Authors:Tadashi Ochiai; Kazuma Shimomoto Abstract: The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over \({\mathcal {O}}[[x_1,\ldots ,x_d]]\) , where \({\mathcal {O}}\) is the ring of integers of a finite extension of the field of p-adic integers \({\mathbb {Q}}_p\) . The specialization method is a technique that recovers the information on the characteristic ideal \({\text {char}}_R (M)\) from \({\text {char}}_{R/I}(M/IM)\) , where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen–Macaulay normal domains by combining the main results in Ochiai (Nagoya Math J 218:125–173, 2015) and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in Ochiai (Compos Math 142:1157–1200, 2006). PubDate: 2018-02-20 DOI: 10.1007/s40316-018-0099-0

Authors:Bernadette Faye; Florian Luca; Pieter Moree Abstract: We consider the family of Lucas sequences uniquely determined by \(U_{n+2}(k)=(4k+2)U_{n+1}(k) -U_n(k),\) with initial values \(U_0(k)=0\) and \(U_1(k)=1\) and \(k\ge 1\) an arbitrary integer. For any integer \(n\ge 1\) the discriminator function \(\mathcal {D}_k(n)\) of \(U_n(k)\) is defined as the smallest integer m such that \(U_0(k),U_1(k),\ldots ,U_{n-1}(k)\) are pairwise incongruent modulo m. Numerical work of Shallit on \(\mathcal {D}_k(n)\) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every \(k\ge 1\) there is a constant \(n_k\) such that \({\mathcal D}_{k}(n)\) has a simple characterization for every \(n\ge n_k\) . The case \(k=1\) turns out to be fundamentally different from the case \(k>1\) . PubDate: 2018-02-12 DOI: 10.1007/s40316-017-0097-7

Authors:André Boivin; Paul M. Gauthier; Myrto Manolaki Abstract: In this paper we study some questions related to the zero sets of harmonic and real analytic functions in \({\mathbb {R}}^N\) . We introduce the notion of analytic uniqueness sequences and, as an application, we show that the zero set of a non-constant real analytic function on a domain always has empty fine interior. We also prove that, for a certain category of sets \(E\subset {\mathbb {R}}^N\) (containing the finely open sets), each function f defined on E is the restriction of a real analytic (respectively harmonic) function on an open neighbourhood of E if and only if f is “analytic (respectively harmonic) at each point” of E. PubDate: 2018-02-01 DOI: 10.1007/s40316-018-0098-1

Authors:Kevin Buzzard; Alan Lauder Pages: 213 - 219 Abstract: We report on a computation of holomorphic cuspidal modular forms of weight one and small level (currently level at most 1500) and classification of them according to the projective image of their attached Artin representations. The data we have gathered, such as Fourier expansions and projective images of Hecke newforms and dimensions of space of forms, is available in both Magma and Sage readable formats on a webpage created in support of this project. PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0072-8 Issue No:Vol. 41, No. 2 (2017)

Authors:Stéphane Dellacherie; Olivier Lafitte Pages: 221 - 264 Abstract: We study a monodimensional stationary system coupling a simplified thermohydraulic model to a simplified neutronic model based on the diffusion approximation with one energy group. We observe that this non-linear coupled model can be studied under two points of view and we show that solving this model is equivalent to the resolution of a non-linear scalar autonomous ordinary differential equation of order 1. As a consequence, it is possible to obtain an explicit solution without using an iterative coupling algorithm solving successively the thermohydraulics and the neutronics. Moreover, we obtain an analytic solution in a simple case. The explicit results obtained with our analytical solutions confirm the numerical results obtained with the iterative classical thermohydraulics–neutronics algorithm. PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0073-7 Issue No:Vol. 41, No. 2 (2017)

Authors:S. Mirvakili; P. Ghiasvand; B. Davvaz Pages: 265 - 276 Abstract: In this paper, we introduce and analyze a fundamental strongly regular equivalence relation on a hypermodule over a hyperring which is the smallest equivalence relation such that the quotient is cyclic module over a (fundamental) ring. Then we state the conditions that is equivalent with the transitivity of this relation. Finally, a characterization of the derived hypermodule (with canonical hypergroup) over a Krasner hyperring has been considered. PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0074-6 Issue No:Vol. 41, No. 2 (2017)

