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Abstract: Let p be an odd prime. In this paper we compute the Fourier coefficients of the Siegel Eisenstein series of degree 2, level p with the trivial or the quadratic character, associated to a certain cusp. For that we need to define the p-factor of the special type of Siegel series with character. PubDate: 2022-01-18

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Abstract: We already have the concept of isotropicity of a minimal surface in a Riemannian 4-manifold and a space-like or time-like surface in a neutral 4-manifold with zero mean curvature vector. In this paper, based on the understanding of it, we define and study isotropicity of a space-like or time-like surface in a Lorentzian 4-manifold N with zero mean curvature vector. If the surface is space-like, then the isotropicity means either the surface has light-like or zero second fundamental form or it is an analogue of complex curves in Kähler surfaces. In addition, if N is a space form, then the isotropicity means that the surface has both the properties. If the surface is time-like and if N is a space form, then the isotropicity means that the surface is totally geodesic. PubDate: 2022-01-03

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Abstract: It is proved that decomposable graphs are set recognizable and that the index graph of the canonical decomposition as well as the graphs induced on the maximal autonomous sets of vertices are set reconstructible. From these results, we obtain set reconstructibility for many decomposable graphs as well as a concise description of the decomposable graphs for which set reconstruction remains an open problem. PubDate: 2022-01-03

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Abstract: We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight. PubDate: 2021-12-14

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Abstract: Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties. PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00251-1

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Abstract: In this paper, we consider a kind of area-preserving flow for closed convex planar curves which will decrease the length of the evolving curve and make the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle as time \(t\rightarrow +\infty \) . PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00249-9

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Abstract: We introduce a model for random chain complexes over a finite field. The randomness in our complex comes from choosing the entries in the matrices that represent the boundary maps uniformly over \(\mathbb {F}_q\) , conditioned on ensuring that the composition of consecutive boundary maps is the zero map. We then investigate the combinatorial and homological properties of this random chain complex. PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00248-w

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Abstract: Let X be a smooth quintic hypersurface in \(\mathbb {P}^3\) , let C be a smooth hyperplane section of X, and let \(H=\mathcal {O}_X(C)\) . In this paper, we give a necessary and sufficient condition for the line bundle given by a non-zero effective divisor on X to be initialized and aCM with respect to H. PubDate: 2021-10-01 DOI: 10.1007/s12188-021-00250-2

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Abstract: 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 off in a natural manner from the p-adic Dirichlet series. We also provide an alternative proof of the expansion using p-adic measures and give an explicit formula for the values of the regularized Bernoulli distribution. The result is particularly simple for \(c=2\) , where we obtain a Dirichlet series expansion that is similar to the complex case. PubDate: 2021-08-30 DOI: 10.1007/s12188-021-00244-0

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Abstract: 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”. PubDate: 2021-08-21 DOI: 10.1007/s12188-021-00246-y

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Abstract: We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in \({\mathbb {Q}}(\sqrt{-7})\) and \({\mathbb {Q}}(\sqrt{-11})\) . In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms. PubDate: 2021-08-12 DOI: 10.1007/s12188-021-00245-z

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Abstract: 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 \(F(\varLambda )\) is independent of p. PubDate: 2021-08-09 DOI: 10.1007/s12188-021-00243-1

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Abstract: Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors. PubDate: 2021-07-22 DOI: 10.1007/s12188-021-00240-4

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Abstract: 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 \(0<h(n) \le h(n+1)\) . We put \(P_0^{g,h}(x)=1\) and $$\begin{aligned} P_n^{g,h}(x) := \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). \end{aligned}$$ As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind \(\eta \) -function. Here, g is the sum of divisors and h the identity function. Kostant’s result on the representation of simple complex Lie algebras and Han’s results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein’s j-invariant, and Chebyshev polynomials of the second kind. PubDate: 2021-06-14 DOI: 10.1007/s12188-021-00241-3

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Abstract: 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 subspaces tangent to M, G, \(G_i\) , respectively, then we study G-actions that satisfy the condition \(m_0=n_0^1 + \cdots + n_0^{l}\) . This condition implies that the orbits are non-degenerate for the pseudo Riemannian metric on M and this allows us to consider the normal bundle to the orbits. Using the properties of the normal bundle to the G-orbits we obtain an isometric splitting of M by considering natural metrics on each \(G_i\) . PubDate: 2021-06-07 DOI: 10.1007/s12188-021-00242-2

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Abstract: A correction to this paper has been published: https://doi.org/10.1007/s12188-021-00239-x PubDate: 2021-04-01 DOI: 10.1007/s12188-021-00239-x

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Abstract: A comparison among different constructions in \(\mathbb {H}^2 \cong {\mathbb {R}}^8\) of the quaternionic 4-form \(\Phi _{\text {Sp}(2)\text {Sp}(1)}\) and of the Cayley calibration \(\Phi _{\text {Spin}(7)}\) shows that one can start for them from the same collections of “Kähler 2-forms”, entering both in quaternion Kähler and in \(\text {Spin}(7)\) geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension 16, similar constructions allow to write explicit formulas in \(\mathbb {R}^{16}\) for the canonical 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\) , associated with Clifford systems related with the subgroups \(\text {Spin}(8)\) and \(\text {Spin}(7)\text {U}(1)\) of \(\text {SO}(16)\) . We characterize the calibrated 4-planes of the 4-forms \(\Phi _{\text {Spin}(8)}\) and \(\Phi _{\text {Spin}(7)\text {U}(1)}\) , extending in two different ways the notion of Cayley 4-plane to dimension 16. PubDate: 2021-04-01 DOI: 10.1007/s12188-020-00229-5

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Abstract: 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 function and the Riemann zeta function. PubDate: 2021-04-01 DOI: 10.1007/s12188-021-00232-4