Dans cet article, on consid\`ere un groupe semi-simple $\rmG$ classique r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un sous-groupe de Cartan compact. On d\'efinit une famille de sous-alg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$. Soit $\g'$ une telle sous-alg\`ebre de Lie. On construit alors une application de transfert des int\'egrales orbitales de $\g'$ dans l'espace des int\'egrales orbitales de $\g$. On montre que cette application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$. Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette application de transfert. On utilise cette application de transfert pour obtenir une formule de r\'eduction de l'int\'egrale de Cauchy Harish-Chandra pour les paires duales d'alg\`ebres de Lie r\'eductives $\bigl( \Ug(p,q),\Ug(r,s) \bigr)$ et $\bigl( \Sp(p,q),\Og^*(2n) \bigr)$ 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--Neg{\aa}rd 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 $\Q(\sqrt{2},\sqrt{\delta})$ 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 $H_{\alpha}(i_2,i_3,\dots)$ of ways a given permutation (with cycles described by the partition $\a$) 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 $H_{\alpha}(i_2,i_3,\dots)$ when $\a$ 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\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \`a la dimension $n$.

We construct covering maps from infinite blowups of the $n$-dimensional Sierpinski gasket $SG_n$ to certain compact fractafolds based on $SG_n$. These maps are fractal analogs of the usual covering maps from the line to the circle. The construction extends work of the second author in the case $n=2$, but a different method 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.