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Eisenstein Cohomology and Critical Values of Certain $L$-Functions: The Case $G_2$

Farid HosseiniJafari

TL;DR

The paper extends the Harder–Raghuram program to the exceptional group $G_2$ by linking Eisenstein cohomology to the rationality of ratios of critical Langlands–Shahidi $L$-values, notably for the adjoint cube $L$-functions and the central character $L$-functions arising in the constant term of Eisenstein series. It develops a full cohomological framework—strong purity, strongly inner cohomology, and Tate twists—together with a detailed treatment of both non-archimedean and archimedean intertwining operators, culminating in a rank-one Eisenstein cohomology theorem and a Manin–Drinfeld principle for $G_2$. The main results establish arithmeticity: the ratios of consecutive critical values are algebraic up to powers of the discriminant $|\,\delta_{F/\mathbb Q} frac{5}{2}|$ and transform predictably under $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$, matching motivic expectations for the corresponding cohomological automorphic data. This work thus generalizes the HR approach to a setting with multiple $L$-functions and an exceptional ambient group, paving the way for broader applications to rationality questions in Langlands–Shahidi theory and Deligne's conjecture in automorphic contexts.

Abstract

We establish results on the rationality of ratios of successive critical values of Langlands-Shahidi $L$-functions, as they appear in the constant term of the Eisenstein series associated with the exceptional group of type $G_2$ over a totally imaginary number field. Furthermore, we prove the rationality of the critical values for each $L$-function in the products, such as the symmetric cube $L$-functions. Our method generalizes the Harder-Raghuram method to cases where multiple $L$-functions appear in the constant term and involve an exceptional group. Finally, our results on the automorphic version of Deligne's conjecture align with its motivic counterpart, as demonstrated in the recent work of Deligne and Raghuram.

Eisenstein Cohomology and Critical Values of Certain $L$-Functions: The Case $G_2$

TL;DR

The paper extends the Harder–Raghuram program to the exceptional group by linking Eisenstein cohomology to the rationality of ratios of critical Langlands–Shahidi -values, notably for the adjoint cube -functions and the central character -functions arising in the constant term of Eisenstein series. It develops a full cohomological framework—strong purity, strongly inner cohomology, and Tate twists—together with a detailed treatment of both non-archimedean and archimedean intertwining operators, culminating in a rank-one Eisenstein cohomology theorem and a Manin–Drinfeld principle for . The main results establish arithmeticity: the ratios of consecutive critical values are algebraic up to powers of the discriminant and transform predictably under , matching motivic expectations for the corresponding cohomological automorphic data. This work thus generalizes the HR approach to a setting with multiple -functions and an exceptional ambient group, paving the way for broader applications to rationality questions in Langlands–Shahidi theory and Deligne's conjecture in automorphic contexts.

Abstract

We establish results on the rationality of ratios of successive critical values of Langlands-Shahidi -functions, as they appear in the constant term of the Eisenstein series associated with the exceptional group of type over a totally imaginary number field. Furthermore, we prove the rationality of the critical values for each -function in the products, such as the symmetric cube -functions. Our method generalizes the Harder-Raghuram method to cases where multiple -functions appear in the constant term and involve an exceptional group. Finally, our results on the automorphic version of Deligne's conjecture align with its motivic counterpart, as demonstrated in the recent work of Deligne and Raghuram.
Paper Structure (45 sections, 23 theorems, 303 equations, 15 figures, 3 tables)

This paper contains 45 sections, 23 theorems, 303 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The base field
  4. The groups
  5. Locally symmetric spaces
  6. Characters of the torus and its parametrizations
  7. Cohomology of the arithmetic groups in $G_2$
  8. Cohomology of arithmetic groups as a sheaf cohomology
  9. Cohomology of boundary components
  10. Kostant theorem
  11. Inner cohomology
  12. Cohomology at transcendental level
  13. Langlands-Shahidi method: Case $G_2$
  14. The case $P_{\beta}$
  15. The case $P_{\alpha}$
  16. ...and 30 more sections

Key Result

Theorem 1

Assume that $^{\iota}\sigma$ is non-monomial. For any two successive critical points $m$ and $m+1$ of both $L$-functions $L(s,Ad^{3}({^{\iota}\sigma}))$ and $L(2s,\omega_{{^{\iota}\sigma}})$, we have:

Figures (15)

  • Figure 1: Cohomological Langlands-Shahidi method via Eisenstein cohomology
  • Figure 2: Root Lattice of $\mathbf{G_2}$, Dominant chamber is shaded
  • Figure 3: Parametrizations of the maximal torus $T_0(F)$
  • Figure 4: Embedding of Fields in Totally Imaginary Case
  • Figure 5: Simple Factors $P_{\beta}$ of $G_2$
  • ...and 10 more figures

Theorems & Definitions (56)

  • Theorem 1: Theorem \ref{['thm:mainbeta']}
  • Theorem 2: Theorem \ref{['thm:mainalpha']}
  • Remark
  • Remark
  • Remark
  • Definition 3.1
  • Definition 4.1
  • Remark
  • Definition 4.2
  • Proposition 4.1
  • ...and 46 more