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Intersections in Lubin-Tate space and biquadratic fundamental lemmas

Benjamin Howard, Qirui Li

Abstract

We compute the intersection multiplicities of special cycles in Lubin-Tate spaces, and formulate a new arithmetic fundamental lemma relating these intersections to derivatives of orbital integrals.

Intersections in Lubin-Tate space and biquadratic fundamental lemmas

Abstract

We compute the intersection multiplicities of special cycles in Lubin-Tate spaces, and formulate a new arithmetic fundamental lemma relating these intersections to derivatives of orbital integrals.

Paper Structure

This paper contains 33 sections, 42 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. An intersection problem
  3. Invariant polynomials
  4. The intersection formula
  5. Matching and fundamental lemmas
  6. Global analogues
  7. Acknowledgements
  8. Invariants of algebra embeddings
  9. Noether-Skolem plus epsilon
  10. Invariant polynomials
  11. The functional equation
  12. The elements $\mathbf{w}$ and $\mathbf{z}$
  13. Regular semisimple pairs
  14. The biquadratic fundamental lemma
  15. Orbital integrals for $(\Phi_1,\Phi_2)$
  16. ...and 18 more sections

Key Result

Theorem A

If $(\Phi_1,\Phi_2)$ is regular semisimple then $|R(g)| \neq 0$ for all $g\in \mathrm{GL}_{2h}(\mathcal{O}_F)$, and the intersection multiplicity intro intersection satisfies In particular, the left hand side is finite. Here $d_i \in \mathcal{O}_F$ is any generator of the discriminant of $K_i/F$, and

Theorems & Definitions (113)

  • Theorem A
  • Remark 1.4.1
  • Theorem 2.1.1: Noether-Skolem
  • proof
  • Definition 2.1.2
  • Corollary 2.1.3
  • proof
  • Remark 2.1.4
  • Lemma 2.2.1
  • proof
  • ...and 103 more