We consider an arbitrary pair of closed geodesics and the corresponding period integrals for the eigenfunctions of the Laplacian on a compact hyperbolic surface. A summation formula that relates geometric information about the geodesics (namely, the angles of intersection and lengths of common perpendiculars between them) to the period integrals is proved. As a corollary, an asymptotic is obtained for the second moment of the period integrals for a fixed geodesic as an average over the eigenvalue with an error term that can be interpreted in terms of the geometric data

Let $p$ be an odd prime number, let $E$ be an elliptic curve over a number field $k$, and let $F/k$ be a Galois extension of degree twice a power of $p$. We study the $Z_p$-corank ${\mathrm{rk}}_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for ${\mathrm{rk}}_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions.

If $K$ is the (unique) quadratic extension of $k$ in $F$, if $G=\mathrm{Gal}(F/K)$, if $G^{+}$ is the subgroup of elements of $G$ commuting with a choice of involution of $F$ over $k$, and if ${\mathrm{rk}}_p(E/K)$ is odd, then we show that (under mild hypotheses) ${\mathrm{rk}}_p(E/F)\ge [G:G^{+}]$.

As a very specific example of this, suppose that $A$ is an elliptic curve over $Q$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $Q$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $F_p^{\times }$ and such that the negative of the conductor of $E$ is not a square modulo $p$, then there is a positive constant $B$ depending on $A$ but not on $E$ or $n$ such that ${\mathrm{rk}}_p(E/Q(A[p^n]))\ge B{p}^{2n}$ for every $n$

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudodifferential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised nondegenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,ħ)$ and gives improvements and new proofs for many of the results in the field. In the completely resonant case, we show that the pseudodifferential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained

We discuss algebraic vector bundles on smooth $k$-schemes $X$ contractible from the standpoint of ${\mathbb{A}}^1$-homotopy theory; when $k=\mathbb{C}$, the smooth manifolds $X(\mathbb{C})$ are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are those of $\mathrm{Spec}k$. One may hope that, furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the nonaffine case, this is false: we show that (essentially) every smooth ${\mathbb{A}}^1$-contractible strictly quasi-affine scheme that admits a $U$-torsor whose total space is affine, for $U$ a unipotent group, possesses a nontrivial vector bundle. Indeed, we produce explicit arbitrary-dimensional families of nonisomorphic ${\mathbb{A}}^1$-contractible schemes, with each scheme in the family equipped with “as many” (i.e., arbitrary-dimensional moduli of) nonisomorphic vector bundles, of every sufficiently large rank $n$, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible but that are not necessarily ${\mathbb{A}}^1$-contractible

Building upon our arithmetic duality theorems for $1$-motives, we prove that the Manin obstruction related to a finite subquotient $Б(X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shafarevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate-type dual exact sequence for $1$-motives and give an application to weak approximation

On montre que certains groupes de Lie réels ou $p$-adiques vérifient une propriété (T) renforcée, pour des actions à petite croissance exponentielle dans des espaces de Hilbert ou des espaces de Banach uniformément convexes. On montre que les groupes hyperboliques n'ont pas cette propriété. On construit des familles d'expanseurs ne se plongeant uniformement dans aucun espace de Banach uniformement convexe.

We show a strong form of property (T) for some real or $p$-adic semisimple groups. This strong form of property (T) means that the trivial representation is isolated among representations with a small exponential growth in Hilbert spaces or even in uniformly convex Banach spaces.

We show that hyperbolic groups do not have strong property (T). We construct families of expanders which do not imbed uniformly in any uniformly convex Banach space