Elliptic units and sign functions

In the first part of this paper we give a new definition of the elliptic analogue of Sinnott’s group of circular units. In this we essentially use the ideas discussed in Oukhaba (in Ann Inst Fourier, 55(33):753–772, 2005). In the second part of the paper we are interested in computing the index of this group of elliptic units. This question is closely related to the behaviour of the universal signed ordinary distributions introduced in loc. cit. Such distributions have a natural resolution discovered by Anderson. Consequently, we can apply Ouyang’s general index formula and the powerful Anderson’s theory of double complex to make the computations     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

K2, K3 and nilpotent ideals

In this note is given an explicit computation method to decide about the minimality of a word in the free group of rank two. A cohomological use of this method is to yield words defining one relator duality groups.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Discrete module categories and operations in K-theory

We study the categories of discrete modules for topological rings arising as the rings of operations in various kinds of topological K-theory. We prove that for these rings the discrete modules coincide with those modules which are locally finitely generated over the ground ring.     

Powers of ideals associated to (C4,2K2)-free graphs

Let G be a $(C_4,2K_2)$-free graph with edge ideal $I(G)?\mathbb{k}[x_1,\dots ,x_n]$. We show that $I{(G)}^s$ has linear resolution for every $s\ge 2$. Also, we show that every power of the vertex cover ideal of G has linear quotients. As a result, we describe the Castelnuovo–Mumford regularity of powers of $I{(G)}^{\vee }$ in terms of the maximum degree of G.     

The genus of Kn − K2

The minimal genus of K_n ? K₂ is determined using the method of index two current graphs.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Rationality problem of GL4 group actions

Let K be any field which may not be algebraically closed, V be a four-dimensional vector space over K, σ∈GL(V) where the order of σ may be finite or infinite, f(T)∈K[T] be the characteristic polynomial of σ. Let α, αβ₁, αβ₂, αβ₃ be the four roots of f(T)=0 in some extension field of K.Theorem 1.BothK(V)^〈σ〉andare rational (=purelytranscendental) overKif at least one of the following conditions is satisfied: (i) charK=2, (ii) f(T) is a reducible or inseparable polynomial inK[T], (iii) not all ofβ₁,β₂,β₃are roots of unity, (iv) iff(T) is separable irreducible, then the Galois group off(T) overKis not isomorphic to the dihedral group of order 8 or the Klein four group.Theorem 2.Suppose that allβ_iare roots of unity andf(T)∈K[T] is separable irreducible. (a) If the Galois group off(T) is isomorphic to the dihedral group of order 8, then bothK(V)^〈σ〉andare not stably rational overK. (b) When the Galois group off(T) is isomorphic to the Klein four group, then a necessary and sufficient condition for rationality ofK(V)^〈σ〉andis provided. (See Theorem 1.5. for details.)     

Triangular imbedding of Kn − K6

For n = 12s + 9 (s ≥ 4) we imbed K_n ? K₆ in an orientable surface of genus 12s² + 11s.     

Graphs without K4 and well-quasi-ordering

It is proved that given an infinite sequence G₁, G₂, G₃,…, of series-parallel graphs there are indices i < j such that G_j contains an induced subgraph contractable onto G_i. An example is given showing that for planar graphs the preceding theorem fails.     

The reduced measure algebra and a K1-space which is not K0

The reduced measure algebra is used to construct, under CH, a hereditarily Lindelöf separable K₁-space X which is not a K₀-space.     

A remark on K-theory and S-categories

It is now well known that the K-theory of a Waldhausen category depends on more than just its (triangulated) homotopy category (Invent. Math. 150 (2002) 111). The purpose of this note is to show that the K-theory spectrum of a (good) Waldhausen category is completely determined by its Dwyer-Kan simplicial localization, without any additional structure. As the simplicial localization is a refined version of the homotopy category which also determines the triangulated structure, our result is a possible answer to the general question: “To which extent K-theory is not an invariant of triangulated derived categories? ”     

Orbits of points on certain K3 surfaces

In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is 0.6527±0.0012.     

The 1-type of a Waldhausen K-theory spectrum

We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.