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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Abstract: 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 22(4):583–607, 1982/83) is solved. PubDate: 2021-04-01
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Abstract: 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. PubDate: 2021-04-01
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Abstract: Let \(pod_3(n)\) 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 \(pod_3(n)\) using the theory of Hecke eigenforms. We also study the divisibility properties of \(pod_3(n)\) using arithmetic properties of modular forms. PubDate: 2021-04-01
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Abstract: 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 k. In particular they conjectured that if a Kähler manifold satisfies the \(\Delta \) -property then it is a complex space form. This paper is dedicated to the proof of the validity of this conjecture. PubDate: 2021-04-01
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Abstract: 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. PubDate: 2021-04-01
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Abstract: 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 Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over \({{\mathbb {Z}}}\) , we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift. PubDate: 2021-04-01
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Abstract: 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 an infinite order linear differential equation. Recently, Prado and Klinger-Logan (J Number Theory 217:422–442, 2020) extended Van Gorder’s result to show that the Hurwitz zeta function \(\zeta (s,a)\) is also formally satisfies a similar differential equation $$\begin{aligned} T\left[ \zeta (s,a) - \frac{1}{a^s}\right] = \frac{1}{(s-1)a^{s-1}}. \end{aligned}$$ But unfortunately in the same paper they proved that the operator T applied to Hurwitz zeta function \(\zeta (s,a)\) does not converge at any point in the complex plane \({\mathbb {C}}\) . In this paper, by defining \(T_{p}^{a}\) , a p-adic analogue of Van Gorder’s operator T, we establish an analogue of Prado and Klinger-Logan’s differential equation satisfied by \(\zeta _{p,E}(s,a)\) which is the p-adic analogue of the Hurwitz-type Euler zeta functions $$\begin{aligned} \zeta _E(s,a)=\sum _{n=0}^\infty \frac{(-1)^n}{(n+a)^s}. \end{aligned}$$ In contrast with the complex case, due to the non-archimedean property, the operator \(T_{p}^{a}\) applied to the p-adic Hurwitz-type Euler zeta function \(\zeta _{p,E}(s,a)\) is convergent p-adically in the area of \(s\in {\mathbb {Z}}_{p}\) with \(s\ne 1\) and \(a\in K\) with \( a _{p}>1,\) where K is any finite extension of \({\mathbb {Q}}_{p}\) with ramification index over \({\mathbb {Q}}_{p}\) less than \(p-1.\) PubDate: 2021-04-01
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Abstract: 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 \(L(1,{\mathrm {ad}}^0(g)\otimes \chi _D)\) for some \(g\in S_k(\Gamma _0(D),\chi _D)\) . PubDate: 2020-12-01
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