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 $\mbox{GL}(2, \mathbb{Q})$. Altu\u{g} 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.