Authors:Jeehoon Park Abstract: Abstract G. Stevens (http://math.bu.edu/people/ghs/research.html) constructed a modular symbol taking values in circular K-groups, which is intimately related to Eisenstein series. We make precise a relationship between his Milnor K-theoretic modular symbol \(\Phi _{MK}\) and the period integrals of Eisenstein series. The main goal here is to extract from \(\Phi _{MK}\) a group 1-cocyle on \({{\mathrm{SL}}}_2(\mathbb {Q})\) with values in differential form valued distributions and use this to construct a p-adic locally analytic distribution which gives a p-adic partial zeta function of a real quadratic field. PubDate: 2017-02-22 DOI: 10.1007/s40316-017-0079-9

Authors:Philippe Charron; Bernard Helffer; Thomas Hoffmann-Ostenhof Abstract: Abstract In 1956, Pleijel gave his celebrated theorem showing that the inequality in Courant’s theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunction as the eigenvalue tends to \(+\infty \) . A similar question occurs naturally for the case of the Schrödinger operator. The first significant result has been obtained recently by the first author for the case of the harmonic oscilllator. The purpose of this paper is to consider more general potentials which are radial. We will analyze either the case when the potential tends to \(+\infty \) or the case when the potential tends to zero, the considered eigenfunctions being associated with the eigenvalues below the essential spectrum. PubDate: 2017-02-01 DOI: 10.1007/s40316-017-0078-x

Authors:Kâzım Büyükboduk Pages: 251 - 302 Abstract: Abstract Ochiai has previously proved that the Beilinson–Kato Euler systems for modular forms interpolate in nearly-ordinary p-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai’s work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson–Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus), assuming that the deformation problem at hand is unobstructed in the sense of Mazur. We then use what we call universal Kolyvagin systems to attempt a main conjecture over the eigencurve. Along the way, we utilize these objects in order to define a quasicoherent sheaf on the eigencurve that behaves like a p-adic L-function (in a certain sense of the word, in 3-variables). PubDate: 2016-03-02 DOI: 10.1007/s40316-015-0044-4 Issue No:Vol. 40, No. 2 (2016)

Authors:Francesc Castella; Matteo Longo Pages: 303 - 324 Abstract: Abstract In Longo and Vigni (Manuscr Math 135:273–328, 2011), Howard’s construction of big Heegner points on modular curves was extended to general Shimura curves over the rationals. In this paper, we relate the higher weight specializations of the big Heegner points of Longo and Vigni (Manuscr Math 135:273–328, 2011) in the definite setting to certain higher weight analogues of the Bertolini–Darmon theta elements (Bertolini and Darmon in Invent Math 126:413–456, 1996). As a consequence of this relation, some of the conjectures in Longo and Vigni (Manuscr Math 135:273–328, 2011) are deduced from recent results of Chida and Hsieh (J Reine Angew Math, 2015). PubDate: 2016-08-01 DOI: 10.1007/s40316-015-0045-3 Issue No:Vol. 40, No. 2 (2016)

Authors:Henri Darmon; Alan Lauder; Victor Rotger Pages: 325 - 354 Abstract: Abstract This article can be read as a companion and sequel to the authors’ earlier article on Stark points and p-adic iterated integrals attached to modular forms of weight one, which proposes a conjectural expression for the so-called p -adic iterated integrals attached to a triple (f, g, h) of classical eigenforms of weights (2, 1, 1). When f is a cusp form, this expression involves the p-adic logarithms of so-called Stark points: distinguished points on the modular abelian variety attached to f, defined over the number field cut out by the Artin representations attached to g and h. The goal of this paper is to formulate an analogous conjecture when f is a weight two Eisenstein series rather than a cusp form. The resulting formula involves the p-adic logarithms of units and p-units in suitable number fields, and can be seen as a new variant of Gross’s p-adic analogue of Stark’s conjecture on Artin L-series at \(s=0\) . PubDate: 2016-08-01 DOI: 10.1007/s40316-015-0042-6 Issue No:Vol. 40, No. 2 (2016)

Authors:Samit Dasgupta; Michael Spieß Pages: 355 - 376 Abstract: Abstract This paper is an announcement of the following result, whose proof will be forthcoming. Let F be a totally real number field, and let \(F \subset K \subset L\) be a tower of fields with L / F a finite abelian extension. Let I denote the kernel of the natural projection from \(\mathbf {Z}[\mathrm{Gal}(L/F)]\) to \(\mathbf {Z}[\mathrm{Gal}(K/F)]\) . Let \(\Theta \in \mathbf {Z}[\mathrm{Gal}(L/F)]\) denote the Stickelberger element encoding the special values at zero of the partial zeta functions of L / F, taken relative to sets S and T in the usual way. Let r denote the number of places in S that split completely in K. We show that \(\Theta \in I^{r}\) , unless K is totally real in which case we obtain \(\Theta \in I^{r-1}\) and \(2\Theta \in I^r\) . This proves a conjecture of Gross up to the factor of 2 in the case that K is totally real and \(\#S \ne r\) . In this article we sketch the proof in the case that K is totally complex. PubDate: 2016-08-01 DOI: 10.1007/s40316-015-0046-2 Issue No:Vol. 40, No. 2 (2016)

