Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the derived Tate curve and global smooth Tate $K$-theory

Jack Morgan Davies, Sil Linskens

Abstract

The interplay between equivariant stable homotopy theory and spectral algebraic geometry is used to construct a derived Tate curve over $\mathrm{KU}((q))$, a lift of the classical elliptic curve of Tate over $\mathbf{Z}((q))$. Applications of both an algebro-geometric and a topological flavour follow. First, we construct a spectral algebro-geometric model for the compactification of the moduli stack of oriented elliptic curves, giving a canonical choice of holomorphic topological $q$-expansion map. Then we define globally equivariant forms of Tate $K$-theory $\mathbf{KO}((q))$ and $\mathbf{KU}((q))$, and equip them with globally equivariant meromorphic topological $q$-expansion maps from global topological modular forms. Finally, we explore $C_2$-equivariant versions of global Tate $K$-theory and connect them with $C_2$-equivariant global topological modular forms with level structures.

On the derived Tate curve and global smooth Tate $K$-theory

Abstract

The interplay between equivariant stable homotopy theory and spectral algebraic geometry is used to construct a derived Tate curve over , a lift of the classical elliptic curve of Tate over . Applications of both an algebro-geometric and a topological flavour follow. First, we construct a spectral algebro-geometric model for the compactification of the moduli stack of oriented elliptic curves, giving a canonical choice of holomorphic topological -expansion map. Then we define globally equivariant forms of Tate -theory and , and equip them with globally equivariant meromorphic topological -expansion maps from global topological modular forms. Finally, we explore -equivariant versions of global Tate -theory and connect them with -equivariant global topological modular forms with level structures.

Paper Structure

This paper contains 12 sections, 27 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Cyclification of global spaces
  3. Cyclification of oriented equivariant cohomology theories
  4. A spectral lift of the Katz--Mazur $\mathbf{P}$-divisible group
  5. The derived Tate curve
  6. Compactifying the moduli stack of oriented elliptic curves
  7. Equivariant and global Tate $K$-theory
  8. Global smooth Tate $K$-theory
  9. Global quasi-elliptic cohomology
  10. Two $C_2$-equivariant global meromorphic topological $q$-expansion maps
  11. The unramified cusp
  12. The ramified cusp

Key Result

Theorem 1

There exists a unique oriented elliptic curve $\mathrm{T}$ over $\mathop{\mathrm{KU}}\nolimits\llparenthesis q\rrparenthesis$ such that:

Theorems & Definitions (75)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • ...and 65 more