Abstract: We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight

Abstract: We establish optimal Lebesgue estimates for a class of generalized Radon transforms defined by averaging functions along polynomial-like curves. The presence of an essentially optimal weight allows us to prove uniform estimates, wherein the Lebesgue exponents are completely independent of the curves and the operator norms depend only on the polynomial degree. Moreover, our weighted estimates

Abstract: Let $p$ be a prime, $K$ a finite extension of $\\Bbb\{Q\}_p$, and let $G_K$ be the absolute Galois group of $K$. The category of \\'etale $(\\varphi,\\tau)$-modules is equivalent to the category of $p$-adic Galois representations of $G_K$. In this paper, we show that all \\'etale $(\\varphi,\\tau)$-modules are overconvergent; this answers a question of Caruso. Our result is an analogue of

Abstract: We study the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of \{\\it a priori\} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally

Abstract: Let $X$ be a smooth projective variety defined over $\\overline\{\\Bbb\{Q\}\}$, and $f\\colon X\\dashrightarrow X$ be a dominant rational map. Let $\\delta_f$ be the first dynamical degree of $f$ and $h_X\\colon X(\\overline\{\\Bbb\{Q\}\})\\rightarrow [1,\\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds

Abstract: We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp $L^\{n-1\}$-estimates for the regularity scale of the level set flow with two-convex initial data. The results, which seem new even in the most classical case of mean convex surfaces evolving by mean curvature

Abstract: We use a weighted variant of the frequency functions introduced by Almgren to prove sharp asymptotic estimates for almost eigenfunctions of the drift Laplacian associated to the Gaussian weight on an asymptotically conical end. As a consequence, we obtain a purely elliptic proof of a result of L. Wang on the uniqueness of self-shrinkers of the mean curvature flow asymptotic to a given cone

Abstract: In an earlier paper, one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincar\\'e duality groups for each $n\\geq 4$. We show that those groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous

Abstract: We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\\xi|<\\epsilon$ for a generic ternary form $Q$ is $|x|\\ll\\epsilon^\{-1\}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable

Abstract: We obtain compact orientable embedded surfaces with constant mean curvature $0

AMERICAN JOURNAL OF MATHEMATICS Founded in 1878 by Johns Hopkins University INDEX TO VOLUME 142 2020 PAGE BECEANU, M. and W. SCHLAG. Structure formulas for wave operators . 751 BERGER, TOBIAS, KRZYSZTOF KLOSIN, CRIS POOR, JERRY SHURMAN , and DAVID S. YUEN. Deformations of Saito-Kurokawa type and the Paramodular Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1821 BERNDTSSON

Abstract: We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=\xi_0+\xi_1 x+\cdots+{1\over\sqrt{n!}}\xi_n x^n$, where $\xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real

Abstract: Let $F$ be a number field and let $\Bbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $\nu:B\to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by $$ M(R):=\big\{\big(\gamma_1,\gamma_2\big)\in\big(B\otimes_F R\big)^2:\nu\big(\gamma_1\big)=\nu\big(\gamma_2\big)\big\}. $$ Motivated by an influential

Abstract: The primal cohomology $\Bbb{K}_{\Bbb{Q}}$ of the theta divisor $\Theta$ of a principally polarized abelian fivefold (ppav) is the direct sum of its invariant and anti-invariant parts $\Bbb{K}_{\Bbb{Q}}^{+1}$, resp. $\Bbb{K}_{\Bbb{Q}}^{-1}$ under the action of $-1$. For smooth $\Theta$, these have dimension $6$ and $72$ respectively. We show that $\Bbb{K}_{\Bbb{Q}}^{+1}$ consists of Hodge

Abstract: We consider the nonlinear heat equation $u_t-\Delta u=|u|^\alpha u$ on $\Bbb{R}^N$, where $\alpha>0$ and $N\ge 1$. We prove that in the range $0<\alpha<{4\over {N-2}}$, for every $\mu>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu|x|^{-{2\over\alpha}}$. The construction is based on the analysis of the related inverted

