We study p-adic L-functions \(L_p(s,\chi )\) for Dirichlet characters \(\chi \). We show that \(L_p(s,\chi )\) has a Dirichlet series expansion for each regularization parameter c that is prime to p and the conductor of \(\chi \). The expansion is proved by transforming a known formula for p-adic L-functions and by controlling the limiting behavior. A finite number of Euler factors can be factored

The moduli spaces of flat \({\text{SL}}_2\)- and \({\text{PGL}}_2\)-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by Tamás Hausel in Remark 3.30 of “Global topology of the Hitchin system”.

We identify the p-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic p as the eigenvalues of a product of special values of a certain matrix of p-adic series. That matrix is a product \(F(\varLambda ^p)^{-1}F(\varLambda )\), where the entries in the matrix \(F(\varLambda )\) are A-hypergeometric series with integral coefficients and \(F(\varLambda

In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and \(0

Let M be a compact connected smooth pseudo-Riemannian manifold that admits a topologically transitive G-action by isometries, where \(G = G_1 \ldots G_l\) is a connected semisimple Lie group without compact factors whose Lie algebra is \({\mathfrak {g}}= {\mathfrak {g}}_1 \oplus {\mathfrak {g}}_2 \oplus \cdots \oplus {\mathfrak {g}}_l\). If \(m_0,n_0,n_0^i\) are the dimensions of the maximal lightlike

A correction to this paper has been published: https://doi.org/10.1007/s12188-021-00239-x

Suppose \(q=p^r\), where p is a prime congruent to 3 or 5 modulo 8 and r is odd or \(q = 2^r\) for any r. Then every closed smooth \({\text {PSL}}(2,q)\) manifold has a strongly algebraic model.

By introducing a new approximation technique in the \(L^2\) theory of the \(\bar{\partial }\)-operator, Hörmander’s \(L^2\) variant of Andreotti-Grauert’s finiteness theorem is extended and refined on q-convex manifolds and weakly 1-complete manifolds. As an application, a question on the \(L^2\) cohomology suggested by a theory of Ueda (Tohoku Math J (2) 31(1):81–90, 1979), Ueda (J Math Kyoto Univ

We classify the Seifert fibrations of lens spaces where the base orbifold is non-orientable. This is an addendum to our earlier paper ‘Seifert fibrations of lens spaces’. We correct Lemma 4.1 of that paper and fill the gap in the classification that resulted from the erroneous lemma.

In 1900, at the international congress of mathematicians, Hilbert claimed that the Riemann zeta function \(\zeta (s)\) is not the solution of any algebraic ordinary differential equations on its region of analyticity. In 2015, Van Gorder (J Number Theory 147:778–788, 2015) considered the question of whether \(\zeta (s)\) satisfies a non-algebraic differential equation and showed that it formally satisfies

We look at genera of even unimodular lattices of rank 12 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{5})\) and of rank 8 over the ring of integers of \({{\mathbb {Q}}}(\sqrt{3})\), using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and

Inspired by the work of Lu and Tian (Duke Math J 125:351--387, 2004), Loi et al. address in (Abh Math Semin Univ Hambg 90: 99-109, 2020) the problem of studying those Kähler manifolds satisfying the \(\Delta \)-property, i.e. such that on a neighborhood of each of its points the k-th power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer

Let \(s,t,u \in {{\mathbb {C}}}\) and T(s, t, u) be the Tornheim double zeta function. In this paper, we investigate some properties of symmetric Tornheim double zeta functions which can be regarded as a desingularization of the Tornheim double zeta function. As a corollary, we give explicit evaluation formulas or rapidly convergent series representations for T(s, t, u) in terms of series of the gamma

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of \(E_\infty \)-ring spectra in various ways. In this work we first establish, in the context of \(\infty \)-categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of \(E_\infty

Let $$pod_3(n)$$ p o d 3 ( n ) denote the number of 3-regular partitions of n with distinct odd parts (and even parts are unrestricted). In this article, we prove an infinite family of congruences for $$pod_3(n)$$ p o d 3 ( n ) using the theory of Hecke eigenforms. We also study the divisibility properties of $$pod_3(n)$$ p o d 3 ( n ) using arithmetic properties of modular forms.

A comparison among different constructions of the quaternionic $4$-form $\Phi_{Sp(2)Sp(1)}$ and of the Cayley calibration $\Phi_{Spin(7)}$ shows that one can start for them from the same collections of "Kahler 2-forms", entering in dimension 8 both in quaternion Kahler and in $Spin(7)$ geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $\pi^n(M)$ of $M$. Moreover it is known that $\pi^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the $\mathbb Z_2$ part in $\pi^n(M)$ analogous to Pontryagin's computation of the

We take another look at the so-called quasi-derivation relations in the theory of multiple zeta values, by giving a certain formula for the quasi-derivation operator. In doing so, we are not only able to prove the quasi-derivation relations in a simpler manner but also give an analog of the quasi-derivation relations for finite multiple zeta values.

