No abstract

The Lichtenbaum–Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni–Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the n n -th Greek letter family is detected by a commutative ring spectrum R R , then we conjecture that the n + 1 n+1 -st Greek letter family will be detected by the algebraic K-theory of R R . We prove this in the case n = 1 n=1 for R = K ( F q ) R=\mathrm {K}(\mathbb {F}_q) modulo ( p , v 1 ) (p,v_1) where p ≥ 5 p\ge 5 and q = ℓ k q=\ell ^k is a prime power generator of the units in Z / p 2 Z \mathbb {Z}/p^2\mathbb {Z} . In particular, we prove that the commutative ring spectrum K ( K ( F q ) ) \mathrm {K}(\mathrm {K}(\mathbb {F}_q)) detects the part of the p p -primary β \beta -family that survives mod ( p , v 1 ) (p,v_1) . The method of proof also implies that these β \beta elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences.

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order 2 to the orthogonal group O(2). From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order 2m. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of p-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero D2m-Mackey functor homotopy groups of THR(Z) for m odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest.