Abstract: Assume a and \(b=na+r\) with \(n \ge 1\) and \(0<r<a\) are relatively prime integers. In case C is a smooth curve and P is a point on C with Weierstrass semigroup equal to \(<a;b>\) then C is called a \(C_{a;b}\) -curve. In case \(r \ne a-1\) and \(b \ne a+1\) we prove C has no other point \(Q \ne P\) having Weierstrass semigroup equal to \(<a;b>\) , in which case we say that the Weierstrass semigroup \(<a;b>\) occurs at most once. The curve \(C_{a;b}\) has genus \((a-1)(b-1)/2\) and the result is generalized to genus \(g<(a-1)(b-1)/2\) . We obtain a lower bound on g (sharp in many cases) such that all Weierstrass semigroups of genus g containing \(<a;b>\) occur at most once. PubDate: 2019-02-15

Abstract: We prove the existence of meromorphic continuation and the functional equation of the real analytic Jacobi Eisenstein series of degree m and matrix index T in case T is a kernel form. PubDate: 2019-01-14

Authors:J. J. Vásquez Pages: 23 - 50 Abstract: We give a systematic way to construct almost conjugate pairs of finite subgroups of \(\mathrm {Spin}(2n+1)\) and \({{\mathrm{Pin}}}(n)\) for \(n\in {\mathbb {N}}\) sufficiently large. As a geometric application, we give an infinite family of pairs \(M_1^{d_n}\) and \(M_2^{d_n}\) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions \(d_n>6\) . We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0185-2 Issue No:Vol. 88, No. 1 (2018)

Authors:Hidenori Katsurada Pages: 67 - 86 Abstract: We give a period formula for the adelic Ikeda lift of an elliptic modular form f for U(m, m) in terms of special values of the adjoint L-functions of f. This is an adelic version of Ikeda’s conjecture on the period of the classical Ikeda lift for U(m, m). PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0178-1 Issue No:Vol. 88, No. 1 (2018)

Authors:Lucas Dahinden Pages: 87 - 96 Abstract: The classical Bott–Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric space. This result on geodesic flows has been generalized to Reeb flows and partially to positive Legendrian isotopies by Frauenfelder–Labrousse–Schlenk. We prove the full theorem for positive Legendrian isotopies. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0180-7 Issue No:Vol. 88, No. 1 (2018)

Authors:Christopher Deninger Pages: 189 - 192 Abstract: It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0179-0 Issue No:Vol. 88, No. 1 (2018)

Authors:Lucio Cadeddu; Maria Antonietta Farina Pages: 193 - 199 Abstract: In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into \({\mathbb {R}}\) ) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to \(C^\infty \) functions. The gap appears only for the non generic set of non Morse functions. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0183-4 Issue No:Vol. 88, No. 1 (2018)

Authors:D. Huybrechts Pages: 201 - 207 Abstract: We observe that derived equivalent K3 surfaces have isomorphic Chow motives. The result holds more generally for arbitrary surfaces, as pointed out by Charles Vial. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0182-5 Issue No:Vol. 88, No. 1 (2018)

Authors:Michael Müger Pages: 209 - 216 Abstract: Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for \(f:[1,\infty )\rightarrow {\mathbb R}\) non-negative and non-decreasing we prove \(f(x)-x=O(x^\gamma )\) with \(\gamma <1\) under certain assumptions on f. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more. PubDate: 2018-04-01 DOI: 10.1007/s12188-017-0187-0 Issue No:Vol. 88, No. 1 (2018)

Abstract: In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups for F a finite étale Galois module on K of order invertible in K and with \(F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))\) . Furthermore, we prove that \(\mathrm {H}^1(K,G) = 0\) for G a simply connected, quasisplit semisimple group over K not of type \(E_8\) . PubDate: 2018-10-01

Abstract: We discuss generalizations of classical theta series, requiring only some basic properties of the classical setting. As it turns out, the existence of a generalized theta transformation formula implies that the series is defined over a quasi-symmetric Siegel domain. In particular the exceptional symmetric tube domain does not admit a theta function. PubDate: 2018-10-01

Abstract: We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature \((n-1,1)\) and trans-Lorentzian signature \((n-2,2)\) . PubDate: 2018-10-01

Abstract: Generalized Burniat surfaces are surfaces of general type with \(p_g=q\) and Euler number \(e=6\) obtained by a variant of Inoue’s construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer et al. in (J Math Sci Univ Tokyo 22(2–15):55–111, 2015. arXiv:1409.1285v2). This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura. PubDate: 2018-10-01

Abstract: In this paper, we give a method to construct a classical modular form from a Hilbert modular form. By applying this method, we can get linear formulas which relate the Fourier coefficients of the Hilbert and classical modular forms. The paper focuses on the Hilbert modular forms over real quadratic fields. We will state a construction of relations between the special values of L-functions, especially at 0, and arithmetic functions. We will also give a relation between the sum of squares functions with underlying fields \(\mathbb {Q}(\sqrt{D})\) and \(\mathbb {Q}\) . PubDate: 2018-10-01

Abstract: In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct \(\chi _{5}\) , the cusp form of lowest weight for the group \({\text {Sp}}(2,\mathbb {Z})\) . In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice \(A_{1}\) and Igusa’s form \(\chi _{5}\) appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates. PubDate: 2018-10-01

Authors:Winfried Kohnen Abstract: We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight. PubDate: 2018-05-16 DOI: 10.1007/s12188-018-0194-9

Authors:Anneleen De Schepper; N. S. Narasimha Sastry; Hendrik Van Maldeghem Abstract: The original version of this article unfortunately contained a mistake in the author’s name N. S. Narasimha Sastry. The corrected name is given above. PubDate: 2018-02-09 DOI: 10.1007/s12188-018-0192-y

Authors:Ljuben Mutafchiev Abstract: Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\) . The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series. PubDate: 2018-02-06 DOI: 10.1007/s12188-018-0191-z

Authors:Victor Alexandrov Abstract: We choose some special unit vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) in \({\mathbb {R}}^3\) and denote by \({\mathscr {L}}\subset {\mathbb {R}}^5\) the set of all points \((L_1,\ldots ,L_5)\in {\mathbb {R}}^5\) with the following property: there exists a compact convex polytope \(P\subset {\mathbb {R}}^3\) such that the vectors \({\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5\) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal \({\mathbf {n}}_k\) is equal to \(L_k\) for all \(k=1,\ldots ,5\) . Our main result reads that \({\mathscr {L}}\) is not a locally-analytic set, i.e., we prove that, for some point \((L_1,\ldots ,L_5)\in {\mathscr {L}}\) , it is not possible to find a neighborhood \(U\subset {\mathbb {R}}^5\) and an analytic set \(A\subset {\mathbb {R}}^5\) such that \({\mathscr {L}}\cap U=A\cap U\) . We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces. PubDate: 2017-12-11 DOI: 10.1007/s12188-017-0189-y

Authors:Hansjörg Geiges; Christian Lange Abstract: We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers \(\alpha _1^0,\alpha _2^0\) , we give an algorithmic construction of a Seifert fibration \(L(p,q)\rightarrow S^2(\alpha \alpha _1^0 ,\alpha \alpha _2^0 )\) , where the natural number \(\alpha \) is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models. PubDate: 2017-10-23 DOI: 10.1007/s12188-017-0188-z