This article studies the orthogonal hypergeometric groups of degree five. We establish the thinness of 12 out of the 19 hypergeometric groups of type O(3, 2) from [4, Table 6]. Some of these examples are associated with Calabi-Yau 4-folds. We also establish the thinness of 9 out of the 17 hypergeometric groups of type O(4, 1) from [13], where the thinness of 7 other cases was already proven. The O(4, 1) type groups were predicted to be all thin and our result leaves just one case open.
In this paper, we show that for a sequence of orientable complete uniformly asymptotically flat 3-manifolds ((M_i, g_i)) with nonnegative scalar curvature and ADM mass (m(g_i)) tending to zero, by subtracting some open subsets (Z_i), whose boundary area satisfies (textrm{Area}(partial Z_i) le C m(g_i)^{frac{1}{2}- varepsilon }), for any base point (p_i in M_i{setminus } Z_i), ((M_i{setminus } Z_i, g_i, p_i)) converges to the Euclidean space (({mathbb {R}}^3, g_E, 0)) in the (C^0) modulo negligible volume sense. Moreover, if we assume that the Ricci curvature is uniformly bounded from below, then ((M_i, g_i, p_i)) converges to (({mathbb {R}}^3, g_E, 0)) in the pointed Gromov–Hausdorff topology.
We investigate the question of sharp upper bounds for the Steklov eigenvalues of a hypersurface of revolution in Euclidean space with two boundary components, each isometric to ({mathbb {S}}^{n-1}). For the case of the first non zero Steklov eigenvalue, we give a sharp upper bound (B_n(L)) (that depends only on the dimension (n ge 3) and the meridian length (L>0)) which is reached by a degenerated metric (g^*) that we compute explicitly. We also give a sharp upper bound (B_n) which depends only on n. Our method also permits us to prove some stability properties of these upper bounds.
We give a description of the growth rates of (L^2)-normalized Laplace eigenfunctions on the unit disk with Dirichlet and Neumann boundary conditions. In particular, we show that the growth rates of both Dirichlet and Neumann eigenfunctions are bounded away from zero. Our approach starts with P. Sarnak growth exponents and uses several key asymptotic formulas for Bessel functions or their zeros.
We show that if a diffeomorphism of a symplectic manifold ((M^{2n},omega )) preserves the form (omega ^{k}) for (0< k < n) and is connected to identity through such diffeomorphisms then it is indeed a symplectomorphism.
For a given Coleman family of modular forms, we construct a formal model and prove the existence of a family of Galois representations associated to the Coleman family. As an application, we study the variations of Iwasawa (lambda )- and (mu )-invariants of dual fine (strict) Selmer groups over the cyclotomic (mathbb {Z}_p)-extension of (mathbb {Q}) in Coleman families of modular forms. This generalizes an earlier work of Jha and Sujatha for Hida families.
We give a new description of the logarithm matrix of a modular form in terms of distributions, generalizing the work of Dion and Lei for the case (a_p=0). What allows us to include the case (a_pne 0) is a new definition, that of a distribution matrix, and the characterization of this matrix by p-adic digits. One can apply these methods to the corresponding case of distributions in multiple variables.
In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm–Liouville eigenvalue problem where the density is a function h(x) whose some power is concave. We prove existence of a maximizer for (mu _k(h)) and we completely characterize it. Then we consider the Neumann eigenvalues (for the Laplacian) of a domain (Omega subset {mathbb {R}}^d) of given diameter and we assume that its profile function (defined as the (d-1) dimensional measure of the slices orthogonal to a diameter) has also some power that is concave. This includes the case of convex domains in ({mathbb {R}}^d), containing and generalizing previous results by P. Kröger. On the other hand, in the last section, we give examples of domains for which the upper bound fails to be true, showing that, in general, (sup D^2(Omega )mu _k(Omega )= +infty ).
Bleher et al. began studying higher codimension Iwasawa theory for classical Iwasawa modules. Subsequently, Lei and Palvannan studied an analogue for elliptic curves with supersingular reduction. In this paper, we obtain a vast generalization of the work of Lei and Palvannan. A key technique is an approach to the work of Bleher et al. that the author previously proposed. For this purpose, we also study the structure of ±-norm subgroups and duality properties of multiply-signed Selmer groups.
Let p be a rational prime, let F denote a finite, unramified extension of (mathbb {Q}_p), let K be the completion of the maximal unramified extension of (mathbb {Q}_p), and let (overline{K}) be some fixed algebraic closure of K. Let A be an abelian variety defined over F, with good reduction, let (mathcal {A}) denote the Néron model of A over (textrm{Spec}(mathcal {O}_F)), and let (widehat{mathcal {A}}) be the formal completion of (mathcal {A}) along the identity of its special fiber, i.e. the formal group of A. In this work, we prove two results concerning the ramification of p-power torsion points on (widehat{mathcal {A}}). One of our main results describes conditions on (widehat{mathcal {A}}), base changed to (text {Spf}(mathcal {O}_K) ), for which the field (K(widehat{mathcal {A}}[p])/K) i s a tamely ramified extension where (widehat{mathcal {A}}[p]) denotes the group of p-torsion points of (widehat{mathcal {A}}) over (mathcal {O}_{overline{K}}). This result generalizes previous work when A is 1-dimensional and work of Arias-de-Reyna when A is the Jacobian of certain genus 2 hyperelliptic curves.
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