[Paper Review] Zeta Functions of F1-buildings

Anton Deitmar, Ming-Hsuan Kang|arXiv (Cornell University)|Mar 27, 2013

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper develops a zeta function for $\mathbb{F}_1$-buildings—specifically, the apartment of a $\mathrm{PGL}_n(\mathbb{Q}_1)$-type group—using trace formulas and group-theoretic methods. It establishes a generalized Ihara formula linking the zeta function to the Langlands $L$-function of the quotient group, with the reciprocal of the zeta function expressed as a determinant involving adjacency operators and regular representations, valid even in the $q=1$ limit.

ABSTRACT

The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.

Motivation & Objective

• To extend the classical Ihara zeta function from graphs to the geometry of $\mathbb{F}_1$-buildings, particularly the apartment of a $\mathrm{PGL}_n$-type group.
• To formulate a trace formula for $\mathrm{PGL}_n(\mathbb{Q}_1)$, treating it as the analog of a $p$-adic group in $\mathbb{F}_1$-geometry.
• To define a several-variable zeta function $S_{\Gamma}(u)$ encoding the group-theoretic structure of the building quotient $\Gamma\backslash\mathcal{B}$, and to interpret it geometrically via closed geodesics.
• To generalize the Ihara formula to the $\mathbb{F}_1$-setting by relating the zeta function to the Langlands $L$-function of the quotient group $\Lambda/\Gamma$.
• To provide a geometric and representation-theoretic foundation for zeta functions in the context of the field with one element, particularly in the absence of a full building structure at $q=1$.

Proposed method

• The authors use the $\mathbb{F}_1$-approach via monoids to define $\mathbb{Q}_1$ as the infinite cyclic group, enabling the construction of the building of $\mathrm{PGL}_n(\mathbb{Q}_1)$ as an apartment with extended affine Weyl group symmetry.
• They derive a trace formula for $\mathrm{PGL}_n(\mathbb{Q}_1)$, computing its unitary dual and constructing the several-variable zeta function $S_{\Gamma}(u)$ from the trace of the regular representation.
• The zeta function $Z_+(X_\Gamma, u)$ is defined as an Euler product over positive closed geodesics in the quotient graph $\Gamma\backslash\mathcal{B}$, with lengths determined by the order of group elements in $\Lambda/\Gamma$.
• The key technical step involves expressing the reciprocal of $Z_+(X_\Gamma, u)$ as a determinant: $Z_+^{-1} = \det(I_N - A_1u + \cdots + (-1)^n u^n I_N)$, where $A_i$ are adjacency operators associated with $i$-tuples of generators.
• The authors define the Langlands $L$-function $L(\Lambda/\Gamma, u)$ as a product over characters $\rho$ of $\Lambda/\Gamma$, with $L(\rho, u) = \prod_{j=1}^n (1 - \rho_j u)^{-1}$, and show that $Z_+(X_\Gamma, u) = L(\Lambda/\Gamma, u)$.
• They prove that when $u = (x, 0, \dots, 0)$, the function $S_\Gamma(x, 0, \dots, 0)$ equals $(n-1)! \cdot \frac{Z_+^\prime}{Z_+}(x)$, linking the group-theoretic zeta function to the logarithmic derivative of the geometric zeta function.

Experimental results

Research questions

• RQ1 How can the Ihara zeta function be generalized from graphs to the $\mathbb{F}_1$-building setting, particularly for the apartment of $\mathrm{PGL}_n(\mathbb{Q}_1)$?
• RQ2 What is the role of the trace formula in defining a zeta function for $\mathrm{PGL}_n(\mathbb{Q}_1)$, and how does it relate to the unitary dual and regular representation?
• RQ3 Can the classical Ihara formula be extended to the $q=1$ case via a geometric and representation-theoretic construction?
• RQ4 How does the zeta function $S_\Gamma(u)$ encode the structure of the quotient building $\Gamma\backslash\mathcal{B}$, and what is its geometric interpretation in terms of closed geodesics?
• RQ5 What is the precise relationship between the zeta function $Z_+(X_\Gamma, u)$ and the Langlands $L$-function of the quotient group $\Lambda/\Gamma$ in the $\mathbb{F}_1$-setting?

Key findings

• The reciprocal of the zeta function $Z_+(X_\Gamma, u)^{-1}$ converges to a polynomial and is expressed as a determinant: $\det(I_N - A_1u + \cdots + (-1)^n u^n I_N)$, where $A_i$ are adjacency operators on the quotient graph.
• The zeta function $Z_+(X_\Gamma, u)$ is equal to the Langlands $L$-function $L(\Lambda/\Gamma, u)$, defined as the product over characters $\rho$ of $\Lambda/\Gamma$ of $\prod_{j=1}^n (1 - \rho_j u)^{-1}$, establishing a direct link between zeta and $L$-functions in the $\mathbb{F}_1$-setting.
• When $u = (x, 0, \dots, 0)$, the several-variable zeta function satisfies $S_\Gamma(x, 0, \dots, 0) = (n-1)! \cdot \frac{Z_+^\prime}{Z_+}(x)$, showing a precise connection between the group-theoretic and geometric zeta functions.
• The contribution of each primitive geodesic of length $m_i$ to the zeta function is $(1 - u^{m_i})^{N/m_i}$, which equals $\det(I_N - \lambda(s_i)u)$, where $\lambda$ is the regular representation of $\Lambda/\Gamma$, confirming the determinant formula.
• The proof relies on expressing the product $\prod_{i=1}^n \det(I_N - \lambda(s_i)u)$ as the determinant of the characteristic polynomial of the adjacency operators, thereby verifying the generalized Ihara formula.
• The result holds even in the $q=1$ limit, providing a consistent $\mathbb{F}_1$-analogue of the Ihara formula, with the Euler characteristic of the torus $\mathcal{B}_\Gamma$ being zero, so no correction factor is needed in the final identity.