Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Multiplicative Formula of Langlands for Orbital Integrals in GL(2)

Malors Espinosa

Abstract

Langlands has introduced a formula for a specific product of orbital integrals in $\mbox{GL}(2, \mathbb{Q})$. Altuğ employs this formula to manipulate the regular elliptic part of the trace formula, with the aim of eliminating the contribution of the trivial representation from the spectral side. Arthur predicts that this formula coincides with a product of polynomials associated with zeta functions of orders developed by Zhiwei Yun. In a previous paper, the author determined the explicit polynomials for the relevant quadratic orders. This paper demonstrates how these polynomials can effectively generalize Langlands' formula to $GL(2, K)$, for general algebraic number fields $K$. Furthermore, we also use this formula to extend a well-known formula of Zagier to any algebraic number field and explain its applications in the contexts of the strategy of Beyond Endoscopy proposed by Langlands.

The Multiplicative Formula of Langlands for Orbital Integrals in GL(2)

Abstract

Langlands has introduced a formula for a specific product of orbital integrals in . Altuğ employs this formula to manipulate the regular elliptic part of the trace formula, with the aim of eliminating the contribution of the trivial representation from the spectral side. Arthur predicts that this formula coincides with a product of polynomials associated with zeta functions of orders developed by Zhiwei Yun. In a previous paper, the author determined the explicit polynomials for the relevant quadratic orders. This paper demonstrates how these polynomials can effectively generalize Langlands' formula to , for general algebraic number fields . Furthermore, we also use this formula to extend a well-known formula of Zagier to any algebraic number field and explain its applications in the contexts of the strategy of Beyond Endoscopy proposed by Langlands.
Paper Structure (9 sections, 20 theorems, 171 equations)

This paper contains 9 sections, 20 theorems, 171 equations.

Table of Contents

  1. Introduction and Motivation
  2. Context of the problem
  3. The work of this paper
  4. Explicit Zeta Functions of Certain Orders
  5. Multiplicative Formula of Langlands
  6. The Congruence Conditions
  7. Generalization of Zagier's Zeta Function
  8. Relevance of our work in the strategy of BE
  9. Questions and future work

Key Result

Theorem 1

(Theorem generalformula in section Multiplicative Formula of Langlands) Let $k_\mathfrak{q}$ be a nonnegative integer for each prime $\mathfrak{q}$ of $K$ with $k_\mathfrak{q} = 0$ for all but finitely many $\mathfrak{q}$. Let and $\gamma$ a contributing matrix for this $f$. We have the expansion where $\mathfrak{d}' = \dfrac{S_{\gamma}}{\mathfrak{d}}$. It is entire and satisfies the functional

Theorems & Definitions (52)

  • Theorem 1
  • Proposition 1
  • Conjecture 1
  • Definition 1
  • Proposition 2
  • Theorem 2
  • Definition 2
  • Proposition 3
  • Definition 3
  • Definition 4
  • ...and 42 more