vanishing and stable-Bernstein-center matching theorems for $\varphi_{\tau,h}$ to non-quasi-split inner forms of products of Weil restrictions of general linear groups; but the finiteness statement completing Conjecture 6.3, the character identities of Conjectures 7.1/7.2 for general ramified PEL data and arbitrary endoscopic triples, the determination of $N$ in general (the weight-monodromy conjecture for the attached Galois representations), and Question 7.5 all remain open, leaving a precise nonempty open core.

Background

Let $p$ be a prime. A Shimura variety is a higher-dimensional analogue of a modular curve: a projective system of algebraic varieties over a number field $E$ (the reflex field, cut out by the cocharacter data defining the variety) whose complex points parametrize abelian varieties carrying PEL structure - polarization, endomorphisms, and level structure, the acronym encoding these three ingredients - encoded by a reductive group $G$ over $\mathbb Q$. The Langlands-Kottwitz method describes the $\ell$-adic étale cohomology of these varieties, which carries commuting actions of $G(\mathbb A_f)$ (the finite-adele points of $G$, acting through Hecke correspondences; $\mathbb A_f$ denotes the finite adeles of $\mathbb Q$) and of the Galois group $\mathrm{Gal}(\overline E/E)$, in terms of automorphic representations and their local Langlands parameters. At hyperspecial (unramified) level this method is classical. Scholze and Shin studied the ramified case where $p$ is a split place: the local group $G_{\mathbb Q_p}$ is quasi-split (in their unconditional theorems, a product of Weil restrictions of general linear groups, realized by EL or quasi-EL data) while the PEL data themselves may ramify. Their main theorem describes the cohomology of the Shimura variety as a virtual $G(\mathbb A_f^p)\times G(\mathbb Z_p)\times W_{E_p}$-representation, where $W_{E_p}$ is the local Weil group at a prime $\mathfrak p\mid p$ of $E$; the restriction of the Hecke action at $p$ to $G(\mathbb Z_p)$ (the group of integral points of a smooth $\mathbb Z_p$-model of $G$, a special maximal compact subgroup of $G(\mathbb Q_p)$) is a feature of their method, and removing it is part of the open problem. The method passes through the Newton stratification of the special fibre of the integral model: strata indexed by isomorphism classes of the underlying $p$-divisible groups, the basic (supersingular) stratum being the one where this group is isoclinic, i.e. all slopes of its Frobenius are equal - the higher-dimensional generalization of the supersingular points of a modular curve.

The local input at $p$ is carried by test functions $\varphi_{\tau,h}\in C_c^\infty(G(\mathbb Q_{p^r}))$ defined by Scholze via the Lefschetz trace formula on deformation spaces of $p$-divisible groups: $\tau$ is an element of the local Weil group $W_E$ acting as the $j$-th power of geometric Frobenius ($r=jf$, with $f$ the inertia degree of $E/\mathbb Q_p$), and $h\in C_c^\infty(G(\mathbb Z_p))$ is a cut-off function. In the endoscopic decomposition of the cohomology, the relevant local packets are the $A$-packets of Arthur: finite multi-sets $\Pi(\psi)$ of irreducible admissible representations of $G(\mathbb Q_p)$ attached to local Arthur parameters, whose stable character $S\Theta_\varphi$ (the endoscopically stabilized combination of the characters of the members of a packet) is expected to be a stable distribution. The transfer of the test functions to endoscopic groups is controlled by the Bernstein center $Z(G)$ of $G(\mathbb Q_p)$ - the center of the category of smooth complex representations, realizable as the ring of invariant distributions acting by convolution - and by its stable variant $Z^{st}(G)$, the ring of regular functions on the variety $\Omega^{st}(G)$ of semisimple $L$-parameters (conjugacy classes of Weil-group homomorphisms into the $L$-group of $G$ with semisimple image); this relates to $\Omega(G)$, the Bernstein variety parametrizing supercuspidal supports of irreducible representations. A distribution is called stable if it kills every function whose stable orbital integrals vanish, and very stable if convolving any such function with it yields another such function. Fargues and Scholze associate to $G$ a spectral Bernstein center $\mathcal Z^{spec}(G)$, the ring of $\widehat G$-invariant functions on the moduli stack of $L$-parameters, together with a natural map $\Psi_G\colon\mathcal Z^{spec}(G)\to Z(G)$ to the Bernstein center (arXiv:2102.13459). Scholze-Shin's Conjecture 6.3 asserts that there is a natural finite map $\Omega(G)\to\Omega^{st}(G)$ such that the induced map $Z^{st}(G)\to Z(G)$ preserves vanishing stable orbital integrals. Their Conjectures 7.1/7.2 assert the stable character identities for the endoscopic transfers: for an endoscopic triple $(H,s,\eta)$ of $G$ - a quasi-split group $H$ over $\mathbb Q_p$ with an embedding $\eta\colon{}^L H\to{}^L G$ of $L$-groups and a semisimple element $s$ in the dual group of $H$ - the transfer $f^H_{\tau,h}$ of $\varphi_{\tau,h}$ satisfies, for every tempered $L$-parameter $\varphi_H$ with semisimple parameter $\lambda_H$ and every $\eta$-transfer $h^H$ of $h$,

