Soient $p$ un nombre premier et $\mathcal{C}$ la courbe de Hecke $p$-adique de $\mathrm{GL}_2$ de niveau moderé $1$ introduite par Coleman et Mazur. Nous montrons que $\mathcal{C}$ est lisse en ses points Eisenstein critiques et nous donnons une condition nécessaire et suffisante pour que l’application poids soit étale en un tel point en terme de certaines valeurs de la fonction $\zeta$$p$-adique. Un point important est la détermination en ces points du lieu de réductibilité schématique du pseudo-caractère Galoisien porté par $\mathcal{C}$, et ce restreint à un groupe de décomposition en $p$. L’assertion de lissité découle alors de ce que les fonctions $L$ de Dirichlet ont au plus des zéros simples aux entiers.

Let $p$ be a prime number and $\mathcal{C}$ be the $p$-adic tame level $1$ eigencurve introduced by Coleman and Mazur. We prove that $\mathcal{C}$ is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of $p$-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by $\mathcal{C}$ restricted to a decomposition group at $p$. Then, the smoothness appears to be a consequence of the fact that the Dirichlet $L$-functions only have simple zeros at integers.

Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.

Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$. When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$. For $r=\dim \overline{X}$, we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$-cycles with modulus.

The aim of this article is to study the Bloch–Kato exponential map and the Perrin-Riou big exponential map purely in terms of $(\varphi , \Gamma )$-modules over the Robba ring. We first generalize the definition of the Bloch–Kato exponential map for all the $(\varphi , \Gamma )$-modules without using Fontaine’s rings ${\mathbf{B} }_{\mathrm{crys} } $, ${\mathbf{B} }_{\mathrm{dR} } $ of $p$-adic periods, and then generalize the construction of the Perrin-Riou big exponential map for all the de Rham $(\varphi , \Gamma )$-modules and prove that this map interpolates our Bloch–Kato exponential map and the dual exponential map. Finally, we prove a theorem concerning the determinant of our big exponential map, which is a generalization of theorem $\delta (V)$ of Perrin-Riou. The key ingredients for our study are Pottharst’s theory of the analytic Iwasawa cohomology and Berger’s construction of $p$-adic differential equations associated to de Rham $(\varphi , \Gamma )$-modules.

Nous étudions la nature arithmétique de $q$-analogues des valeurs $\zeta(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)=\sum_{k=1}^{\infty}q^k\sum_{d\mid k}d^{s-1}$, $s=1,2,\dots$, où $q$ est un nombre complexe, $|q|<1$ (ces fonctions sont intimement liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta_q(3),\zeta_q(5),\dots,\zeta_q(M)$ est au moins $c_1\sqrt{M}$, avec $c_1=0,3358$. Ce résultat peut être considéré comme un $q$-analogue du résultat de Rivoal et de Ball et Rivoal, qui affirme que la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta(3),\zeta(5),\dots,\zeta(M)$ est au moins $c_2\log M$, avec $c_2=0,5906$. Pour les mêmes valeurs de $q$, une minoration similaire pour les valeurs $\zeta_q(s)$ aux entiers $s$ pairs nous permet de redémontrer un cas particulier d’un résultat de Bertrand qui affirme la transcendance sur $\mathbb{Q}$ de l’une des deux séries d’Eisenstein $E_4(q)$ et $E_6(q)$ pour tout nombre complexe $q$ tel que $0<|q|<1$.

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

We consider the inverse connection problem consisting of determining a gauge field on $\mathbb{R}^d$ from its non-abelian Radon transform along oriented straight lines. The determination is considered modulo gauge transformations. Our results include: global uniqueness theorems for $d\geq3$, new local uniqueness theorems for $d=2$, constructive proofs (i.e. proofs containing reconstruction procedures), counterexamples to the global uniqueness for $d=2$, a reduction to the attenuated X-ray transform.

AMS 2000 Mathematics subject classification: Primary 53C65; 81U40

We describe the structure of ‘K-approximate subgroups’ of torsion-free nilpotent groups, paying particular attention to Lie groups.

Three other works, by Fisher et al., by Sanders and by Tao, have appeared that independently address related issues. We comment briefly on some of the connections between these papers.

We construct a bad field in characteristic zero. That is, we construct an algebraically closed field which carries a notion of dimension analogous to Zariski-dimension, with an infinite proper multiplicative subgroup of dimension one, and such that the field itself has dimension two. This answers a longstanding open question by Zilber.