We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko and Zagier and the usual sum formulas for classical double zeta values.

We study analogues of Sturm’s bounds for vector valued Siegel modular forms of type (k, 2), which was already studied by Sturm in the case of an elliptic modular form and by Choi–Choie–Kikuta, Poor–Yuen and Raum–Richter in the case of scalar valued Siegel modular forms.

We show that a dihedral congruence prime for a normalised Hecke eigenform f in $$S_k(\Gamma _0(D),\chi _D)$$ , where $$\chi _D$$ is a real quadratic character, appears in the denominator of the Bloch–Kato conjectural formula for the value at 1 of the twisted adjoint L-function of f. We then use a formula of Zagier to prove that it appears in the denominator of a suitably normalised $$L(1,{\mathrm {ad}}^0(g)\otimes

We show that a hypersurface of the regularized, spatial circular restricted three-body problem is of contact type whenever the energy level is below the first critical value (the energy level of the first Lagrange point) or if the energy level is slightly above it. A dynamical consequence is that there is no blue sky catastrophe in this energy range.

We study finite dimensional almost- and quasi-effective prolongations of nilpotent $${\mathbb {Z}}$$ -graded Lie algebras, especially focusing on those having a decomposable reductive structural subalgebra. Our assumptions generalize effectiveness and algebraicity and are appropriate to obtain Levi–Malcev and Levi–Chevalley decompositions and precisions on the heigth and other properties of the prolongations

We derive the existence of p-adic Hurwitz–Lerch L-function by means of a method provided by Washington. This function is a generalization of the one-variable p-adic L-function of Kubota and Leopoldt, and two-variable p-adic L-function of Fox. We also deduce divisibility properties of generalized Apostol–Bernoulli polynomials, in particular establish Kummer-type congruences for them.

Inspired by the work of Lu and Tian (Duke Math J 125(2):351–387, 2004), in this paper we address the problem of studying those Kähler manifolds satisfying the $$\Delta$$ Δ -property, i.e. such that on a neighborhood of each of its points the k th power of the Kähler Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer k (see below for its definition). We prove

In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff orthogonality, we get an analogue of the Gauss map. Then we can define concepts of principal, Gaussian, and mean curvatures in terms of the eigenvalues of the differential

Fatighenti and Mongardi have defined Fano varieties of type S6 as zero loci of a certain vector bundle on the Grassmannian $\hbox{Gr}(2,10)$. These varieties have 3 Hodge structures of K3 type in their cohomology. We show that the Chow ring of these varieties also displays "K3 type" behaviour.

In this paper, we sketch some constructions providing a link between complex-analytic geometry and nonstandard algebraic geometry via a categorical point of view. The analytic category is seen as a completed fiber of a family of nonstandard algebraic geometries by applying a standard part functor. We indicate how various notions of analytic objects fit into this context (as for example banachanalytic

Let $p$ be an odd prime number. We construct explicit uniformizers for the totally ramified extension $\mathbb{Q}_p(\zeta_{p^2},\sqrt[p]{p})$ of the field of $p$-adic numbers $\mathbb{Q}_p$, where $\zeta_{p^2}$ is a primitive $p^2$-th root of unity.

For an arbitrary real connected Lie group G we define $$\mathrm {p}(G)$$ p ( G ) as the maximal integer p such that $$\mathbb {Z}^p$$ Z p is isomorphic to a discrete subgroup of G and $$\mathrm {q}(G)$$ q ( G ) is the maximal integer q such that $$\mathbb {R}^q$$ R q is isomorphic to a closed subgroup of G . The aim of this paper is to investigate properties of these two invariants. We will show that

There are several inaccurate points and misprints in our article [1].

Completely replicable functions play an important role in number theory and finite group theory, in particular the Monstrous Moonshine. In this paper, we give a characterization of completely replicable functions by certain symmetries.

We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and Bianchi), a parity result of orders of vanishing, and invariance of Iwasaswa invariants under conjugate twists of the p-adic L-functions.

A list of possible holonomy groups with indecomposable holonomy representation contained in the exceptional, non-compact Lie group $${\mathrm {G}}_2^{*}$$ was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1, 2 or 3, respectively. It was also shown by Fino and Kath that all

We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$, where $p$ is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and

Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder as a new concept of a smooth structure in infinite dimensions. We prove that scale structures on mapping spaces are completely determined by the dimension of domain manifolds. As a consequence, we give a complete description of the local invariant introduced by U. Frauenfelder for mapping spaces. Product mapping spaces and relative

In this paper, we study the non-vanishing of the Miyawaki type lift in various situations. In the case of GSpin(2, 10) constructed in Kim and Yamauchi (Math Z 288(1–2):415–437, 2018), we use the fact that the Fourier coefficient at the identity is closely related to the Rankin–Selberg L-function of two elliptic cusp forms. In the case of the original Miyawaki lifts of Siegel cusp forms, we use the

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of $24$ modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight $k$ with $4\mid k$

We prove the meromorphic continuation and the functional equation of a twisted real-analytic Hermitain Eisenstein series of Klingen type, and as a consequence, deduce similar properties for the twisted Dirichlet series associated to a pair of Hermitian modular forms involving their Fourier–Jacobi coefficients. As an application of our result, we prove that infinitely many of the Fourier–Jacobi coefficients

AbstractThe paper builds a DPW approach of Willmore surfaces via conformal Gauss maps. As applications, we provide descriptions of minimal surfaces in $${\mathbb {R}}^{n+2}$$Rn+2, isotropic surfaces in $$S^4$$S4 and homogeneous Willmore tori via the loop group method. A new example of a Willmore two-sphere in $$S^6$$S6 without dual surfaces is also presented.

