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Abstract: R\'esum\'e We correct the faulty formulas given in a previous article and we compute the defect group for the Iwasawa \(\lambda \) invariants attached to the S-ramified T-decomposed abelian pro- \(\ell \) -extensions over the \({{\mathbb {Z}}_\ell }\) -cyclotomic extension of a number field. As a consequence, we extend the results of Itoh, Mizusawa and Ozaki on tamely ramified Iwasawa modules for the cyclotomic \({{\mathbb {Z}}_\ell }\) -extension of abelian fields. PubDate: 2024-04-10 DOI: 10.1007/s40316-024-00223-w
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Abstract: Abstract We show that extended graph 4-manifolds (as defined by Frigerio–Lafont–Sisto in [12]) do not support Einstein metrics. PubDate: 2024-04-01 DOI: 10.1007/s40316-021-00192-4
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Abstract: Abstract Let \(p\ge 5\) be a prime. We construct modular Galois representations for which the \(\mathbb {Z}_p\) -corank of the p-primary Selmer group (i.e., its \(\lambda \) -invariant) over the cyclotomic \(\mathbb {Z}_p\) -extension is large. More precisely, for any natural number n, one constructs a modular Galois representation such that the associated \(\lambda \) -invariant is \(\ge n\) . The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form \(f_1\) satisfying suitable conditions, one constructs a congruent modular form \(f_2\) for which the \(\lambda \) -invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakhruddin–Khare–Patrikis, which extends previous work of Ramakrishna. The results are illustrated by explicit examples. PubDate: 2024-04-01 DOI: 10.1007/s40316-023-00212-5
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Abstract: Abstract The group \(Ham(M,\omega )\) of all Hamiltonian diffeomorphisms of a symplectic manifold \((M,\omega )\) plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry of \(Ham(M,\omega )\) , in the case where M is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in \(Ham(M,\omega )\) arbitrarily far from being a k-th power, with respect to the metric, for any \(k \ge 2\) . This part generalizes previous work by Polterovich and Shelukhin. Second, we show that the free group on two generators embeds into the asymptotic cone of \(Ham(M,\omega )\) . This part extends previous work by Alvarez-Gavela et al. Both extensions are based on two results from geometric group theory regarding incompressibility of surface embeddings. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00202-z
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Abstract: Abstract Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use a framework of variational eigenvalues and provide a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximisers (except for simply connected surfaces) are always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00207-8
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Abstract: Abstract This paper begins a new approach to the r-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic L-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur’s noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00200-1
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Abstract: R\'esum\'e We obtain non-Euclidean versions of classical theorems due to Hardy and Littlewood concerning smoothness of the boundary function of an analytic mapping on the unit disk with an appropriate growth condition. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00205-w
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Abstract: Abstract We extend the definition of the bounded reduction property to endomorphisms of automatic group and find conditions for it to hold. We study endomorphisms with L-quasiconvex image and prove that those with finite kernel satisfy a synchronous version of the bounded reduction property. Finally, we use these techniques to prove L-quasiconvexity of the equalizer of two endomorphisms under certain (strict) conditions. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00196-8
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Abstract: Abstract Let K be an imaginary quadratic field and \(K_\infty \) be the \({\textbf{Z}}_p^2\) -extension of K. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group over \(K_\infty \) associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of \(\Gamma \) -system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung’s chromatic Selmer groups of supersingular elliptic curves along \(K_\infty \) . PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00210-z
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Abstract: Abstract Let \(\ell \) be a rational prime. Previously, abelian \(\ell \) -towers of multigraphs were introduced which are analogous to \(\mathbb {Z}_{\ell }\) -extensions of number fields. It was shown that for towers of bouquets, the growth of the \(\ell \) -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for \(\mathbb {Z}_{\ell }\) -extensions of number fields). In this paper, we extend this result to abelian \(\ell \) -towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in \(\mathbb {Z}_\ell \) arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at \(u=1\) of the Artin–Ihara L-function, when the base multigraph is not necessarily a bouquet. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00194-w
