Ihara-Langlands-Kottwitz Method

• The Ihara-Langlands-Kottwitz method is a unified framework combining geometric stratification and harmonic analysis to compute cohomological invariants in Shimura varieties.
• It constructs central test functions in Hecke algebras using nearby cycles and deformation theory to equate geometric fixed-point counts with automorphic trace formulas.
• The method yields explicit evaluations of zeta factors, orbital integrals, and local Langlands correspondence elements, thus advancing arithmetic geometry and the Langlands program.

The Ihara-Langlands-Kottwitz method is a framework unifying geometric and harmonic analysis techniques to compute traces of Frobenius and Hecke operators on the cohomology of Shimura varieties and related moduli spaces in terms of automorphic representation theory. It synthesizes earlier approaches by Ihara (point counting), Langlands (linking zeta functions to automorphic L-functions), and Kottwitz (trace formulas via orbital integrals), achieving explicit formulas for zeta functions and cohomological invariants even at places of bad reduction. Central to the method is the construction of certain test functions in the Bernstein center of Hecke algebras—often via nearby cycles or deformation theory—whose twisted orbital integrals encode the arithmetic and automorphic structure of the underlying moduli problem.

1. Geometric and Harmonic Analysis Foundations

The method begins by studying integral models of Shimura varieties or related moduli spaces (e.g., modular curves, Siegel modular varieties, Rapoport-Zink spaces). The geometry is analyzed via stratification of special fibers and the paper of nearby cycles, elucidating the degeneration of cohomology.

Key tools:

• Definition of “depth” parameters $k(g)$ and supplementary invariants (e.g., $\ell(g)$ for $g\in GL_2(\mathbb Q_{p^r})$), encapsulating distances in the Bruhat-Tits building and spectral data (Scholze, 2010).
• Decomposition of nearby cycles reflecting moduli stratification, such as $$\pi_*R\Psi_1 = \bigoplus_{\chi: T(\mathbb F_p)^\times} R\Psi_\chi$$ in the Drinfeld case, indexed by characters of the covering torus (Haines et al., 2010).
• Construction of integral models and their stratification, enabling combinatorial descriptions of cohomology contributions at points in the special fiber via deformation theory and representation-theoretic data (Scholze, 2010).

This geometric description provides the input for interpreting counts and traces via harmonic analysis, specifically through orbital integrals and the Arthur–Selberg trace formula.

2. Test Functions and the Bernstein Center

A crucial ingredient is the explicit construction of test functions in the center of the Hecke algebra which mirror the geometry:

• For modular curves of bad reduction, the function $\varphi_{p,n}$ is defined piecewise by $p$-adic valuations of determinant and trace, and shown to equal the convolution $\varphi_p \ast e_{\Gamma(p^n)}$, lying in the Bernstein center (Scholze, 2010).
• In Drinfeld-type Shimura varieties with $\Gamma_1(p)$-level, one defines functions $\varphi_{r,\chi}$:

$$\varphi_{r,\chi}(t\,w^{-1}) = \delta^1(w^{-1},\chi)\, \chi_r^{-1}(t)\, k_{\mu^*,r}(w),$$

where $k_{\mu^*,r}(w)$ is a Bernstein/Kottwitz function reflecting admissible elements in the affine Weyl group, ultimately centralized by averaging over torus characters (Haines et al., 2010).

• Via Hecke algebra isomorphisms (Goldstein-Morris-Roche), these functions are transferred to affine Hecke algebras associated to Levi subgroups, allowing explicit spectral calculations using the Satake/Bernstein isomorphism.

Properties established include:

• Centrality with respect to convolution,
• Compatibility relations under induction and deformation,
• Explicit orbital integrals matching those associated to geometric fixed points.

3. Trace Formula Comparison and Orbital Integrals

The heart of the method lies in equating two expressions for arithmetic traces:

1. Geometric side: Fixed-point counts or traces of Frobenius-Hecke correspondences computed via Grothendieck-Lefschetz or Lefschetz-Verdier fixed-point formulas applied to the cohomology (or intersection cohomology) of Shimura varieties and their strata (e.g., Newton strata) (Liu, 5 Jul 2025, Keranen, 2016).
2. Spectral side: Arthur–Selberg trace formula expansions over automorphic representations, involving sums of orbital integrals of chosen test functions and weighted by multiplicities and volumes (Scholze, 2010, Keranen, 2016).

Explicit formulas include:

$$\operatorname{tr}^{\mathrm{ss}}(\Phi_{p^r}|H^*) = \sum_{\gamma_0}\sum_{\gamma,\delta} c(\gamma_0;\gamma,\delta) O_\gamma(f^p) \cdot TO_{\delta,\sigma}(\varphi),$$

where $O_\gamma(f^p)$ are orbital integrals away from $p$, $TO_{\delta,\sigma}(\varphi)$ are twisted orbital integrals at $p$—the twist arising from Frobenius automorphisms or base change.

