Authors:Samuel Baumard; Leila Schneps Pages: 43 - 62 Abstract: Richard Hain and Makoto Matsumoto constructed a category of universal mixed elliptic motives, and described the fundamental Lie algebra of this category: it is a semi-direct product of the fundamental Lie algebra \({{\mathrm{Lie}}}\pi _1(\mathsf {MTM})\) of the category of mixed Tate motives over \(\mathbb {Z}\) with a filtered and graded Lie algebra \(\mathfrak {u}\) . This Lie algebra, and in particular \(\mathfrak {u}\) , admits a representation as derivations of the free Lie algebra on two generators. In this paper we study the image \(\mathscr {E}\) of this representation of \(\mathfrak {u}\) , starting from some results by Aaron Pollack, who determined all the relations in a certain filtered quotient of \(\mathscr {E}\) , and gave several examples of relations in low weights in \(\mathscr {E}\) that are connected to period polynomials of cusp forms on \({{\mathrm{SL}}}_2(\mathbb {Z})\) . Pollack’s examples lead to a conjecture on the existence of such relations in all depths and all weights, that we state in this article and prove in depth 3 in all weights. The proof follows quite naturally from Ecalle’s theory of moulds, to which we give a brief introduction. We prove two useful general theorems on moulds in the appendices. PubDate: 2017-04-01 DOI: 10.1007/s40316-015-0040-8 Issue No:Vol. 41, No. 1 (2017)

Authors:Michael Brandenbursky; Jarek Kędra Pages: 63 - 65 Abstract: We prove that the autonomous norm on the group of compactly supported Hamiltonian diffeomorphisms of the standard \({\mathbf R}^{2n}\) is bounded. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0065-7 Issue No:Vol. 41, No. 1 (2017)

Authors:Maxim Braverman; Boris Vertman Pages: 67 - 96 Abstract: We prove that refined analytic torsion on a manifold with boundary is a weakly holomorphic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a unitary point. Finally we provide a new proof of Brüning-Ma gluing formula for the Ray–Singer torsion associated to a non-Hermitian connection. Our proof is quite different from the one given by Brüning and Ma and uses a temporal gauge transformation. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0062-x Issue No:Vol. 41, No. 1 (2017)

Authors:Cornelius Greither Pages: 97 - 104 Abstract: To every abelian Galois extension K / k of number fields with group G, one can associate a so-called Brumer element \(\theta _{K/k,S}\) of the rational group ring \(\mathbb Q[G]\) (depending on another technical parameter S). In a certain sense this element can be thought of as an equivariant version of the class number \(h_K\) . There is a slight problem since this element is only “almost” integral (that is, it may have non-integral coefficients). The potential denominators always divide \(w_K\) , the number of roots of unity in K. Hayes (Contemp Math 358:193–205, 2004) raised the question whether the Brumer element will be p-integral under certain suitable hypotheses, one of which implies that p divides \(h_K\) ; of course only situations with \(p w_K\) are of interest. This paper answers this question in the negative. We start with an “almost counterexample” over \(\mathbb Q\) (for \(k=\mathbb {Q}\) there are no true counterexamples) and manufacture true counterexamples in which k is a suitable real quadratic field. Using deep recent results on the distribution of class numbers (Bhargava et al.) one can show that the method in fact yields infinitely many counterexamples. PubDate: 2017-04-01 DOI: 10.1007/s40316-015-0039-1 Issue No:Vol. 41, No. 1 (2017)

Authors:Suhail Gulzar; N. A Rather; K. A. Thakur Pages: 105 - 110 Abstract: In this paper, we present certain results on the bounds for the moduli of the zeros of a polynomial with complex coefficients which among other things contain some generalizations and refinements of classical results due to Cauchy, Tôya, Carmichael and Mason, Williams and others. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0064-8 Issue No:Vol. 41, No. 1 (2017)