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Abstract: Abstract We introduce multi-torsion, a spectral invariant generalizing Ray–Singer analytic torsion. We define multi-torsion for compact manifolds with a certain local geometric product structure that gives a bigrading on differential forms. We prove that multi-torsion is metric-independent in a suitable sense. Our definition of multi-torsion is inspired by an interpretation of each of analytic torsion and the eta invariant as a regularized integral of a closed differential form on a space of metrics on a vector bundle or on a space of elliptic operators. We generalize the Stokes’ theorem argument explaining the dependence of torsion and eta on the geometric data used to define them to the local product setting to prove our metric-independence theorem for multi-torsion. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00199-5
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Abstract: Abstract We give all normal integral bases for the simplest cubic field \(L_n\) generated by the roots of Shanks’ cubic polynomial when these bases exist, that is, \(L_n/{\mathbb {Q}}\) is tamely ramified. Furthermore, as an application of the result, we give an explicit relation between the roots of Shanks’ cubic polynomial and the Gaussian periods of \(L_n\) in the case that \(L_n/{\mathbb {Q}}\) is tamely ramified, which is a generalization of the work of Lehmer, Châtelet and Lazarus in the case that the conductor of \(L_n\) is equal to \(n^2+3n+9\) . PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00204-x
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Abstract: Abstract We study the adjoint Bloch–Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with \(\textrm{GSp}_{2g}\) . Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier. PubDate: 2024-04-01 DOI: 10.1007/s40316-022-00209-6
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Abstract: R\'esum\'e In this article, we give computable lower bounds for the first non-zero Steklov eigenvalue \(\sigma _1\) of a compact connected 2-dimensional Riemannian manifold M with several cylindrical boundary components. These estimates show how the geometry of M away from the boundary affects this eigenvalue. They involve geometric quantities specific to manifolds with boundary such as the extrinsic diameter of the boundary. In a second part, we give lower and upper estimates for the low Steklov eigenvalues of a hyperbolic surface with a geodesic boundary in terms of the length of some families of geodesics. This result is similar to a well known result of Schoen, Wolpert and Yau for Laplace eigenvalues on a closed hyperbolic surface. PubDate: 2024-03-26 DOI: 10.1007/s40316-024-00221-y
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Abstract: R\'esum\'e 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. PubDate: 2024-03-25 DOI: 10.1007/s40316-024-00222-x
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Abstract: Abstract We associate to a filtered \(\varphi \) -module \(\mathcal {D}\) a sub- \(\mathbb Z_p[[x]]\) -module of convergent series on the open unit disk in which the p-adic L-functions of the Galois representation associated to \(\mathcal {D}\) live (if they exist). This generalizes the already known case where \(\mathcal {D}\) is of dimension 2, for example associated to an elliptic curve or a modular form. PubDate: 2024-03-16 DOI: 10.1007/s40316-024-00224-9
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Abstract: Abstract 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. PubDate: 2024-03-14 DOI: 10.1007/s40316-024-00226-7
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Abstract: R'esum'e 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. PubDate: 2024-03-06 DOI: 10.1007/s40316-024-00225-8
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Abstract: Abstract This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds (M, g) whose boundary \(\partial M\) consists in two distinct connected components \(\Gamma _0\) and \(\Gamma _1\) . First, we show that the Steklov eigenvalues can be divided into two families \((\lambda _m^\pm )_{m \ge 0}\) which satisfy accurate asymptotics as \(m \rightarrow \infty \) . Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Steklov eigenfunctions are exponentially localized on the whole boundary \(\partial M\) as \(m \rightarrow \infty \) . When we add an asymmetric perturbation of the metric to a symmetric warped product, we observe in almost all cases a flea on the elephant effect. Roughly speaking, we prove that “half” the Steklov eigenfunctions are exponentially localized on one connected component of the boundary, say \(\Gamma _0\) , and the other half on the other connected component \(\Gamma _1\) as \(m \rightarrow \infty \) . PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00185-3
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Abstract: Given a compactly supported area-preserving diffeomorphism of the disk, we prove an integral formula relating the asymptotic action to the asymptotic winding number. As a corollary, we obtain a new proof of Fathi’s integral formula for the Calabi homomorphism on the disk. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00173-7
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