Niko Naumann

Commutativity conditions for truncated Brown-Peterson spectra of height 2

An algebraic criterion, in terms of closure under power operations, is determined for the existence and uniqueness of generalized truncated Brown-Peterson spectra of height 2 as E1-ring spectra. The criterion is checked for an example at the prime 2 derived from the universal elliptic curve equipped with a level 1(3) structure. 2000MSC: 55P42, 55P43, 55N22 and 14L05

Nilpotence and descent in equivariant stable homotopy theory

Abstract Let G be a finite group and let F be a family of subgroups of G. We introduce a class of G-equivariant spectra that we call F -nilpotent. This definition fits into the general theory of torsion, complete, and nilpotent objects in a symmetric monoidal stable ∞-category, with which we begin. We then develop some of the basic properties of F -nilpotent G-spectra, which are explored further in the sequel to this paper. In the rest of the paper, we prove several general structure theorems for ∞-categories of module spectra over objects such as equivariant real and complex K-theory and Borel-equivariant MU. Using these structure theorems and a technique with the flag variety dating back to Quillen, we then show that large classes of equivariant cohomology theories for which a type of complex-orientability holds are nilpotent for the family of abelian subgroups. In particular, we prove that equivariant real and complex K-theory, as well as the Borel-equivariant versions of complex-oriented theories, have this property.

Algebraic independence in the Grothendieck ring of varieties

We give sufficient cohomological criteria for the classes of given varieties over a field k to be algebraically independent in the Grothendieck ring of varieties over k and construct some examples.

Beta-elements and divided congruences

The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of β-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem.

On a nilpotence conjecture of J.P. May

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., H1-ring spectra. Using an explicit nilpotence bound on the torsion elements in K(n)-local H1-algebras over En, we reduce the conjecture to the nilpotence theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence results in various bordism rings including MSpin and MString , results about the behavior of the Adams spectral sequence for E1-ring spectra, and the non-existence of E1-ring structures on certain complex oriented ring spectra.

On the irreducibility of the two variable zeta-function for curves over finite fields

R. Pellikaan (Arithmetic, Geometry and Coding Theory, Vol. 4, Walter de Gruyter, Berlin, 1996, pp. 175–184) introduced a two variable zeta-function Z(t,u) for a curve over a finite field Fq which, for u=q, specializes to the usual zeta-function and he proved rationality: Z(t,u)=(1−t)−1(1−ut)−1P(t,u) with P(t,u)∈Z[t,u]. We prove that P(t,u) is absolutely irreducible. This is motivated by a question of J. Lagarias and E. Rains about an analogous two variable zeta-function for number fields. To cite this article: N. Naumann, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Linear Relations Among the Values of Canonical Heights from the Existence of Non-Trivial Endomorphisms

We study the interplay between canonical heights and endomorphisms of an abelian variety A over a number field k. In particular we show that wheneverthe ring of endomorphisms defined over k is strictly largerthan Z there will be Q-linear relations among the values of a canonical height pairing evaluated at a basis modulo torsion of A(k).