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Let N be a finitely generated nilpotent group. The subgroup zeta function ζ N ⩽ (s) and the normal zeta function ζ N ⊲ (s) of N are Dirichlet series enumerating the finite index subgroups or the finite index normal subgroups of N. We present results about their abscissae of convergence α N ⩽ and α N ⊲ , also known as the degrees of polynomial subgroup growth and polynomial normal subgroup growth of N, respectively. We first prove some upper bounds for the functions N ↦ α N ⩽ and N ↦ α N ⊲ when restricted to the class of torsion-free nilpotent groups of a fixed Hirsch length. We then show that if two finitely generated nilpotent groups have isomorphic C -Mal’cev completions, then their subgroup (resp. normal) zeta functions have the same abscissa of convergence. This follows, via the Mal’cev correspondence, from a similar result that we establish for zeta functions of rings. This result is obtained by proving that the abscissa of convergence of an Euler product of certain Igusa-type local zeta functions introduced by du Sautoy and Grunewald remains invariant under base change. We also apply this methodology to formulate and prove a version of our result about nilpotent groups for virtually nilpotent groups. As a side application of our result about zeta functions of rings, we present a result concerning the distribution of orders in number fields. [ABSTRACT FROM AUTHOR]

We introduce the notion of approximation type for the partial, and in certain cases the total description of extensions of a given valuation from a field K to the rational function field K(x). To every extension, a unique approximation type of x over K is associated, while x may be the limit of many pseudo Cauchy sequences. Approximation types also provide information in cases where the extensions are not immediate, and we prove that they correspond bijectively to the extensions when K is algebraically closed or, more generally, lies dense in its algebraic closure with respect to the topology induced by the valuation. [ABSTRACT FROM AUTHOR]

In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer–Cartan equation. Following the classical theory of Kodaira–Spencer–Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of (p, q)-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology. [ABSTRACT FROM AUTHOR]

Let X be a proper, smooth, and geometrically connected curve of genus g (X) ≥ 1 over a p-adic local field. We prove that there exists an effectively computable open affine subscheme U ⊂ X with the property that period (X) = 1 , and index (X) equals 1 or 2 (resp. period (X) = index (X) = 1 , assuming period (X) = index (X) ), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of Usplits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to X, and give a new characterisation of sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to Pic 0 (resp. Pic ∧ ). As a consequence we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi [3] is orthogonal to Pic 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups G ⊂ GL 2 , explicitly computing H 2 (G , k ∗) , and then matching these up with Artin's list of ramification data and Reiten–Van den Bergh's AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let B = k ζ 〚 x , y 〛 be the skew power series ring where ζ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form A = B / (f) where f ∈ Z (B) which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers. [ABSTRACT FROM AUTHOR]

In this paper we prove a family of results connecting the problem of computing cup products in surface bundles to various other objects that appear in the theory of the cohomology of the mapping class group Modg and the Torelli group Ig. We show that Kawazumi’s twisted MMM class m0,k can be used to compute k-fold cup products in surface bundles, and that m0,k provides an extension of the higher Johnson invariant τk-2 to Hk-2(Modg,∗,∧kH1). These results are used to show that the behavior of the restriction of the even MMM classes e2i to H4i(Ig1) is completely determined by im(τ4i)≤∧4i+2H1, and to give a partial answer to a question of D. Johnson. We also use these ideas to show that all surface bundles with monodromy in the Johnson kernel Kg,∗ have cohomology rings isomorphic to that of a trivial bundle, implying the vanishing of all τi when restricted to Kg,∗. [ABSTRACT FROM AUTHOR]

A group action on the input ring or category induces an action on the algebraic K-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic K-theory is, for example, that the map of spectra with G-action induced by a G-map of G-rings is not equivariant. We define a version of equivariant algebraic K-theory which encodes a group action on the input in a functorial way to produce a genuine algebraic K-theory G-spectrum for a finite group G. The main technical work lies in studying coherent actions on the input category. A payoff of our approach is that it builds a unifying framework for equivariant topological K-theory, Atiyah's Real K-theory, and existing statements about algebraic K-theory spectra with G-action. We recover the map from the Quillen-Lichtenbaum conjecture and the representational assembly map studied by Carlsson and interpret them from the perspective of equivariant stable homotopy theory. [ABSTRACT FROM AUTHOR]

We classify, up to conjugation, all automorphisms of Niemeier lattices to which we can apply Miyamoto's orbifold construction. Using this classification, we prove that the VOAs obtained in Miyamoto (Symmetries, integrable systems and representations. Springer proceedings in mathematics and statistics, vol 40. Springer, London, pp 319-344, ) and Sagaki and Shimakura (Trans Am Math Soc, to appear) are all of holomorphic non-lattice VOAs which we can obtain by applying the $${\mathbb {Z}}_3$$ -orbifold construction to a Niemeier lattice and its automorphism. [ABSTRACT FROM AUTHOR]

A. K. Bousfield's $$H\mathbb {Z}$$ -localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $$H\mathbb {Z}$$ -localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $$G$$ to its Bousfield $$H\mathbb {Z}$$ -localization is not always a $$G$$ -perfect subgroup. In the case of algebraic closure of groups, we prove the analogous result that this kernel is not always a union of invisible subgroups. [ABSTRACT FROM AUTHOR]

