Reduction of structure group of pullbacks of semistable principal bundles

Let C be an irreducible smooth projective curve defined over an algebraically closed field k. Let G be a semisimple linear algebraic group defined over the field k and P⊂G a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P. Let EG be a semistable principal G-bundle over C. If the characteristic of k is positive, then EG is assumed to be strongly semistable. Take any real number ?>0. Then there is an irreducible smooth projective curve defined over k, a nonconstant morphism

     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

The Uncountable Lattice of All Subfunctors of the Functor of Rational Representations

In set theory the cardinality of the continuum \(|{\mathbb R}|\) is the cardinal number of some interesting sets, like the Cantor set or the transcendental numbers. We will prove that the cardinal number of all subfunctors of the functor of rational representations \(k \otimes_{{\mathbb Z}} R_{{\mathbb Q}}\), taking values on odd order groups over the field k of characteristic 2, is equal to \(|{\mathbb R}|\). When the characteristic q?>?0 of the field k is not necessarily even, we will present a formula giving the dimension of the evaluations S _C,k(G), of the simple functor S _C,k, at any group G of order prime to q and being associated to a suitable cyclic group C.     

Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map has trivial kernel, denoting the set of places of k.The conjecture is true if G is of type 1A∗, i.e., isomorphic to SL₁(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin [S1] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2A∗, B_n, C_n, D_n (D₄ nontrialitarian), G₂ or F₄; a group is said to be of type 2A∗, if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.     

On differentiable vectors for representations of infinite dimensional Lie groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π:G→GL(V) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V^∞ of smooth vectors. If G is a Banach-Lie group, we define a topology on the space V^∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V^∞ is a Fréchet space. This applies in particular to C^∗-dynamical systems (A,G,α), where G is a Banach-Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈π(g)v,v〉 is smooth. The second class of results concerns criteria for Ck-vectors in terms of operators of the derived representation for a Banach-Lie group G acting on a Banach space V. In particular, we provide for each k∈N examples of continuous unitary representations for which the space of Ck+1-vectors is trivial and the space of Ck-vectors is dense.     

Centerpole sets for colorings of abelian groups

A subset C?G of a group G is called k-centerpole if for each k-coloring of G there is an infinite monochromatic subset G, which is symmetric with respect to a point c??C in the sense that S=cS ^?1 c. By c _k (G) we denote the smallest cardinality c _k (G) of a k-centerpole subset in G. We prove that c _k (G)=c _k (? ^m ) if G is an abelian group of free rank m??k. Also we prove that c ₁(? ⁿ⁺¹)=1, c ₂(? ⁿ⁺²)=3, c ₃(? ⁿ⁺³)=6, 8??c ₄(? ⁿ⁺⁴)??c ₄(?⁴)=12 for all n????, and ${\frac{1}{2}(k^{2}+3k-4)\le c_{k}(\mathbb{Z}^{n})\le2^{k}-1-\max_{s\le k-2}\binom {k-1}{s-1}}$ for all n??k??4.     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

Largest random component of ak-cube

LetC ^k denote the graph with vertices (? ₁, ...,? _k ),? _i =0,1 and vertices adjacent if they differ in exactly one coordinate. We callC ^k thek-cube. LetG=G _{k, p} denote the random subgraph ofC ^k defined by letting $$Prob(\{ i,j\} \in G) = p$$ for alli, j ∈ C ^k and letting these probabilities be mutually independent. We show that forp=λ/k, λ>1,G _{k, p} almost surely contains a connected component of sizec2 ^k ,c=c(λ). It is also true that the second largest component is of sizeo(2 ^k ).     

A characterization of the linear groups L 2(p)

Let G be a finite group and π _e (G) be the set of element orders of G. Let k ∈ π _e (G) and m _k be the number of elements of order k in G. Set nse(G):= {m _k : k ∈ π _e (G)}. In fact nse(G) is the set of sizes of elements with the same order in G. In this paper, by nse(G) and order, we give a new characterization of finite projective special linear groups L ₂(p) over a field with p elements, where p is prime. We prove the following theorem: If G is a group such that |G| = |L ₂(p)| and nse(G) consists of 1, p ² ? 1, p(p + ?)/2 and some numbers divisible by 2p, where p is a prime greater than 3 with p ≡ 1 modulo 4, then G ? L ₂(p).     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

