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arXiv · 2503.04494

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

Jack Morgan Davies·Sil Linskens
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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.

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Jack Morgan Davies, Sil Linskens. 2026-07-01. On the derived Tate curve and global smooth Tate $K$-theory. https://arxiv.org/abs/2503.04494

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