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Zeta Functions of Certain Quadratic Orders

Malors Espinosa

Abstract

Langlands provides a formula for certain product of orbital integrals in $GL(2, \mathbb{Q})$. Its generalization has become an important question for the strategy of Beyond Endoscopy. Arthur predicts this formula should coincide with a product of polynomials associated to zeta functions of orders constructed by Zhiwei Yun. In this paper we compute, for a certain family of orders, explicit formulas for these zeta functions by a recursive method. We use these zeta functions in a further paper to prove that Arthur's prediction is correct.

Zeta Functions of Certain Quadratic Orders

Abstract

Langlands provides a formula for certain product of orbital integrals in . Its generalization has become an important question for the strategy of Beyond Endoscopy. Arthur predicts this formula should coincide with a product of polynomials associated to zeta functions of orders constructed by Zhiwei Yun. In this paper we compute, for a certain family of orders, explicit formulas for these zeta functions by a recursive method. We use these zeta functions in a further paper to prove that Arthur's prediction is correct.
Paper Structure (5 sections, 32 theorems, 125 equations)

This paper contains 5 sections, 32 theorems, 125 equations.

Table of Contents

  1. Introduction and Results of this Article
  2. Zeta Functions of Orders
  3. Arithmetic of the Orders $\mathcal{O}_n$
  4. Study of the Principal Part
  5. Solution of the Recurrence

Key Result

Theorem 1.1

(Theorem solutionrecurrence in section solutionoftherecurrence) For each $n\ge 0$ define the following polynomials: and for $n\ge 1$ define Finally, also put $U_0(X) = S_0(X) = 1$. Explicitly, these polynomials are Then the solution of the ramified, unramified and split case recurrence, respectively, satisfy

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • ...and 58 more