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Characteristic Epsilon Cycles of $\ell$-adic Sheaves on Varieties

Daichi Takeuchi

Abstract

Let $X$ be a smooth variety over a finite field $\mathbb{F}_q$. Let $\ell$ be a rational prime number invertible in $\mathbb{F}_q$. For an $\ell$-adic sheaf $\mathcal{F}$ on $X$, we construct a cycle supported on the singular support of $\mathcal{F}$ whose coefficients are $\ell$-adic numbers modulo roots of unity. It is a refinement of the characteristic cycle $CC(\mathcal{F})$, in the sense that it satisfies a Milnor-type formula for local epsilon factors. After establishing fundamental results on the cycles, we prove a product formula of global epsilon factors modulo roots of unity. We also give a generalization of the results to varieties over general perfect fields.

Characteristic Epsilon Cycles of $\ell$-adic Sheaves on Varieties

Abstract

Let be a smooth variety over a finite field . Let be a rational prime number invertible in . For an -adic sheaf on , we construct a cycle supported on the singular support of whose coefficients are -adic numbers modulo roots of unity. It is a refinement of the characteristic cycle , in the sense that it satisfies a Milnor-type formula for local epsilon factors. After establishing fundamental results on the cycles, we prove a product formula of global epsilon factors modulo roots of unity. We also give a generalization of the results to varieties over general perfect fields.

Paper Structure

This paper contains 20 sections, 62 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Relative Singular Supports and Characteristic Cycles (Bei, relsing, Sai17)
  3. Relative singular support
  4. Flat function and characteristic cycle
  5. Reminder on good pencils
  6. Local Epsilon Factors (cf. Del, Y1, Y3)
  7. Generalities on local epsilon factors
  8. Reminder on oriented product and local Fourier transform (CLE)
  9. Reduction to the case of positive characteristic
  10. Local epsilon factors of convolutions
  11. Epsilon Cycles of $\ell$-adic Sheaves
  12. Group of characters modulo torsion
  13. Constructions of epsilon cycles
  14. Properties of epsilon cycles
  15. Epsilon cycles for tamely ramified sheaves
  16. ...and 5 more sections

Key Result

Theorem 1.1

(Theorem epcygenmil) Let $X$ be a smooth variety over a finite field $k$. Let $\mathcal{F}$ be an element of $D^b_c(X,\overline{\mathbb{Z}_\ell})$. Write $SS({\cal F})=\bigcup_aC_a$ for the decomposition into irreducible components of the singular support. Then there exists a unique cycle called the epsilon cycle, which satisfies the following property. For any diagram of $k$-schemes with $j$ ét

Theorems & Definitions (94)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 84 more