Authors:Takako Fukaya; Kazuya Kato; Romyar Sharifi Pages: 377 - 395 Abstract: Abstract We consider modifications of Manin symbols in first homology groups of modular curves with \(\mathbb {Z}_p\) -coefficients for an odd prime p. We show that these symbols generate homology in primitive eigenspaces for the action of diamond operators, with a certain condition on the eigenspace that can be removed on Eisenstein parts. We apply this to prove the integrality of maps taking compatible systems of Manin symbols to compatible systems of zeta elements. In the work of the first two authors on an Iwasawa-theoretic conjecture of the third author, these maps are constructed with certain bounded denominators. As a consequence, their main result on the conjecture was proven after inverting p, and the results of this paper allow one to remove this condition. PubDate: 2016-03-02 DOI: 10.1007/s40316-016-0059-5 Issue No:Vol. 40, No. 2 (2016)

Authors:Matthew Greenberg; Marco Adamo Seveso Pages: 397 - 434 Abstract: Abstract We show that p-adic families of modular forms give rise to certain p-adic Abel-Jacobi maps at their p-new specializations. We introduce the concept of differentiation of distributions, using it to give a new description of the Coleman-Teitelbaum cocycle that arises in the context of the \(\mathcal {L}\) -invariant. PubDate: 2016-04-07 DOI: 10.1007/s40316-016-0060-z Issue No:Vol. 40, No. 2 (2016)

Authors:Michael Harris Pages: 435 - 452 Abstract: Abstract The Ichino-Ikeda conjecture is an identity that relates a ratio of special values of automorphic L-functions to a ratio of period integrals. Both sides of this identity are expected to satisfy certain equidistribution properties when the data vary, and indeed it has been possible to transfer such properties from one side of the identity to the other in cases where the identity is known. The present article studies parallels between complex-analytic and p-adic equidistribution properties and relates the latter to questions about Galois cohomology. PubDate: 2016-08-01 DOI: 10.1007/s40316-016-0057-7 Issue No:Vol. 40, No. 2 (2016)

Authors:Fabian Januszewski Pages: 453 - 489 Abstract: Abstract We construct p-adic L-functions for Rankin-Selberg convolutions for \({\mathrm {GL}}(n+1)\times {\mathrm {GL}}(n)\) over arbitrary number fields, and show that they satisfy an expected functional equation. PubDate: 2016-04-05 DOI: 10.1007/s40316-016-0061-y Issue No:Vol. 40, No. 2 (2016)

Authors:G. Ander Steele Pages: 491 - 517 Abstract: Abstract We compute the p-adic L-functions of evil Eisenstein series using an explicit Eisenstein modular symbol constructed from Shintani cocycles. PubDate: 2016-08-01 DOI: 10.1007/s40316-015-0055-1 Issue No:Vol. 40, No. 2 (2016)

Authors:Bruno Poizat Abstract: 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: 2016-12-19 DOI: 10.1007/s40316-016-0075-5

Authors:Stéphane Dellacherie; Olivier Lafitte Abstract: 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: 2016-11-22 DOI: 10.1007/s40316-016-0073-7

Authors:Kevin Buzzard; Alan Lauder Abstract: 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: 2016-11-10 DOI: 10.1007/s40316-016-0072-8

Abstract: Abstract Let K be a p-adic field. Restricting to the case of no intermediate extensions, we obtain formulæ counting the number of (totally and wildly) ramified extensions of degree \(p^4\) of K up to K-isomorphism and in particular, we count the number of isomorphism classes of extensions for which the Galois closure has a prescribed Galois group. The principal tool used is a result, proved in Del Corso et al. (On wild extensions of a p-adic field, arXiv:1601.05939v1), which states that there is a one-to-one correspondence between the isomorphism classes of extensions of degree \(p^k\) of K having no intermediate extensions and the irreducible H-sub-modules of dimension k of \(F^*{/}{F^*}^p\) , where F is the composite of certain fixed normal extensions of K and H is its Galois group over K. PubDate: 2016-10-31 DOI: 10.1007/s40316-016-0076-4

Abstract: 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: 2016-10-31 DOI: 10.1007/s40316-016-0074-6

Authors:Jean-François Jaulent Abstract: Abstract We use \(\ell \) -adic class field theory to take a new view on cyclotomic norms and Leopoldt or Gross-Kuz’min conjectures. By the way we recall and complete some classical results. We illustrate the logarithmic approach by various numerical examples and counter-examples obtained with PARI. PubDate: 2016-10-28 DOI: 10.1007/s40316-016-0069-3

Authors:Marine Rougnant 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: 2016-10-28 DOI: 10.1007/s40316-016-0071-9

Authors:Robert Laterveer Abstract: Abstract Let X be a smooth cubic hypersurface, and let F be the Fano variety of lines on X. We establish a relation between the Chow motives of X and F. This relation implies in particular that if X has finite-dimensional motive (in the sense of Kimura), then F also has finite-dimensional motive. This proves finite-dimensionality for motives of Fano varieties of cubics of dimension 3 and 5, and of certain cubics in other dimensions. PubDate: 2016-10-22 DOI: 10.1007/s40316-016-0070-x

Authors:Dylan Attwell-Duval Abstract: Abstract We study the boundary of orthogonal Shimura varieties associated to a positive multiple of a maximal lattice splitting two hyperbolic planes. We provide closed formulas for the number of 0 and 1-dimensional cusps of these spaces and study their configuration within the boundary. This generalizes our earlier results about maximal lattices. PubDate: 2016-08-02 DOI: 10.1007/s40316-016-0067-5