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Characteristic polynomials of isometries of even unimodular lattices

Yuta Takada

Abstract

E. Bayer-Fluckiger gave a necessary and sufficient condition for a polynomial to be realized as the characteristic polynomial of a semisimple isometry of an even unimodular lattice, by describing the local-global obstruction, and the author extended the result. This article presents a systematic way to compute the obstruction. As an application, we give a necessary and sufficient condition for a Salem number of degree $10$ or $18$ to be realized as the dynamical degree of an automorphism of nonprojective K3 surface, in terms of its minimal polynomial.

Characteristic polynomials of isometries of even unimodular lattices

Abstract

E. Bayer-Fluckiger gave a necessary and sufficient condition for a polynomial to be realized as the characteristic polynomial of a semisimple isometry of an even unimodular lattice, by describing the local-global obstruction, and the author extended the result. This article presents a systematic way to compute the obstruction. As an application, we give a necessary and sufficient condition for a Salem number of degree or to be realized as the dynamical degree of an automorphism of nonprojective K3 surface, in terms of its minimal polynomial.
Paper Structure (23 sections, 45 theorems, 123 equations, 1 table)

This paper contains 23 sections, 45 theorems, 123 equations, 1 table.

Table of Contents

  1. Introduction
  2. Recent developments
  3. Obstruction
  4. Main results
  5. Acknowledgments
  6. Inner products and isometries
  7. Inner product spaces and lattices
  8. Symmetric polynomials
  9. Isometries of inner product spaces
  10. Indices of isometries
  11. Local-global principle and obstruction
  12. Local conditions
  13. Local-global principle
  14. Local-global obstruction
  15. Computation of obstruction
  16. ...and 8 more sections

Key Result

Theorem 1.2

Let $r,s$ be non-negative integers with $r\equiv s \bmod 8$, $F\in \mathbb{Z}[X]$ a $*$-symmetric polynomial of degree $r+s$ with the conditions eq:Sign_intro and eq:Square_intro, and $\mathfrak{i}\in \operatorname{Idx}(r,s;F)$ an index map. Assume that each of $m_+$ and $m_-$ is $0$ or at least $3$

Theorems & Definitions (108)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 98 more