We show the existence of group-theoretic sections of the 'étale-by-geometrically abelian' quotient of the arithmetic fundamental group of hyperbolic curves over $$p$$ -adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points. [ABSTRACT FROM AUTHOR]

Let l be an odd prime and K/ k a Galois extension of totally real number fields with Galois group G such that K/ k and $${k/\mathbb{Q}}$$ are finite. We give a full description of the algebraic structure of the semisimple algebra $${\mathcal{Q} G = {\rm Quot} (\Lambda G)}$$ for pro- l Galois groups G with $$\Lambda G = {\mathbb{Z}_l}$$[[G]] the Iwasawa algebra. We moreover compute the cohomological dimension of the centres of the Wedderburn components of $${\mathcal{Q} G}$$ and state some results on the completion of $${\mathcal{Q} G}$$ . [ABSTRACT FROM AUTHOR]

Let A be an excellent regular local ring of dimension two, let T be a finitely generated preordering in A, and let $${\widehat T}$$ be the preordering generated by T in the completion $${\widehat A}$$ of A. We study the question when the property of being saturated descends from $${\widehat T}$$ to T, and establish conditions of geometric nature which allow to decide this question. As an application we classify all principal preorderings in A of degree ≤ 3 which are saturated, in the case where A has a real closed residue field. These results have direct implications for nonnegativity certificates for real polynomials on two-dimensional semi-algebraic sets. [ABSTRACT FROM AUTHOR]

Quadratic forms over fields of characteristic different from two are generalised to virtual forms over arbitrary fields. These objects are related to Milnor’s K-theory for fields. The connection is established by a sequence of maps corresponding to Delzant’s Stiefel–Whitney classes. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, with a suitable condition, we describe the algebraic structure of block extensions of nilpotent blocks over arbitrary fields, thus generalize the main result of B. Külshammer and L. Puig on block extensions of nilpotent blocks over algebraically closed fields. [ABSTRACT FROM AUTHOR]

Abstract By regarding the classical non-abelian cohomology of groups from a 2-dimensional categorical viewpoint, we are led to a non-abelian cohomology of groupoids which continues to satisfy classification, interpretation and representation theorems generalizing the classical ones. This categorical approach is based on the fact that if groups are regarded as categories, then, on the one hand, crossed modules are 2-groupoids, cocycles are lax 2-functors, and the cocycle conditions are precisely the coherence axioms for lax 2-functors, and, on the other hand, group extensions are fibrations of categories. Furthermore, n-simplices in the nerve of a 2-category are lax 2-functors. [ABSTRACT FROM AUTHOR]

Abstract Alperin’s weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in §1 and §2, respectively, and §3 provides a tool for computing the cohomology of the functors arising in §2. Taking as starting point the alternating sum formulation of Alperin’s weight conjecture by Knörr-Robinson [11], the material of the previous sections is applied in §4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17]. [ABSTRACT FROM AUTHOR]

Suppose given a Galois étale cover Y? X of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and sufficient conditions on G to be the Galois group of an étale cover of X dominating Y? X. [ABSTRACT FROM AUTHOR]

In a G-algebra A of a p-group G over a perfect ground field there is a commutative separable semisimple G-subalgebra such that any traces on relative projectivity in A can be controlled in the subalgebra; and in a residually central sense, the maximal such subalgebras form exactly one $(A^G)^*$-conjugacy class. Applying to modules, the induced module of an indecomposable module of a subnormal subgroup of p- primary index is characterized. [ABSTRACT FROM AUTHOR]

We introduce H-fields as ordered differential fields of a certain kind. Hardy fields extending ${\mathbb R}$ , as well as the field of logarithmic-exponential series over ${\mathbb R}$ are H-fields. We study Liouville extensions in the category of H-fields, as a step towards a model theory of H-fields. The main result is that an H-field has at most two Liouville closures. [ABSTRACT FROM AUTHOR]

Infra-nilmanifolds are compact K( G,1)-manifolds with G a torsion-free, finitely generated, virtually nilpotent group. Motivated by previous results of various authors on p-completions of K( G,1)-spaces with G a finite or a nilpotent group, we study the homotopy type of p-completions of infra-nilmanifolds, for any prime p. We prove that the p-completion of an infra-nilmanifold is a virtually nilpotent space which is either aspherical or has infinitely many nonzero homotopy groups. The same is true for p-localization. Moreover, we show by means of examples that rationalizations of infra-nilmanifolds may be elliptic or hyperbolic. [ABSTRACT FROM AUTHOR]

After revisiting the concept of the torsion subgroup of a group with respect to a set of prime numbers P (as introduced by Ribenboim), we show that, for all p in P, p-th roots are unique in a virtually nilpotent group if and only if P-roots are unique in both its Fitting subgroup and its Fitting quotient. This more general notion of torsion also turns out to be sufficient to understand completely the kernel of the P-localization homomorphism of a virtually nilpotent group. Using this result, we characterize the finitely generated virtually nilpotent groups such that, when dividing out the P-torsion subgroup, P-roots exist and are unique in the resulting quotient. [ABSTRACT FROM AUTHOR]