Dans ce papier nous étudions une correspondance de Jacquet–Langlands locale pour toutes les représentations lisses irréductibles. La correspondance est caractérisée par le fait qu’elle respecte la correspondance de Jacquet–Langlands classique et commute avec le foncteur d’induction parabolique. Elle est compatible dans un sens à préciser au foncteur de Jacquet et à l’involution d’Aubert–Schneider–Stuhler. Nous utilisons cette correspondance pour montrer qu’une certaine classe de représentations d’une forme intérieure de $\mathrm{GL}_n$ sur un corps $p$-adique sont unitarisables. C’est le premier pas dans la preuve de la conjecture U1 de Tadić.

We study a local Jacquet–Langlands correspondence for all smooth irreducible representations. This correspondence is characterized by the fact that it respects the classical Jacquet–Langlands correspondence and it commutes with the parabolic induction functor. It has good behavior with respect to the Jacquet’s functor and the involution of Aubert–Schneider–Stuhler. Using this correspondence, we prove some particular cases of the global Jacquet–Langlands correspondence and we deduce that a certain class of representations of an inner form of $\mathrm{GL}_n$ over a $p$-adic field are unitarizable. This is the first step towards the proof of Conjecture U1 of Tadić.

We prove uniform Sobolev estimates $\Vert u\Vert _{{L}^{p\prime } } \leq C\Vert (\Delta - \alpha )u\Vert _{{L}^{p} } $ for $\alpha \in \mathbb{C} $ and $p= 2n/ (n+ 2), {p}^{\prime } = 2n/ (n- 2)$ on non-trapping asymptotically conic manifolds of dimension $n\geq 3$, generalizing to non-constant coefficient Laplacians a result of Kenig, Ruiz and Sogge [13].

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble $\text{CLE}_{4}$ and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the $\text{CLE}_{4}$ carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of $\text{CLE}_{4}$) are in fact measurable functions of the GFF.

Given k ≥ 2, we construct a (2k − 2)-parameter family of properly embedded minimal surfaces in ℍ2 × ℝ invariant by a vertical translation T, called saddle towers, which have total intrinsic curvature 4π(1 − k), genus zero and 2k vertical Scherk-type ends in the quotient by T. Each of those examples is obtained from the conjugate graph of a Jenkins–Serrin graph over a convex polygonal domain with 2k edges of the same (finite) length. As limits of saddle towers, we obtain properly embedded minimal surfaces, called minimal k-noids, which are symmetric with respect to a horizontal slice (in fact they are vertical bi-graphs) and have total intrinsic curvature 4π(1 − k), genus zero and k vertical planar ends.

For a function $f\in L^p(\mathbb{R}d)$, $d\ge 2$, let $A_tf(x)$ be the mean of $f$ over the sphere of radius $t$ centred at $x$. Given a set $E\subset(0,\infty)$ of dilations we prove various endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E}|A_tf(x)|$, under some regularity assumptions on $E$.

AMS 2000 Mathematics subject classification: Primary 42B25. Secondary 42B20

Given a prime $p\gt 2$, an integer $h\geq 0$, and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $, we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$, to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $-coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$-adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope $\leq h$.

Pour chaque entier $d=2,3,4$, il existe un corps $F$ de dimension cohomologique $1$ et une surface de del Pezzo de degré $d$ sur $F$ sans zéro-cycle de degré $1$, en particulier sans point rationnel. Les démonstrations utilisent le théorème de Merkur’ev et Suslin, le théorème de Riemann-Roch sur une surface et la formule du degré de Rost.

We propose a geometric method to measure the wild ramification of a smooth étale sheaf along the boundary. Using the method, we study the graded quotients of the logarithmic ramification groups of a local field of characteristic p > 0 with arbitrary residue field. We also define the characteristic cycle of an ℓ-adic sheaf, satisfying certain conditions, as a cycle on the logarithmic cotangent bundle and prove that the intersection with the 0-section computes the characteristic class, and hence the Euler number.

The admissible representations of a real reductive group G are known by work of Langlands, Knapp, Zuckerman and Vogan. This paper describes an effective algorithm for computing the irreducible representations of G with regular integral infinitesimal character. The algorithm also describes structure theory of G, including the orbits of K(ℂ) (a complexified maximal compact subgroup) on the flag variety. This algorithm has been implemented on a computer by the second author, as part of the ‘Atlas of Lie Groups and Representations’ project.

We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles $F_{p}$ . For $p\in \mathbf{N}$ , the flat vector bundle $F_{p}$ is the direct image of $L^{p}$ , where $L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.

In this paper we study the reduction to characteristic $p$ of the Shimura variety associated to a unitary group which has signature $(n-1,1)$ at its real place. We describe the Newton polygon, the Ekedahl–Oort, and the final stratification. In addition we examine the moduli space of $p$-isogenies using a variant of the local model for Shimura varieties. We apply our results to obtain a proof of the Eichler–Shimura congruence relation for the case that $n$ is even.