Authors:Bernard Helffer Pages: 111 - 118 Abstract: In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points \(\ell _k\) in the boundary set of a minimal k-partition tends to \(+\infty \) as \(k\rightarrow +\infty \) . In this note, we show that \(\ell _k\) increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As in the original proof by Pleijel of his celebrated theorem, this involves Faber-Krahn’s inequality and Weyl’s formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl’s formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles. In a recent paper with Thomas Hoffmann-Ostenhof, we proved that the number of critical points \(\ell _k\) in the boundary set of a k-minimal partition tends to \(+\infty \) as \(k\rightarrow +\infty \) . In this note, we show that \(\ell _k\) increases linearly with k as suggested by a hexagonal conjecture about the asymptotic behavior of the energy of these minimal partitions. As the original proof by Pleijel, this involves Faber-Krahn’s inequality and Weyl’s formula, but this time, due to the magnetic characterization of the minimal partitions, we have to establish a Weyl’s formula for Aharonov-Bohm operator controlled with respect to a k-dependent number of poles. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0058-6 Issue No:Vol. 41, No. 1 (2017)

Authors:Antonio Lei; Gautier Ponsinet Pages: 155 - 167 Abstract: We study the functional equation for the multi-signed Selmer groups for non-ordinary motives whose Hodge-Tate weights are 0 and 1, defined by Büyükboduk and the first named author. This generalizes simultaneously Greenberg’s result for ordinary motives in and Kim’s result for supersingular elliptic curves. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0063-9 Issue No:Vol. 41, No. 1 (2017)

Authors:Ahmed J. Zerouali Pages: 169 - 197 Abstract: We prove a Hitchin–Thorpe inequality for noncompact 4-manifolds with foliated geometry at infinity by extending on previous work by Dai and Wei. After introducing the objects at hand, we recall some preliminary results regarding the G-signature formula and the rho invariant, which are used to obtain expressions for the signature and Euler characteristic in our geometric context. We then derive our main result, and present examples. PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0066-6 Issue No:Vol. 41, No. 1 (2017)

Authors:Marie José Bertin; Wadim Zudilin Pages: 199 - 211 Abstract: We prove Boyd’s “unexpected coincidence” of the Mahler measures for two families of two-variate polynomials defining curves of genus 2. We further equate the same measures to the Mahler measures of polynomials \(y^3-y+x^3-x+kxy\) whose zero loci define elliptic curves for \(k\ne 0,\pm 3\) . PubDate: 2017-04-01 DOI: 10.1007/s40316-016-0068-4 Issue No:Vol. 41, No. 1 (2017)

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

Authors:Nipen Saikia; Jubaraj Chetry Abstract: For any non-negative integer n and non-zero integer r, let \(p_r(n)\) denote Ramanujan’s general partition function. In this paper, we prove many infinite families of congruences modulo 7 for the general partition function \(p_r(n)\) for negative values of r by using q-identities. PubDate: 2017-05-25 DOI: 10.1007/s40316-017-0084-z

Authors:Andrew Fiori Abstract: In this paper we study the integral structure of lattices over finite extensions of \(\mathbb {Z}_p\) which arise from restriction or transfer from a lattice over a finite extension. We describe explicitly the structure of the resulting lattices. Special attention is given to the case of lattices whose quadratic forms arise from Hermitian forms. Then, in the case of Hermitian lattices where the final lattice is over \(\mathbb {Z}_p\) we focus on the problem of computing the local densities. PubDate: 2017-05-05 DOI: 10.1007/s40316-017-0083-0

Authors:Aida Kh. Asgarova Abstract: Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra \(A\subset C(X)\) , which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved. PubDate: 2017-04-06 DOI: 10.1007/s40316-017-0081-2

Authors:Jean-Marie De Koninck; Imre Kátai Abstract: We show that some sequences of real numbers involving sharp normal numbers or non-Liouville numbers are uniformly distributed modulo 1. In particular, we prove that if \(\tau (n)\) stands for the number of divisors of n and \(\alpha \) is a binary sharp normal number, then the sequence \((\alpha \tau (n))_{n\ge 1}\) is uniformly distributed modulo 1 and that if g(x) is a polynomial of positive degree with real coefficients and whose leading coefficient is a non-Liouville number, then the sequence \((g(\tau (\tau (n))))_{n \ge 1}\) is also uniformly distributed modulo 1. PubDate: 2017-04-04 DOI: 10.1007/s40316-017-0080-3

Authors:Jeehoon Park 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:Jean-François Jaulent 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:Robert Laterveer 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: 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