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: 2019-04-01

Abstract: Let \(\varphi \) stand for the Euler totient function. Garcia and Luca have proved that, given any positive integer \(\ell \) , the set of those primes p such that \(\varphi (p+\ell )/\varphi (p-\ell )>1\) has the same density as the set of those primes p for which \(\varphi (p+\ell )/\varphi (p-\ell )<1\) . Here we prove this result using classical results from probabilistic and analytic number theory. We then establish similar results for the sum of divisors function and for the k-fold iterate of the Euler function. We also examine the modulus of continuity of some arithmetical functions. Finally, we provide a general result regarding the existence of the distribution function for the function \(s(p):=f(p+\ell )-f(p-\ell )\) for any fixed positive integer \(\ell \) provided the additive function f satisfies certain conditions. PubDate: 2019-04-01

Abstract: In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting (for weakly exact or monotone Lagrangians with large minimal Maslov number) and use it to study the fundamental group of a Lagrangian cobordism \(W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )\) between two Lagrangian submanifolds \(L, L'\subset ( M, \omega )\) . We show that under natural conditions the inclusions \(L,L'\hookrightarrow W\) induce surjective maps \(\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)\) , \(\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)\) and when the previous maps are injective then W is an h-cobordism. In particular, in dimension at least 6, W is topologically trivial in this case. PubDate: 2019-04-01

Abstract: Let \(\rho \) be an even two-dimensional representation of the Galois group \({{\mathrm{Gal}}}(\overline{\mathbb Q}/\mathbb Q)\) which is induced from a character \(\chi \) of odd order of the absolute Galois group of a real quadratic field K. After imposing some additional conditions on \(\chi \) , we attach \(\rho \) to a Hecke eigenclass in the cohomology of \(\mathrm{GL}(2,\mathbb Z)\) with coefficients in a certain infinite-dimensional vector space V over an arbitrary field of characteristic not equal to 2. The space V is defined purely algebraically starting from the field K. PubDate: 2019-04-01

Abstract: We provide several inequalities between eigenvalues of some classical eigenvalue problems on compact Riemannian manifolds with \(C^2\) boundary. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kuttler and Sigillito from subsets of \(\mathbb {R}^2\) to the manifold setting. PubDate: 2019-03-09

Abstract: Let K be a number field of degree n over \(\mathbb {Q}\) . Then the 4-rank of the strict class group of K is at least \(\text {rank}_2 \, ( E_{K}^{+} / E_K^2) - \lfloor n /2 \rfloor \) where \(E_K\) and \( E_{K}^{+} \) denote the units and the totally positive units of K, respectively, and \(\text {rank}_2\) is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field K with a totally positive system of fundamental units contains at least \((n-1)/2\) (n odd) or \(n/2 -1\) (n even) independent elements of order 4. We also investigate when units in K are sums of two squares in K or are squares mod 4 in K. PubDate: 2019-02-25

Abstract: We show that all CM-types of Galois CM-fields without proper CM-subfields are nondegenerate. As a consequence, the Hodge conjecture is true for abelian varieties with complex multiplication by such CM-fields. PubDate: 2019-01-07

Abstract: In this paper, we prove a p-adic analogous of the Kulikov-Persson-Pinkham classification theorem for the central fiber of a degeneration of K3-surfaces in terms of the nilpotency degree of the monodromy of the family. Namely, let \(X_K\) be a be a smooth, projective K3-surface over the p-adic field K, which has either, a minimal semistable model X over \(\mathscr {O}_K\) , or combinatorial reduction. If we let \(N_{st}\) be the Fontaine’s monodromy operator on \(D_{st}(H^2_{\acute{{\mathrm{e}}}\mathrm{t}}(X_{\overline{K}},{\mathbb {Q}}_{p}))\) , then we prove that the degree of nilpotency of \(N_{st}\) determines the type of the special fiber of X. As a consequence we give a criterion for the good reduction of the semistable K3-surface \(X_K\) in terms of its p-adic representation \(H^2_{\acute{{\mathrm{e}}}\mathrm{t}}(X_{\overline{K}},{\mathbb {Q}}_{p})\) , which is similar to the criterion of good reduction for p-adic abelian varieties and curves given by Coleman and Iovita and Andreatta-Iovita-Kim. PubDate: 2018-12-11

