On the derived Tate curve and global smooth Tate $K$-theory
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913722274938880 |
|---|---|
| author | Davies, Jack Morgan Linskens, Sil |
| author_facet | Davies, Jack Morgan Linskens, Sil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the derived Tate curve and global smooth Tate $K$-theory Davies, Jack Morgan Linskens, Sil Algebraic Topology Algebraic Geometry K-Theory and Homology 14A30, 19L47, 55P91, 55P42 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. |
| title | On the derived Tate curve and global smooth Tate $K$-theory |
| topic | Algebraic Topology Algebraic Geometry K-Theory and Homology 14A30, 19L47, 55P91, 55P42 |
| url | https://arxiv.org/abs/2503.04494 |