Let $V$ be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of $V$ of height bounded by invariants associated to any variety containing $V$. In particular, we determine whether such a set is or is not dense in $V$. We then prove that these sets can always be written as the intersection of $V$ with a finite union of translates of tori of which we control the sum of the degrees.

As a consequence, we prove a conjecture by Amoroso and David up to a logarithmic factor

Nous étudions la question de la croissance des nombres de Betti de certaines variétés arithmétiques dans des revêtements de congruence. Plus précisement nos résultats portent sur les variétés de Siegel et les variétés associées à des groupes orthogonaux. Nous expliquons comment un théorème de Waldspurger permet de majorer et de minorer ces nombres. Les résultats obtenus vont dans le sens de conjectures de Sarnak et Xue [SX].

We study the question of the growth of Betti numbers of certain arithmetic varieties in a tower of congruence coverings. In fact, our results are about Siegel varieties and varieties associated to orthogonal groups. We explain how a theorem of Waldspurger can be used to obtain lower and upper bounds. Our results are in the direction of conjectures made by Sarnak and Xue [SX]

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic $\mathrm{PSL}(2,C)$-representation of a tame $3$-manifold. If the representation is the geometric representation of a hyperbolic $3$-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume

We prove that for any closed surface of genus at least four, and any punctured surface of genus at least two, the space of ending laminations is connected. A theorem of E. Klarreich [28, Theorem 1.3] implies that this space is homeomorphic to the Gromov boundary of the complex of curves. It follows that the boundary of the complex of curves is connected in these cases, answering the conjecture of P. Storm. Other applications include the rigidity of the complex of curves and connectivity of spaces of degenerate Kleinian groups

We establish long-time stability of multidimensional viscous shocks of a general class of symmetric hyperbolic-parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions $d\ge 2$. This extends the existing result established by Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such stability result for (fast) MHD shocks. At the same time, we are able to drop a technical assumption on the structure of the so-called glancing set that was necessarily used in previous analyses. The key idea to the improvements is to introduce a new simple argument for obtaining an $L^1\to L^p$ resolvent bound in low-frequency regimes by employing the recent construction of degenerate Kreiss symmetrizers by Guès, Métivier, Williams, and Zumbrun. Thus, at the low-frequency resolvent bound level, our analysis gives an alternative to the earlier pointwise Green-function approach of Zumbrun. High-frequency solution operator bounds have been previously established entirely by nonlinear energy estimates