For the shifted convolution sum

$$D_h\left(X\right)={\sum }_{m=1}^{\infty }{\lambda }_1(1,m){\lambda }_2(m+h)V\left(\frac{m}{X}\right),$$

where ${\lambda }_1(1,m)$ are the Fourier coefficients of an $SL(3,\mathbb{Z})$ Maass form ${\pi }_1$, ${\lambda }_2\left(m\right)$ are those of an $SL(2,\mathbb{Z})$ Maass or holomorphic form ${\pi }_2$, and $1\le \left|h\right|\ll X^{1+\epsilon }$, we establish the bound

$$D_h\left(X\right){\ll }_{{\pi }_1,{\pi }_2,\epsilon }{X}^{1-1/20+\epsilon }.$$

The bound is uniform with respect to the shift $h$.

We study the regularity of the Lyapunov exponent for quasiperiodic cocycles $(T_{\omega },A)$ where $T_{\omega }$ is an irrational rotation $x\to x+2\pi \omega$ on ${\mathbb{S}}^1$ and $A\in {\mathcal{C}}^l({\mathbb{S}}^1,SL(2,\mathbb{R}\left)\right)$, $0\le l\le \infty$. For any fixed $l=0,1,2,\dots ,\infty$ and any fixed $\omega$ of bounded type, we construct $D_l\in {\mathcal{C}}^l({\mathbb{S}}^1,SL(2,\mathbb{R}\left)\right)$ such that the Lyapunov exponent is not continuous at $D_l$ in ${\mathcal{C}}^l$-topology. We also construct such examples in a smaller Schrödinger class.

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabilizer of some end is metabelian, then there is a local field $k$ such that ${PSL}_2\left(k\right)\le G\le {PGL}_2\left(k\right)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank $1$ simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and we provide a complete classification in case the root groups are torsion-free.

We prove the existence of secondary terms of order $X^{5/6}$ in the Davenport–Heilbronn theorems on cubic fields and $3$-torsion in class groups of quadratic fields. For cubic fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also prove a variety of generalizations, including to arithmetic progressions, where we discover a curious bias in the secondary term.

Roberts’s conjecture has also been proved independently by Bhargava, Shankar, and Tsimerman. In contrast to their work, our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms, developed by Shintani and Datskovsky–Wright.

We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen–Macaulay.

Étant donné un nombre premier impair $p$ et un corps $p$-adique $K$, on développe dans cet article un analogue de la théorie des $(\phi ,\Gamma )$-modules de Fontaine en remplaçant la $p$-extension cyclotomique par l’extension $K_{\infty }$ de $K$ obtenue en ajoutant un système compatible de racines $p^n$-ièmes d’une uniformisante $\pi$ fixée. Ceci nous conduit à une nouvelle classification des représentations $p$-adiques de $G_K=Gal(\mathrm{K̄}/K)$ via des $(\phi ,\tau )$-modules. Nous établissons ensuite un lien entre la théorie des $(\phi ,\tau )$-modules à celle des $(\phi ,N_{\nabla })$-modules de Kisin. Comme corollaire, nous répondons à une question de Tong Liu en démontrant que, lorsque $K$ est une extension finie de ${\mathbb{Q}}_p$, toute représentation de $E\left(u\right)$-hauteur finie de $G_K$ est potentiellement semi-stable.

Let $p$ be an odd prime number, and let $K$ be a $p$-adic field. In this paper, we develop an analogue of Fontaine’s theory of $(\phi ,\Gamma )$-modules replacing the $p$-cyclotomic extension by the extension $K_{\infty }$ obtained by adding to $K$ a compatible system of $p^n$-th roots of a fixed uniformizer $\pi$ of $K$. As a result, we obtain a new classification of $p$-adic representations of $G_K=Gal(\mathrm{K̄}/K)$ by some $(\phi ,\tau )$-modules. We then make a link between the theory of $(\phi ,\tau )$-modules discussed above and the so-called theory of $(\phi ,N_{\nabla })$-modules developed by Kisin. As a corollary, we answer a question of Tong Liu: we prove that, if $K$ is a finite extension of ${\mathbb{Q}}_p$, then every representation of $G_K$ of $E\left(u\right)$-finite height is potentially semistable.