L-Function as Spectral Determinant

All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) where π is a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases.

L-Function as Spectral Determinant

Summary

All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) where π is a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases.

Statement

\label{def:l-function-spectral-determinant}
Let $\pi$ be an automorphic representation of $\mathrm{GL}_m(\mathbb{A}_\tau)$ with local components $\pi_p$ at each prime $p$. The \textbf{spectral $L$-function} is
\begin{equation}\label{eq:ch27-spectral-l-function}
L(s, \pi) = \prod_{p} L_p(s, \pi_p) = \det\left(I - s^{-1} H_\pi\right)^{-1},
\end{equation}
where $H_\pi$ is the adelic operator on $H_L^{\text{ad}}$ with local eigenvalues $\lambda_p = p^{-\rho_p}$ determined by the Satake parameters $\rho_p$ of $\pi_p$.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-03.jsonl line 76
• Manuscript source: 2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch27-the-grand-grh.tex lines 113-119

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookIII.Doors.GrandGRH
• Name: l_function_spectral_check

Dependencies

• Canonical: III.D16, III.D22

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.