---
abstract: 'We compute topological Hochschild homology mod $p$ and $v_1$ of the connective cover of the $K(1)$ local sphere spectrum for all primes $p\ge 3$. This is accomplished using a May-type spectral sequence in topological Hochschild homology constructed from a filtration of a commutative ring spectrum.'
author:
- 'Gabe Angelini-Knoll'
bibliography:
- 'sources.bib'
title: 'On topological Hochschild homology of the $K(1)$-local sphere'
---

Introduction
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Algebraic K-theory of rings is known to encode deep arithmetic information; for example, algebraic K-groups of rings of integers in totally real number fields are related to special values of Dedekind zeta functions by Lichtenbaum’s conjecture. We would like to explore the arithmetic encoded in the algebraic K-theory of “brave new rings" or, more precisely, commutative ring spectra. To approach this, we use a technique, initiated by Bökstedt in [@bok2], where one approximates algebraic K-theory by topological Hochschild homology. Specifically, topological Hochschild homology is a linear approximation to algebraic K-theory in the sense of Goodwillie’s calculus of functors. The purpose of this paper is to compute topological Hochschild homology of the connective cover of the $K(1)$-local sphere mod $(p,v_1)$.

This computation uses a spectral sequence associated to a multiplicative filtration constructed by the author 