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Arithmetic fundamental lemma for the spherical Hecke algebra.

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This study introduces Hecke operators on unitary RZ spaces, exploring their geometric properties and proposing new conjectures for the spherical Hecke algebra. These conjectures are proven for a specific mathematical case.

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Area of Science:

  • Algebraic Geometry
  • Number Theory
  • Representation Theory

Background:

  • Unitary RZ spaces are fundamental objects in number theory.
  • Hecke operators and correspondences play a crucial role in the study of automorphic forms and related structures.
  • Understanding the geometric and arithmetic properties of these operators is essential for advancing the field.

Purpose of the Study:

  • To define and investigate Hecke correspondences and operators on unitary RZ spaces.
  • To study the fundamental geometric properties of these operators, including a commutativity conjecture.
  • To formulate and prove new conjectures related to the arithmetic fundamental lemma and spherical Hecke functions.

Main Methods:

  • Definition of Hecke correspondences and operators on unitary RZ spaces.
  • Analysis of basic geometric properties, including commutativity.
  • Formulation of arithmetic fundamental lemma and abundance conjectures for spherical Hecke algebra.
  • Proof of conjectures for the specific case of .

Main Results:

  • Established definitions for Hecke correspondences and operators on unitary RZ spaces.
  • Investigated geometric properties and proposed a commutativity conjecture for Hecke operators.
  • Formulated novel conjectures concerning the arithmetic fundamental lemma and spherical Hecke functions.
  • Successfully proved these conjectures for the case.

Conclusions:

  • The study provides a foundational framework for Hecke theory on unitary RZ spaces.
  • The proven conjectures offer significant insights into the arithmetic properties of spherical Hecke algebras.
  • This work opens avenues for further research into the behavior of Hecke operators in more general settings.