The Conner-Floyd Map for Formal A-Modules

The purpose of this article to determine and classify the indecomposable RG-lattices, where R is a certain ring of algebraic integers, and G is a cyclic group of prime order.

The article investigates the classification with precision up to equivalence of involutions in the algebra of upper triangular matrices over the ring of integers of algebraic numbers of quadratic fields.The description of involutions in algebras represents one of the classical problems of ring theory. Standard examples of involutions are transposition in matrix algebra and conjugation in the field of complex numbers and the algebra of quaternions.In the case where the field 𝑃 has a characteristic different from two, a complete description of involutions with precision up to their equivalence in the algebra 𝑇𝑛(𝑃) for any natural number 𝑛 was obtained in [15]. In this work [3] involutions in the algebra of upper triangular matrices over commutative rings are studied. If the ring is a field of characteristic 2 or a Boolean ring, then necessary and sufficient conditions for the finiteness of the number of equivalence classesof involutions were found.This article is a continuation of the work of [3]. In the article [3], in particular, the number of equivalence classes of involutions in the algebras of upper triangular matrices over the ring of integers was found. In this regard, the natural result is the problem of describing involutions with precision up to their equivalence in algebras of upper triangular matrices over the ring of algebraic integers of quadratic fields, to which this work is devoted. In the work, the number of equivalence classes of involutions in such algebras is found and the method of finding representatives in each equivalence class is illustrated with examples. Upon receipt the main results in this work, the apparatus of the theory of Pell’s equations is significantly used.

Given a number field k with ring of algebraic integers o(k), define an equivalence relation by declaring that two ideals a and b lie in the same class if and only if there exist non-zero elements c and d such that ca = db. These classes form an abelian group called the ideal class group of o(k), abbreviated Cl(o(k)). This group, in some sense, measures how different a ring of algebraic integers is from a principal ideal domain.

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height $dn$ formal group law (i.e., the height $dn$ Morava stabilizer group) to the cohomology of the automorphism group of an $A$-height $n$ formal $A$-module, where $A$ is the ring of integers in a degree $d$ field extension of $\mathbb{Q}_p$. We then compute this map for the quadratic extensions of $\mathbb{Q}_p$ and the height $2$ Morava stabilizer group at primes $p>3$. We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an $A$-height $1$-formal $A$-module with the ramified part of the abelianization of the absolute Galois group of $K$, yielding an action of $Gal(K^{ab}/K^{nr})$ on the Lubin-Tate/Morava $E$-theory spectrum $E_2$ for each quadratic extension $K/\mathbb{Q}_p$. Finally, we run the associated descent spectral sequence to compute the $V(1)$-homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the $K(2)$-local homotopy groups of the Smith-Toda complex $V(1)$, either that element or its dual is detected in the Galois cohomology of the abelian closure of some quadratic extension of $\mathbb{Q}_p$.

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences $$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$ for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

Characters and decomposition of a representation in a number field

We explain a new relationship between formal group laws and ring spectra in\nstable homotopy theory. We study a ring spectrum denoted DB which depends on a\ncommutative ring B and is closely related to the topological Andre-Quillen\nhomology of B. We present an explicit construction which to every 1-dimensional\nand commutative formal group law F over B associates a morphism of ring spectra\nF_*: HZ --> DB from the Eilenberg-MacLane ring spectrum of the integers. We\nshow that formal group laws account for all such ring spectrum maps, and we\nidentify the space of ring spectrum maps between HZ and DB. That description\ninvolves formal group law data and the homotopy units of the ring spectrum DB.\n

To a complex oriented cohomology theory $h^*(-)$ one may assign a formal group law $F_h(u,v)$ over $h^*(pt)$ . The purpose of this paper is to show that the converse holds true for the case of Abel's universal formal group law ${\cal F}_{Ab}(u,v)$ , i.e. we will prove the existence of a complex oriented cohomology theory $Ab^*(-)$ whose formal group law $F_{Ab}$ is isomorphic to ${\cal F}_{Ab}(u,v)$ .

Let Cl(OK[G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK[G]-modules, where OK is the ring of algebraic integers in the number field K and G is a finite group. We show how to compute the Swan subgroup, T(OK[G]), of Cl(OK[G]) when K = Q(ζp), ζp a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that hp+ = 1 and 2 is inert in K/Q. We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ˚ we do the computations obtaining for several primes p a nontrivial divisor of Cl(Z[ζp]C2). These calculations give an alternative proof that the fields Q(ζp) for p=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly. In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed. We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.

Geometric representations of the formal affine Hecke algebra

The main result is the construction of a minimal integral model of an algebraic torus defined over a complete non-Archimedean extension of an algebraic number field. The structure of such models is studied. The main problem is the study of the model in the case of a ramified splitting field. Reductions of these models with respect to a simple module are described. Minimal models of tori over the ring of algebraic integers are constructed. The local volumes and class numbers of some models are calculated.

In this note we study a certain formal group law over a complete discrete valuation ring F [[ u n -1 ]] of characteristic p > 0 which is of height n over the closed point and of height n -1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law H n -1 of height n - 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group S n -1 of H n -1 . We show that the automorphism group S n of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of S n -1 to the cohomology of S n with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed.

We give an account of some largely experimental results about the successive minima of the ring of integers of an algebraic number field endowed with its canonical Euclidean norm. This throws light on some interesting facts which could be important for the algorithmic theory of number fields.

This paper shows that the group is boundedly elementary generated for and the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this paper for elementary bounded generation are independent of the underlying global function field and only depend on the integer . Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on .