Unramified cohomology of degree 3 and Noether’s problem

Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W) ^G is purely transcendental over C goes back to Emmy Noether. Using the unramified cohomology group of degree 2 of this field as an invariant, Saltman gave the first examples for which C(W) ^G is not rational over C. Around 1986, Bogomolov gave a formula which expresses this cohomology group in terms of the cohomology of the group G. In this paper, we prove a formula for the prime to 2 part of the unramified cohomology group of degree 3 of C(W) ^G . Specializing to the case where G is a central extension of an F _p -vector space by another, we get a method to construct nontrivial elements in this unramified cohomology group. In this way we get an example of a group G for which the field C(W) ^G is not rational although its unramified cohomology group of degree 2 is trivial. Dedicated to Jean-Louis Colliot-Thélène.     

Independence polynomials of circulants with an application to music

The independence polynomial of a graph G is the generating function I(G,x)=∑_k≥0i_kx^k, where i_k is the number of independent sets of cardinality k in G. We show that the problem of evaluating the independence polynomial of a graph at any fixed non-zero number is intractable, even when restricted to circulants. We provide a formula for the independence polynomial of a certain family of circulants, and its complement. As an application, we derive a formula for the number of chords in an n-tet musical system (one where the ratio of frequencies in a semitone is 2^1/n) without ‘close’ pitch classes.     

p-Rank and semi-stable reduction of curves

Let R be a discrete complete valuation ring, with field of fractions K, and with algebraically closed residue field k of characteristic p > 0. Let X be a germ of an R-curve at an ordinary double point. Consider a finite Galois covering f: Y → X, whose Galois group G is a p-group, such that Y is normal, and which is étale above X_k≔ x × _rk. Asume that Y has a semi-stable model :→ Y over R, and let y be a closed point of Y. If the inertia subgroup I(y) at y is cyclic of order pⁿ, we compute the p-rank of tf⁻¹ (y) by using a result of Raynaud. In particular, we prove that this p-rank is bounded by pⁿ −1.     

Degree conditions for the partition of a graph into cycles, edges and isolated vertices

Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{d_G(x),d_G(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H₁,…,H_k such that V(H₁)∪?∪V(H_k)=V(G) and H_i is a cycle or K₁ or K₂ for each 1≤i≤k, unless k=2 and G=C₅, or k=3 and G=K₁∪C₅.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

On the descent of Levi factors

Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose ? is a finite Galois extension of k for which the extension degree [? : k] is relatively prime to p. If G _/? has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups ${R \simeq Lie(R)}$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If ${G_{{/k}_{sep}}}$ has a Levi decomposition for a separable closure k _sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which ${G_{{/k}_{sep}}}$ has a Levi decomposition, but G itself has no Levi decomposition.     

A mean value theorem for discriminants of abelian extensions of a number field

Let k be an algebraic number field and let N(k,C?;m) denote the number of abelian extensions K of k with G(K/k)≅C?, the cyclic group of prime order ?, and the relative discriminant D(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for _∑m?XN(k,C?;m) using the class field theory and a method, developed by Wright. We show that our result is identical to a result of Cohen, Diaz y Diaz and Olivier, obtained by methods of classical algebraic number theory, although our methods allow for a more elegant treatment and reduce a global calculation to a series of local calculations.     

A note on traversing specified vertices in graphs embedded with large representativity

A graph G is said to have property P(2,k) if given any k+2 distinct vertices a,b,v₁,…,v_k, there is a path P in G joining a and b and passing through all of v₁,…,v_k. A graph G is said to have property C(k) if given any k distinct vertices v₁,…,v_k, there is a cycle C in G containing all of v₁,…,v_k. It is shown that if a 4-connected graph G is embedded in an orientable surface Σ (other than the sphere) of Euler genus eg(G,Σ), with sufficiently large representativity (as a function of both eg(G,Σ) and k), then G possesses both properties P(2,k) and C(k).     

Étude du cas rationnel de la théorie des formes linéaires de logarithmes

We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v₀ be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u∈Lie(G(Cv₀)) a logarithm of a point p∈G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)k_⊗Cv₀. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height loga of p, removing a polynomial term in logloga. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy.