Dans cet article, on considère un groupe semi-simple $\text{G}$ classique réel et connexe. On suppose de plus que $\text{G}$ possède un sous-groupe de Cartan compact. On définit une famille de sous-algèbres de Lie associée à $\mathfrak{g}=\text{Lie(G)}$, de même rang que g dont tous les facteurs simples sont de rang 1 ou 2. Soit $\mathfrak{g}'$ une telle sous-algèbre de Lie. On construit alors une application de transfert des intégrales orbitales de $\mathfrak{g}'$ dans l’espace des intégrales orbitales de $\mathfrak{g}$. On montre que cette application est définie dès que $\mathfrak{g}$ ne possède pas de facteur simple réel de type CI de rang supérieur ouégal à 3. Si de plus, $\mathfrak{g}$ ne possède pas de facteur simple de type BI de rang supérieur à 3, on montre la surjectivité de cette application de transfert.

On utilise cette application de transfert pour obtenir une formule de réduction de l’intégrale de Cauchy Harish-Chandra pour les paires duales d’algèbres de Lie réductives $(\text{U(}p,q\text{),U(}r,s\text{)})$ et $(\text{Sp(}p,q\text{),O*(2n)})$ avec $p+q=r+s=n$.

We give some examples of Calabi–Yau 3-folds with $\rho =1$ and $\rho =2$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of ${{\mathbb{P}}^{7}}$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field ${{\mathbb{F}}_{p}}$. Three of our examples (of degree 17 and 20) are new. The two others (degree 15 and 18) are known, and we recover their well-known invariants with our method. These examples are built out of Gulliksen–Negård and Kustin–Miller complexes of locally free sheaves.

Finally, we give two new examples of Calabi–Yau 3-folds of ${{\mathbb{P}}^{6}}$ of degree 14 and 15 (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $({{H}^{3}},{{c}_{2}}\cdot H,{{c}_{3}})$ and $\rho =1$.

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings, that is, we study the number ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ of ways a given permutation (with cycles described by the partition $\alpha $) can be decomposed into a product of exactly ${{i}_{2}}$ 2-cycles, ${{i}_{3}}$ 3-cycles, etc., with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain descent structure and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to determine ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ when $\alpha $ has one or two parts, extending earlier work of Goulden and Jackson. We also show how these methods are readily modified to count inequivalent factorizations, where equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits us to generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of their analysis.

Pour toute sous-vari ét é géométriquement irréductible $V$ du groupe multiplicatif $\mathbb{G}_{m}^{n}$, on sait qu’en dehors d’un nombre fini de translat és de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minorée par une certaine quantité $q{{(V)}^{-1}}>0$. On connaît de plus une borne supérieure pour la somme des degrés de ces translatés de tores pour des valeurs de $q(V)$ polynomiales en le degré de $V$. Ceci n’est pas le cas si l’on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement linéaire en l’inverse du degré.

Nous apportons ici une réponse partielle à ce problème : nous donnons une majoration de la somme des degrés de ces translat és de sous-tores de codimension 1 d’une hypersurface $V$. Les résultats, obtenus dans le cas de $\mathbb{G}_{m}^{3}$,mais complètement explicites, peuvent toutefois s’étendre à $\mathbb{G}_{m}^{n}$,moyennant quelques petites complications inhérentes à la dimension $n$.

We construct covering maps from infinite blowups of the $n$-dimensional Sierpinski gasket $S{{G}_{n}}$ to certain compact fractafolds based on $S{{G}_{n}}$. These maps are fractal analogs of the usual covering maps fromthe line to the circle. The construction extends work of the second author in the case $n=2$, but a differentmethod of proof is needed, which amounts to solving a Sudoku-type puzzle. We can use the covering maps to define the notion of periodic function on the blowups. We give a characterization of these periodic functions and describe the analog of Fourier series expansions. We study covering maps onto quotient fractalfolds. Finally, we show that such covering maps fail to exist for many other highly symmetric fractals.

We define periodic functions on infinite blow-ups of the Sierpinski gasket as lifts of functions defined on certain compact fractafolds via covering maps. This is analogous to defining periodic functions on the line as lifts of functions on the circle via covering maps. In our setting there is only a countable set of covering maps. We give two different characterizations of periodic functions in terms of repeating patterns. However, there is no discrete group action that can be used to characterize periodic functions. We also give a Fourier series type description in terms of periodic eigenfunctions of the Laplacian. We define almost periodic functions as uniform limits of periodic functions.