Matching of orbital integrals (and twisted invariants) between geometric and automorphic functions, often via base-change identities, proves crucial (e.g., $\phi_{n,p}*h$ has orbital integrals matching those from nearby cycles (Scholze, 2010)).

4. Galois Representations and Zeta Functions

The method gives direct expressions for the local (and global) factors of zeta functions and Euler characteristics:

• Zeta factors: The semisimple local factor of the Hasse–Weil zeta function is expressed in automorphic terms, e.g.,

$$\zeta_{\mathfrak{p}}^{\mathrm{ss}}(Sh_K,s) = \prod_{\pi_f} L^{\mathrm{ss}}(s - \frac{n-1}{2}, \pi_p, r)$$

with $L$-functions and exponents determined by multiplicities and dimensions of $K$-invariants (Scholze, 2010).

• Galois–automorphic correspondence: Intersection cohomology and (compactly supported) Euler characteristics of local systems on moduli stacks are related to Galois representations through spectral decompositions indexed by Arthur parameters and spin/half-spin representations (Taïbi, 1 Oct 2025).
• Endoscopy and transfer: Endoscopic contributions are systematically included via stable trace formulas, as in Morel’s stabilization and the appearance of endoscopic transfer factors (Taïbi, 1 Oct 2025, Tam, 2012).

These results yield unconditional existence of GSpin-valued Galois representations for Siegel modular forms (level one), producing higher genus analogues of Deligne’s and Weissauer’s theorems (Taïbi, 1 Oct 2025).

5. Non-Hyperspecial Levels, Deformations, and Extensions

The method generalizes beyond hyperspecial level structures:

• At deeper level subgroups (e.g., Drinfeld level structures, pro-unipotent radicals), central test functions are constructed through stratification and combinatorial averaging, still yielding stable matches between geometric and automorphic data (Haines et al., 2010).
• Deformation spaces of $p$-divisible groups (in PEL/EL settings) are analyzed via universal deformation rings and rigid-analytic covers, giving rise to test functions $\varphi_{\tau,h}$ defined through traces on étale cohomology of the deformation space. These functions encode local geometric contributions even for bad reduction (Scholze, 2011).
• Compatibility of geometric and automorphic data is ensured through transfer (base change, Hecke algebra isomorphisms) and through identification of central elements in Hecke algebras via explicit formulas:

$$\mathrm{tr}\,\pi_p(f_{r,1}) = \dim(\pi_p^{I^+})\, p^{r\langle\rho,\mu^*\rangle}\, \mathrm{Tr}\big(r_{\mu^*}(\varphi'_{\pi_p}(\Phi^r)), V_{\mu^*}^{I_p}\big)$$

6. Transfer Relations, Endoscopic Matching, and Explicit Correspondence

The local Langlands correspondence, especially in the essentially tame case, is described through explicit “rectifier” factors (products of $x$-data) matched to endoscopic transfer factors. The automorphic induction character identity and the spectral transfer character identity coincide when normalized by standard Whittaker data (Tam, 2012). This yields “geometric” expressions making the local Langlands correspondence fully explicit, compatible with trace formula calculations and orbital integral matching—a cornerstone of the method’s efficacy.

In the context of non-quasi-split groups and inner forms, the B(G)-parametrization lifts the basic-packet approach to a full Kottwitz set parametrization, with compatibility ensured through highest-weight theory and Newton/Kottwitz maps, and supporting endoscopic character identities regularized for geometric lemma contributions (Meli et al., 2022).

7. Applications, Numerical Formulas, and Future Directions

Applications include:

• Explicit computation of the compactly supported Euler characteristic and point counts on Siegel and Picard modular varieties (Taïbi, 1 Oct 2025, Keranen, 2016), leveraging weight truncation and stable trace techniques.
• Extension to moduli of local shtukas, convolution and duality morphisms for mixed characteristic situations, and new cases of the Kottwitz conjecture for inner forms and minuscule cocharacters (Imai, 2019).
• Formulation of the method in families, giving a modular interpretation of the Ihara lemma via local Langlands correspondence in deformation settings (Sorensen, 2014).
• Geometric realization and spectral decomposition of cohomology of stacks of shtukas, confirming Arthur-Kottwitz conjectures for elliptic Langlands parameters (Lafforgue et al., 2018).

The method robustly extends to cases of bad reduction, deep level structures, and refined cohomological decompositions. Its intrinsic harmonization of geometry, harmonic analysis, and arithmetic representation theory continues to underpin advances in the Langlands program and arithmetic geometry.