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Lubin's conjecture for height-one $p$-adic dynamical systems over $(p^2-p)$-tame extensions

Martin Debaisieux

Abstract

To a height-$1$ formal group defined over the ring of integers of a finite extension $K$ of $\mathbb{Q}_p$ is attached its $p$-adic Tate module, which is a crystalline character of $\mathrm{Gal}(\overline K/K)$ of Hodge-Tate weight $1$. This association is an equivalence. We prove, over extensions whose ramification index is relatively prime to $p^2-p$, that the set of consistent sequences attached to a height-$1$ commuting pair $(f, u)$ of noninvertible and invertible formal power series is a crystalline character of weight $1$, for which $f$ is an endomorphism. As a result, we deduce a proof of a conjecture of Lubin in new cases.

Lubin's conjecture for height-one $p$-adic dynamical systems over $(p^2-p)$-tame extensions

Abstract

To a height- formal group defined over the ring of integers of a finite extension of is attached its -adic Tate module, which is a crystalline character of of Hodge-Tate weight . This association is an equivalence. We prove, over extensions whose ramification index is relatively prime to , that the set of consistent sequences attached to a height- commuting pair of noninvertible and invertible formal power series is a crystalline character of weight , for which is an endomorphism. As a result, we deduce a proof of a conjecture of Lubin in new cases.
Paper Structure (27 sections, 11 theorems, 43 equations)

This paper contains 27 sections, 11 theorems, 43 equations.

Table of Contents

  1. The dynamical system
  2. Logarithm
  3. Torsion points
  4. The latent Tate module
  5. The Galois action on f-consistent sequences
  6. Action on cells
  7. Action on the whole set
  8. Fontaine's period rings
  9. The integers of the tilt of Cp
  10. Blank period rings
  11. The de Rham period field
  12. The crystalline period ring
  13. Application to formal groups
  14. The universal fr-consistent ring
  15. Regularity of the Galois action
  16. ...and 12 more sections

Key Result

Proposition 1.1

For every integer $n \geqslant 1$, the endpoints of the decreasing part of the Newton polygon of $f^{\circ n}\!$ are the $(p^i, e(n-i))$ for each integer $i \in \{0, \dots, n\}$.

Theorems & Definitions (25)

  • Proposition 1.1
  • proof
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 15 more