The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

Lehmer's totient problem over Fq[x]

In this paper, we consider the function field analogue of the Lehmer's totient problem. Let $p(x)\in {\mathbb{F}}_q[x]$ and $\phi (q,p(x))$ be the Euler's totient function of $p(x)$ over ${\mathbb{F}}_q[x]$, where ${\mathbb{F}}_q$ is a finite field with q elements. We prove that $\phi (q,p(x))|(q^{\mathrm{deg}(p(x))}?1)$ if and only if (i) $p(x)$ is irreducible; or (ii) $q=3$, $p(x)$ is the product of any 2 non-associate irreducibles of degree 1; or (iii) $q=2$, $p(x)$ is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.     

A topological nullstellensatz for tensor-triangulated categories

Let $\mathrm{Spec}(\mathbb{T})$ be the spectrum of a tensor-triangulated category $(\mathbb{T},?,\mathbf{1})$. We show that there is a homeomorphism between the spectral space of radical thick tensor ideals in $(\mathbb{T},?,\mathbf{1})$ and the collection of open subsets of $\mathrm{Spec}(\mathbb{T})$ in inverse topology. In fact, we prove a more general result in terms of supports on $(\mathbb{T},?,\mathbf{1})$ and work by combining methods from commutative algebra, topology and tensor triangular geometry.     

Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let ${\mathbb{F}}_q$ be a field of q elements, where q is a power of an odd prime p. The polynomial $f(y)\in {\mathbb{F}}_q[y]$ defined by$f(y):={(1+\sqrt{y})}^{(q+1)/2}+{(1-\sqrt{y})}^{(q+1)/2}$ has the property that$f(1-y)=\rho (2)f(y),$ where ρ is the quadratic character on ${\mathbb{F}}_q$. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.     

Une remarque sur les sommes de Ramanujan

We study the asymptotic behaviour of the function $A(q)={\sup }_{x,y}|{\sum }_{x<n?y}{c}_q(n)|$ where $c_q(n)$ denotes the Ramanujan sum. We prove that $q?A(q)?q\sqrt{{\log }_{2}q}$ and determine how sharp this estimation is. To cite this article: A. Chadozeau, C. R. Acad. Sci. Paris, Ser. I 341 (2005).