Abstract: In this paper we consider the defocusing energy critical wave equation with a trapping potential in dimension $3$. We prove that the set of initial data for which solutions scatter to an unstable excited state $(\phi,0)$ forms a finite co-dimensional path connected $C^1$ manifold in the energy space. This manifold is a global and unique center-stable manifold associated with $(\phi,0)$. It

Abstract: We consider a rigidity problem for the spectral gap of the Laplacian on an ${\rm RCD}(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng-Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional

Abstract: Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline{x})$ over a finite field of characteristic $p$ defines an Artin-Schreier-Witt tower of varieties whose Galois group is isomorphic to $\Bbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a nontrivial finite character of $\Bbb{Z}_p$ and a generic polynomial whose convex hull

Abstract: We give a simple sufficient condition for a Verdier quotient $\scr{T}/\scr{S}$ of a triangulated category $\scr{T}$ by a thick subcategory $\scr{S}$ to be realized inside of $\scr{T}$ as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot-Guo-Keller.

abstract: We study the $p$-adic (generalized) hypergeometric equations by using the theory of multiplicative convolution of arithmetic $\scr{D}$-modules. As a result, we prove that the hypergeometric isocrystals with suitable rational parameters have a structure of overconvergent $F$-isocrystals.

abstract: We present new free-curvature one-cycle sweepout estimates in Riemannian geometry, both on surfaces and in higher dimension. More precisely, we derive upper bounds on the length of one-parameter families of one-cycles sweeping out essential surfaces in closed Riemannian manifolds. In particular, we show that there exists a homotopically substantial one-cycle sweepout of the essential sphere

abstract: We establish an improvement of flatness result for critical points of Ginzburg-Landau energies with long-range interactions. It applies in particular to solutions of $(-\Delta)^{s/2}u=u-u^3$ in $\Bbb{R}^n$ with $s\in(0,1)$. As a corollary, we establish that solutions with asymptotically flat level sets are $1$D and prove the analogue of the De Giorgi conjecture (in the setting of minimizers)

abstract: S. Seto, L. Wang, and G. Wei proved that the gap between the first two Dirichlet eigenvalues of a convex domain in the unit sphere is at least as large as that for an associated operator on an interval with the same diameter, provided that the domain has the diameter at most $\pi/2$. In this paper, we extend Seto-Wang-Wei's result to convex domains in the unit sphere with diameter less than

abstract: We extend work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram. More precisely we establish sharp, uniform bounds for complete exponential sums and the number of polynomial congruences for general quasi-homogeneous polynomials in two variables

abstract: Let $K$ be a discretely valued field with Henselian valuation ring and separably closed (but not necessarily perfect) residue field of characteristic $p$, $H$ a connected reductive $K$-group, and $\Theta$ a finite group of automorphisms of $H$. We assume that $p$ does not divide the order of $\Theta$ and Bruhat-Tits theory is available for $H$ over $K$ with $\scr{B}(H/K)$ the Bruhat-Tits

abstract: In this paper, we prove that there are no solutions for the curvature equation $$ \Delta u+e^u=8\pi n\delta_0\ {\rm on}\ E_{\tau},\quad n\in\Bbb{N}, $$ where $E_{\tau}$ is a flat rectangular torus and $\delta_0$ is the Dirac measure at the lattice points. This confirms a conjecture of Lin and Wang and also improves a result of Eremenko and Gabrielov. The nonexistence is a delicate problem

abstract: We establish quantitative estimates for sampling (dominating) sets in model spaces associated with meromorphic inner functions, i.e., those corresponding to de Branges spaces. Our results encompass the Logvinenko-Sereda-Panejah (LSP) Theorem including Kovrijkine's optimal sampling constants for Paley-Wiener spaces. It also extends Dyakonov's LSP theorem for model spaces associated with bounded