Abstract: Let k be a totally real number field and let \(k_\infty \) be its cyclotomic \(\mathbb {Z}_p\) -extension. This work continues our article « Approche p-adique de la conjecture de Greenberg pour les corps totalement réels» by means of heuristics on the p-adic behavior of ideal norms in \(k_\infty /k\) ; indeed, this conjecture (on the nullity of the Iwasawa invariants \(\lambda \) , \(\mu \) ) depends on some images of these norms in the torsion group \({{\mathcal {T}}}_k\) of the Galois group of the maximal abelian p-ramified pro-p-extension of k, thus of their Artin symbols in a finite extension F / k obtained by Galois descent of \({{\mathcal {T}}}_k\) . A natural assumption of distribution of these Artin symbols implies \(\lambda =\mu =0\) . Statistics confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg’s conjecture in the sole framework of Iwasawa’s theory. PubDate: 2018-10-17

Abstract: We study the local behaviour of generalised Iwasawa invariants attached to multiple \(\mathbb {Z}_p\) -extensions of a number field K, with respect to a suitable topology on the sets of \(\mathbb {Z}_p^d\) -extensions of K, \(d \in \mathbb {N}\) . These invariants generalise the classical Iwasawa invariants which describe the asymptotic growth of the ideal class groups of the intermediate fields in a \(\mathbb {Z}_p\) -extension. Generalising work for \(\mathbb {Z}_p\) -extensions and classical Iwasawa invariants, we show that, under certain assumptions, generalised Iwasawa invariants are locally maximal. In other words, the generalised Iwasawa invariants of certain \(\mathbb {Z}_p^d\) -extensions \(\mathbb {K}\) of K bound the generalised Iwasawa invariants for all \(\mathbb {Z}_p^d\) -extensions of K close to \(\mathbb {K}\) . The main ingredient in the proof is the precise investigation of the Galois module structure of the ideal class groups, which is much more involved than in the one-dimensional case. PubDate: 2018-10-03

Abstract: Let X be a simply-connected space, \((\Lambda V, d)\) its minimal Sullivan model and \(d_k\) ( \(k\ge 2\) ) the first non-zero homogeneous part of the differential d. In this paper, assuming that \((\Lambda V, d_k)\) is elliptic, we show that \(H(\Lambda V,d)\) has no \(e_0\) -gap and consequently we confirm the Hilali conjecture when \(V = V^{odd}\) or else when \(k\ge 3\) . PubDate: 2018-10-03

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: 2018-10-01

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: 2018-10-01

Authors:Rita Gitik Abstract: We prove that the free product of two finitely presented locally tame groups is locally tame and describe many examples of tame subgroups of finitely presented groups. We also include some open problems related to tame subgroups. PubDate: 2018-04-21 DOI: 10.1007/s40316-018-0102-9

Authors:Daniel Delbourgo; Qin Chao Abstract: We completely describe \(\hbox {K}_{1}({\mathbb {Z}}_p[\![{\mathcal {G}}_{\infty }]\!])\) and its localisations by using an infinite family of p-adic congruences, where \({\mathcal {G}}_{\infty }\) is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when \(\hbox {dim}({\mathcal {G}}_{\infty })=2\) , and of the first named author and Lloyd Peters when \({\mathcal {G}}_{\infty } \cong {\mathbb {Z}}_p^{\times }\ltimes {\mathbb {Z}}_p^d\) with a scalar action of \({\mathbb {Z}}_p^{\times }\) . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. PubDate: 2018-04-16 DOI: 10.1007/s40316-018-0100-y

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: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: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: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