We prove that at least one of the six numbers $$\beta (2i)$$β(2i) for $$i=1,\ldots ,6$$i=1,…,6 is irrational. Here $$\beta (s)=\sum _{k=0}^{\infty }(-1)^k(2k+1)^{-s}$$β(s)=∑k=0∞(-1)k(2k+1)-s denotes Dirichlet’s beta function, so that $$\beta (2)$$β(2) is Catalan’s constant.

We consider the Siegel upper half space $$H_{2m}$$H2m of degree 2m and a subset $$H_m\times H_m$$Hm×Hm of $$H_{2m}$$H2m consisting of two $$m\times m$$m×m diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$Sp(m,R)×Sp(m,R)⊂Sp(2m,R), one is the action on holomorphic functions on $$H_{2m}$$H2m defined by the automorphy factor of

Assume a and $$b=na+r$$b=na+r with $$n \ge 1$$n≥1 and $$0

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.

In the 1960s Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $$\chi _{5}$$χ5, the cusp form of lowest weight for the group $${\text {Sp}}(2,\mathbb {Z})$$Sp(2,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

Generalized Burniat surfaces are surfaces of general type with $$p_g=q$$pg=q and Euler number $$e=6$$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

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 $$F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))$$F′=Hom(F,Q/Z(1)). Furthermore, we prove that $$\mathrm

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

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

We prove a non-vanishing result in weight aspect for the product of two Fourier coefficients of a Hecke eigenform of half-integral weight.

This paper is devoted, first of all, to give a complete unified proof of the characterization theorem for compact generalized Kähler manifolds. The proof is based on the classical duality between “closed” positive forms and “exact” positive currents. In the last part of the paper we approach the general case of non compact complex manifolds, where “exact” positive forms seem to play a more significant

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.

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 $$Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}$$Q(x)=∏j=1∞(1-xj)-j is

A symplectic polarity of a building {\Delta} of type E6 is a polarity whose fixed point structure is a building of type F4 containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type F4). In this paper, we show in a geometric way that every building of type E6 contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line

We choose some special unit vectors $${\mathbf {n}}_1,\ldots ,{\mathbf {n}}_5$$n1,…,n5 in $${\mathbb {R}}^3$$R3 and denote by $${\mathscr {L}}\subset {\mathbb {R}}^5$$L⊂R5 the set of all points $$(L_1,\ldots ,L_5)\in {\mathbb {R}}^5$$(L1,…,L5)∈R5 with the following property: there exists a compact convex polytope $$P\subset {\mathbb {R}}^3$$P⊂R3 such that the vectors $${\mathbf {n}}_1,\ldots ,{\mathbf

We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers $$\alpha _1^0,\alpha _2^0$$α10,α20, we give an algorithmic construction of a Seifert fibration $$L(p,q)\rightarrow S^2(\alpha |\alpha _1^0|,\alpha |\alpha _2^0|)$$L(p,q)→S2(α|α10|,α|α20|), where the natural number $$\alpha $$α is determined by the algorithm. This algorithm produces

Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for $$f:[1,\infty )\rightarrow {\mathbb R}$$f:[1,∞)→R non-negative and non-decreasing we prove $$f(x)-x=O(x^\gamma )$$f(x)-x=O(xγ) with $$\gamma <1$$γ<1 under certain assumptions on f. We state a conjecture concerning the weakest natural assumptions and

Let p be an odd prime number and $$\ell $$ℓ an odd prime number dividing $$p-1$$p-1. We denote by $$F=F_{p,\ell }$$F=Fp,ℓ the real abelian field of conductor p and degree $$\ell $$ℓ, and by $$h_F$$hF the class number of F. For a prime number $$r \ne p,\,\ell $$r≠p,ℓ, let $$F_{\infty }$$F∞ be the cyclotomic $$\mathbb {Z}_r$$Zr-extension over F, and $$M_{\infty }/F_{\infty }$$M∞/F∞ the maximal pro-r

We give a systematic way to construct almost conjugate pairs of finite subgroups of $$\mathrm {Spin}(2n+1)$$Spin(2n+1) and $${{\mathrm{Pin}}}(n)$$Pin(n) for $$n\in {\mathbb {N}}$$n∈N sufficiently large. As a geometric application, we give an infinite family of pairs $$M_1^{d_n}$$M1dn and $$M_2^{d_n}$$M2dn of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing

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.