$$S\Theta_{\varphi_H}(f^H_{\tau,h})=\mathrm{tr}\bigl(s^{-1}\tau\mid(r_{-\mu}\circ\eta\,\lambda_H)|_{W_E}\bigr)\cdot|E|^{\langle-\rho,\mu\rangle}\,S\Theta_{\varphi_H}(h^H),$$

where $\mu$ is the minuscule cocharacter attached to the Shimura datum, $r_{-\mu}$ the irreducible representation of the dual group of highest weight $-\mu$, $\rho$ the half-sum of the positive roots, and $|E|$ the size of the residue field of $E$ at $p$; Conjecture 7.2 equivalently states that $z^H_\tau\ast h^H$, with $z^{H,st}_\tau\in Z^{st}(H)$ the function taking a semisimple parameter $\lambda$ to the same trace factor and $z^H_\tau$ its image in $Z(H)$, is the twisted endoscopic transfer of $\varphi_{\tau,h}$.

On the Galois side, the cohomology realizes $\ell$-adic Galois representations $R_{\ell,\iota}(\Pi)$ attached to automorphic representations $\Pi$ (relative to an isomorphism $\iota\colon\overline{\mathbb Q}_\ell\cong\mathbb C$). At finite places $y\nmid\ell$ the comparison with the local Langlands correspondence produces a Weil-Deligne representation - a pair consisting of a representation of the Weil group together with a nilpotent monodromy operator $N$ satisfying the standard commutation relation - and Scholze-Shin's Theorem 11.1 establishes this compatibility except possibly for the determination of $N$; deciding $N$ in general is the weight-monodromy conjecture for these Galois representations, and in the closing remarks of their Section 11 Scholze-Shin reduce it to a stratum-by-stratum description of the cohomology of the Newton strata: replacing the pseudo-stabilized test function $\varphi_{\tau,h}$ by $\varphi_{\tau,h}\chi_b$, where $\chi_b$ is the characteristic function of the $\sigma$-conjugacy class of $b$ (conjugacy for the Frobenius-twisted action on $G(\mathbb Q_{p^r})$; these classes form Kottwitz's set $B(G)$ indexing the Newton strata), describes the cohomology of the stratum indexed by $b$; the paper leaves the full execution of this stratum-by-stratum description, and with it the determination of $N$ in general, as further work. Fargues-Scholze's geometrization of the local Langlands correspondence - the construction of excursion operators on the moduli stack of $L$-parameters, from which every smooth irreducible representation of a $p$-adic reductive group can be recovered - supplies the modern framework in which several of these local inputs have recently been settled.

Problem Statement

Prove the Langlands-Kottwitz description of the cohomology of a compact PEL Shimura variety of type A at ramified split places, at the full $G(\mathbb A_f)$-level, unconditionally, by establishing the local inputs that Scholze-Shin (arXiv:1110.0232, Journal of the American Mathematical Society 26 (2013), 261-294) reduced it to. Their theorem describes the full cohomology as a virtual $G(\mathbb A_f^p)\times G(\mathbb Z_p)\times W_{E_p}$-representation for EL and quasi-EL data and, for general ramified PEL data of type A, is conditional (their conditional verification covers type A or C, assuming global hypotheses on the A-packet classification and the stable trace formula). The problem is to establish the remaining inputs for general ramified PEL data of type A: (1) prove Conjectures 7.1/7.2, the stable character identities satisfied by the pseudo-stabilized functions $\varphi_{\tau,h}$ and their endoscopic transfers $f^H_{\tau,h}$ [the identity displayed in the background], for arbitrary endoscopic triples $(H,s,\eta)$ and tempered $L$-parameters, beyond the established cases (EL and quasi-EL data, all data unramified, and the non-quasi-split inner forms of products of Weil restrictions of general linear groups treated by Chi-Haines); (2) prove the remaining parts of Conjecture 6.3, namely the existence of a natural finite map $\Omega(G)\to\Omega^{st}(G)$ from the Bernstein variety of supercuspidal supports to the variety of semisimple $L$-parameters such that the induced map $Z^{st}(G)\to Z(G)$ preserves vanishing stable orbital integrals, and prove in the geometrized setting the identification of the stable (equivalently very stable) Bernstein center with the image, inside $Z(G)$, of the Fargues-Scholze map $\Psi_G\colon\mathcal Z^{spec}(G)\to Z(G)$, beyond Hansen's theorem that $\Psi_G$ factors through very stable central distributions; (3) determine the monodromy operator $N$ of the Weil-Deligne representation attached to $R_{\ell,\iota}(\Pi)$ at places $y\nmid\ell$ of the reflex field, i.e. establish the weight-monodromy conjecture for these Galois representations beyond the known square-integrable, non-slightly-regular, rank-two, and density-one cases; and (4) resolve Question 7.5 affirmatively: for a connected reductive group $G$ over a $p$-adic field $F$, a special maximal compact subgroup $K\subset G(F)$, and a local $A$-packet $\Pi(\psi)=\{\pi_1,\dots,\pi_k\}$, prove that the restrictions $\pi_1|_K,\dots,\pi_k|_K$ are linearly independent in the Grothendieck group of admissible representations of $K$ - thereby upgrading the cohomology description from the $G(\mathbb A_f^p)\times G(\mathbb Z_p)$-level to the full $G(\mathbb A_f)$-level. All four components are jointly needed: the cohomology description at full level requires the stable transfer (2), the character identities (1), the Galois-side input (3), and the Hecke isolation (4) simultaneously.

References

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