Authors:Bruno Poizat Pages: 277 - 307 Abstract: We define in the infinitely generated free models of an arbitrary equational class an independence relation, which is necessarily the model-theoretic independence over the empty set when these structures happen to be \(\upomega \) -homogeneous stable groups. We establish the basic properties of this independence relation, give some examples, and ask some questions concerning its model-theoretic behaviour (many of them dealing with the treatment of the free models in Positive Logic). PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0075-5 Issue No:Vol. 41, No. 2 (2017)

Authors:Marine Rougnant Pages: 309 - 335 Abstract: Résumé Nous nous intéressons dans ce travail aux pro-p groupes \(G_S\) , groupes de Galois de pro-p extensions maximales de corps de nombres non ramifiées en dehors d’un ensemble fini S de places ne divisant pas p, et plus particulièrement à la propagation de la propriété mild au-dessus d’une extension quadratique imaginaire. Notre point de départ est le critère de Labute-Schmidt (Schmidt, Doc Math 12:441–471, 2007), basé sur l’étude du cup-produit sur le groupe de cohomologie \(H^1(G_S,\mathbb {F}_p)\) . Dans un contexte favorable, nous montrons par le calcul que le groupe étudié vérifie souvent une version faible ( \(LS_f\) ) du critère de Labute-Schmidt. Un critère théorique est ensuite établi, permettant de montrer le caractère mild de certains groupes auxquels le critère ( \(LS_f\) ) ne s’applique pas. Ce critère théorique est enfin appliqué à des exemples pour \(p=3\) et comparé aux travaux de Labute et Vogel (Labute, J Reine Angew Math 596:155–182, 2006 et Vogel, Circular sets of primes of imaginary quadratic number fields, 2006). PubDate: 2017-10-01 DOI: 10.1007/s40316-016-0071-9 Issue No:Vol. 41, No. 2 (2017)

Authors:Antoine Métras Abstract: We consider the Dirichlet eigenvalue problem on a polytope. We use the Rellich identity to obtain an explicit formula expressing the Dirichlet eigenvalue in terms of the Neumann data on the faces of the polytope of the corresponding eigenfunction. The formula is particularly simple for polytopes admitting an inscribed ball tangent to all the faces. Our result could be viewed as a generalization of similar identities for simplices recently found by Christianson (Equidistribution of Neumann data mass on simplices and a simple inverse problem, ArXiv e-prints, 2017, Equidistribution of Neumann data mass on triangles. ArXiv e-prints, 2017). PubDate: 2017-11-06 DOI: 10.1007/s40316-017-0096-8

Authors:Jay Jorgenson; Anna-Maria von Pippich; Lejla Smajlović Abstract: We develop two applications of the Kronecker’s limit formula associated to elliptic Eisenstein series: A factorization theorem for holomorphic modular forms, and a proof of Weil’s reciprocity law. Several examples of the general factorization results are computed, specifically for certain moonshine groups, congruence subgroups, and, more generally, non-compact subgroups with one cusp. In particular, we explicitly compute the Kronecker limit function associated to certain elliptic fixed points for a few small level moonshine groups. PubDate: 2017-10-31 DOI: 10.1007/s40316-017-0094-x

Authors:Eudes Leite de Lima; Henrique Fernandes de Lima Abstract: We obtain a sharp estimate to the scalar curvature of stochastically complete hypersurfaces immersed with constant mean curvature in a locally symmetric Riemannian space obeying standard curvature constraints (which includes, in particular, a Riemannian space with constant sectional curvature). For this, we suppose that these hypersurfaces satisfy a suitable Okumura-type inequality recently introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014), which is a weaker hypothesis than to assume that they have two distinct principal curvatures. Our approach is based on the equivalence between stochastic completeness and the validity of the weak version of the Omori–Yau’s generalized maximum principle, which was established by Pigola et al. (Proc Am Math Soc 131:1283–1288, 2002; Mem Am Math Soc 174:822, 2005). PubDate: 2017-10-28 DOI: 10.1007/s40316-017-0095-9

Authors:Thong Nguyen Quang Do Abstract: Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants \(\lambda \) and \(\mu \) associated to the cyclotomic \({\mathbb {Z}}_p\) -extension of any totally real number field F should vanish. In his foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when p is undecomposed, resp. totally decomposed in F. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg. PubDate: 2017-10-20 DOI: 10.1007/s40316-017-0093-y

