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On the Linear AFL: The Non-Basic Case

Qirui Li, Andreas Mihatsch

Abstract

The linear Arithmetic Fundamental Lemma (AFL) conjecture compares intersection numbers on Lubin--Tate deformation spaces with derivatives of orbital integrals. It has been introduced for elliptic orbits in arXiv:1803.07553 and arXiv:2010.07365. In these cases, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the theory to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which is achieved by exploiting the connected-étale sequence. Our theory will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.

On the Linear AFL: The Non-Basic Case

Abstract

The linear Arithmetic Fundamental Lemma (AFL) conjecture compares intersection numbers on Lubin--Tate deformation spaces with derivatives of orbital integrals. It has been introduced for elliptic orbits in arXiv:1803.07553 and arXiv:2010.07365. In these cases, the relevant intersection problem is formulated for the basic isogeny class. In the present article, we extend the theory to all orbits and all isogeny classes. Our main result is a reduction of the non-basic cases of the AFL to the basic ones, which is achieved by exploiting the connected-étale sequence. Our theory will be relevant in the global setting, where also locally non-elliptic orbits may contribute in a non-trivial way.
Paper Structure (19 sections, 26 theorems, 133 equations)

This paper contains 19 sections, 26 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. The Fundamental Lemma
  3. Invariants
  4. Matching
  5. Orbital integrals for $(E_1, E_2)$
  6. $η$-Twisted orbital integrals for $(E_0, E_3)$
  7. Fundamental Lemma
  8. Reduction formula
  9. Vanishing order
  10. The Arithmetic Fundamental Lemma
  11. Quadratic cycles
  12. Hecke correspondences
  13. Intersection numbers
  14. The AFL
  15. Reduction to elliptic invariant
  16. ...and 4 more sections

Key Result

Theorem 1.2

Write $β = (β^0, β^1)$ for the two components of $β$ with respect to $\mathbb{X} = \mathbb{X}^0\times \mathbb{X}^1$. Then Conj. conj:intro holds for $β$ if and only if its hold for the connected component $β^0$.

Theorems & Definitions (61)

  • Conjecture 1.1: Linear AFL
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: HL
  • Remark 2.4
  • Proposition 2.5: HL*Prop. 2.5.6
  • proof
  • Definition 2.6
  • ...and 51 more