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Global B(G) with adelic coefficients and transfer factors at non-regular elements

Alexander Bertoloni Meli

Abstract

The goal of this paper is extend Kottwitz's theory of $B(G)$ for global fields. In particular, we show how to extend the definition of "$B(G)$ with adelic coefficients" from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for non-regular semisimple elements of non-quasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties.

Global B(G) with adelic coefficients and transfer factors at non-regular elements

Abstract

The goal of this paper is extend Kottwitz's theory of for global fields. In particular, we show how to extend the definition of " with adelic coefficients" from tori to all connected reductive groups. As an application, we give an explicit construction of certain transfer factors for non-regular semisimple elements of non-quasisplit groups. This generalizes some results of Kaletha and Taibi. These formulas are used in the stabilization of the cohomology of Shimura and Igusa varieties.

Paper Structure

This paper contains 19 sections, 28 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Global B(G) with Adelic Coefficients
  3. Naturality with respect to M,Y, Y-bar, X
  4. Changing G
  5. The map Phi(f,g,h)
  6. Psi(g,h)
  7. Compatibility of Phi and Psi
  8. Localization
  9. Total Localization Map
  10. Basic Subsets
  11. Some Maps of Global Cohomology
  12. Key global diagram: Gder simply connected case
  13. Key global diagram: general case
  14. Inflation
  15. Normalizing Transfer Factors
  16. ...and 4 more sections

Key Result

Theorem \oldthetheorem

Suppose $F$ is a number field and $G$ is a connected reductive group over $F$ that satisfies the Hasse principle and has simply connected derived subgroup. Then the theory of $H^1_{\mathrm{alg}}(\mathcal{E}_2, G(\overline{\mathbb{A}_F}))$ gives an explicit normalization of the transfer factors betwe

Theorems & Definitions (63)

  • Theorem \oldthetheorem: Imprecise version of Theorem \ref{['localtrans']}
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 53 more