Authors:Maia Fraser; Leonid Polterovich; Daniel Rosen Abstract: For certain contact manifolds admitting a 1-periodic Reeb flow we construct a conjugation-invariant norm on the universal cover of the contactomorphism group. With respect to this norm the group admits a quasi-isometric monomorphism of the real line. The construction involves the partial order on contactomorphisms and symplectic intersections. This norm descends to a conjugation-invariant norm on the contactomorphism group. As a counterpoint, we discuss conditions under which conjugation-invariant norms for contactomorphisms are necessarily bounded. PubDate: 2017-10-16 DOI: 10.1007/s40316-017-0092-z

Authors:A. Iacobucci; S. Olla; G. Stoltz Abstract: We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap. PubDate: 2017-10-06 DOI: 10.1007/s40316-017-0091-0

Authors:V. Nestoridis Abstract: We give a simple proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire’s Category Theorey and Montel’s Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of functions \(X=X({\varOmega })\subset H({\varOmega })\) . We show that the set of non-extendable functions not only contains a \(G_{\delta }\) -dense subset of \(X({\varOmega })\) , but it is itself a \(G_{\delta }\) -dense set. We give an example of a domain in \(\mathbb {C}\) which is a \(H({\varOmega })\) -domain of holomorphy but not a \(A({\varOmega })\) -domain of holomorphy. PubDate: 2017-09-21 DOI: 10.1007/s40316-017-0089-7

Authors:Hui Gao Abstract: In this note, we study the relation between Fontaine–Laffaille modules and strongly divisible modules, without assuming the main theorem of Fontaine–Laffaille (but we need to assume the main results concerning strongly divisible modules). This in particular gives a new proof for the main theorem of Fontaine–Laffaille (for \(p>2\) ). PubDate: 2017-09-16 DOI: 10.1007/s40316-017-0090-1

Authors:Xiuxiong Chen Abstract: In this note, we introduce a new continuity path of fourth order nonlinear equations connecting the cscK equation to a second order elliptic equation, which is the critical point equation of the J-flow introduced by Donaldson (Asian J Math 3(1):1–16, 1999) and the author (Commun Anal Geom 12(4):837–852, 2004). This is a generalization of the classical Aubin continuity path for Kähler–Einstein metrics. The aim of this new path is to attack the existence problem of cscK metric. The “openness” along this continuity path is proved and a set of open problems associated with this new path is proposed. PubDate: 2017-09-02 DOI: 10.1007/s40316-017-0086-x

Authors:Ming-Lun Hsieh; Kenichi Namikawa Abstract: We prove an inner product formula for vector-valued Yoshida lifts by an explicit calculation of local zeta integrals in the Rallis inner product formula for \({\mathrm{O}}(4)\) and \({\mathrm {Sp}}(4)\) . As a consequence, we obtain the non-vanishing of Yoshida lifts. PubDate: 2017-07-28 DOI: 10.1007/s40316-017-0088-8

Authors:Rupam Barman; Neelam Saikia Abstract: We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting points on the family $$\begin{aligned} Z_{\lambda }: x_1^d+x_2^d=d\lambda x_1x_2^{d-1} \end{aligned}$$ over a finite field \(\mathbb {F}_p\) . Salerno expresses the number of points over a finite field \(\mathbb {F}_p\) on the family \(Z_{\lambda }\) in terms of quotients of p-adic gamma functions under the condition that \(d p-1\) . In this paper, we first express the number of points over a finite field \(\mathbb {F}_p\) on the family \(Z_{\lambda }\) in terms of McCarthy’s p-adic hypergeometric series for any odd prime p not dividing \(d(d-1)\) , and then deduce two summation identities for the p-adic hypergeometric series. We also find certain transformations and special values of the p-adic hypergeometric series. We finally find a summation identity for the Greene’s finite field hypergeometric series. PubDate: 2017-07-11 DOI: 10.1007/s40316-017-0087-9

Authors:Sergei Lanzat; Frol Zapolsky Abstract: We construct an embedding of the full braid group on \(m+1\) strands \(B_{m+1}\) , \(m \ge 1\) , into the contact mapping class group of the contactization \(Q \times S^1\) of the \(A_m\) -Milnor fiber Q. The construction uses the embedding of \(B_{m+1}\) into the symplectic mapping class group of Q due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov–Eliashberg dga for Legendrians which lie above one another in \(Q \times {\mathbb {R}}\) , reducing the proof to Floer homology. As corollaries we obtain a contribution to the contact isotopy problem for \(Q \times S^1\) , as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of Q into the contact mapping class group of \(Q \times S^1\) . PubDate: 2017-07-01 DOI: 10.1007/s40316-017-0085-y