Algebraic K-theory of elliptic cohomology

TL;DR

This paper computes the mod (p, v_1, v_2) homotopy of the topological cyclic homology of a form of elliptic cohomology, revealing a chromatic redshift from v_2 to v_3 periodicity at primes p ≥ 7.

Contribution

It provides the first explicit calculation demonstrating chromatic redshift from v_2 to v_3 in algebraic K-theory of elliptic cohomology spectra.

Findings

Homotopy groups form a finitely generated free F_p[v_3]-module.

Explicit degrees and generators are identified within a specific range.

Results apply to primes p ≥ 7, covering all such primes.

Abstract

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner.