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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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Abstract: Abstract We study the algebraicity of the central critical values of twisted triple product L-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result with Cheng on Deligne’s conjecture for certain automorphic L-functions for \({\text {GL}}_3 \times {\text {GL}}_2\) . PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00169-3

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Abstract: Abstract In a previous article [7], the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter t running through \({\mathbb {Q}}\) . However, a well-known example of Washington has root number \(-1\) for every fiber when t runs through \({\mathbb {Z}}\) . Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. This result depends on the squarefree conjecture and Chowla’s conjecture, and is unconditional in many cases. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00164-8

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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 a certain class of 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 give a generalization to a broader class of regular abelian \(\ell \) -towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in \({\mathbb {Z}}_{\ell }\) and then study the special value at \(u=1\) of the Artin-Ihara L-function \(\ell \) -adically. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00183-5

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Abstract: Abstract Let G be a semisimple compact connected Lie group. An N-fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N-fold reduced product of G. Suzuki and Takakura gave a volume formula for the N-fold reduced product of \( \mathbf {SU}(3) \) in [25] by using geometric quantization and the Riemann–Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of \( \mathbf {SU}(3) \) . PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00171-9

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Abstract: Abstract In this paper, we study an almost coKähler manifold admitting certain metrics such as \(*\) -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKähler 3-manifold (M, g) admitting a \(*\) -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKähler \((\kappa ,\mu )\) -almost coKähler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a \((\kappa , \mu )\) -almost coKähler manifold (M, g) is coKähler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKähler manifolds which are non-coKähler. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00162-w

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Abstract: Abstract We consider an automorphism of an arbitrary CAT(0) cube complex. We study its combinatorial displacement and we show that either the automorphism has a fixed point or it preserves some combinatorial axis. It follows that when a f.g. group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls. As an application Baumslag-Solitar groups and Heisenberg groups provide examples of groups having a proper action on measured spaces with walls, but no proper action on a discrete space with wall. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00186-2

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Abstract: Abstract Let \(G=A *B\) be a free product of freely indecomposable groups. We explicitly construct quasimorphisms on G which are invariant with respect to all automorphisms of G. We also prove that the space of such quasimorphisms is infinite-dimensional whenever G is not the infinite dihedral group. As an application we prove that an invariant analogue of stable commutator length recently introduced by Kawasaki and Kimura is non-trivial for these groups. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00184-4

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Abstract: Abstract Let \(f(z)=q+\sum _{n\ge 2}a(n)q^n\) be a weight k normalized newform with integer coefficients and trivial residual mod 2 Galois representation. We extend the results of Amir and Hong in Amir and Hong (On L-functions of modular elliptic curves and certain K3 surfaces, Ramanujan J, 2021) for \(k=2\) by ruling out or locating all odd prime values \( \ell <100\) of their Fourier coefficients a(n) when n satisfies some congruences. We also study the case of odd weights \(k\ge 1\) newforms where the nebentypus is given by a quadratic Dirichlet character. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00168-4

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Abstract: Abstract In this paper we consider the Diophantine equation \(U_n=p^x\) where \(U_n\) is a linear recurrence sequence, p is a prime number, and x is a positive integer. Under some technical hypotheses on \(U_n\) , we show that, for any p outside of an effectively computable finite set of prime numbers, there exists at most one solution (n, x) to that Diophantine equation. We compute this exceptional set for the Tribonacci sequence and for the Lucas sequence plus one. PubDate: 2023-10-01 DOI: 10.1007/s40316-021-00163-9

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Abstract: Abstract 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. PubDate: 2023-08-03 DOI: 10.1007/s40316-023-00219-y

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Abstract: Abstract 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. PubDate: 2023-08-02 DOI: 10.1007/s40316-023-00220-5

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Abstract: Abstract 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. PubDate: 2023-07-06 DOI: 10.1007/s40316-023-00217-0

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Abstract: Abstract 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 \(a_p\ne 0\) 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. PubDate: 2023-06-21 DOI: 10.1007/s40316-023-00215-2

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Abstract: Abstract 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, \(\sup D^2(\Omega )\mu _k(\Omega )= +\infty \) . PubDate: 2023-06-10 DOI: 10.1007/s40316-023-00218-z

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Abstract: R\'esum\'e 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. PubDate: 2023-05-15 DOI: 10.1007/s40316-023-00216-1

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Abstract: Abstract 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. PubDate: 2023-05-11 DOI: 10.1007/s40316-023-00214-3

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Abstract: Abstract We study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28]. PubDate: 2023-04-01 DOI: 10.1007/s40316-022-00193-x