Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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  • 2024-09-01

    Sign changes of the coefficients of triple product $L$-functions

    Huixue Lao, Fengjiao Qiao

    Abstract : Let $f(z)$ be a primitive holomorphic cusp form and $g(z)$ be a Maass cusp form. In this paper, we give quantitative results for the sign changes of coefficients of triple product $L$-functions $L(f\times f\times f, s)$ and $L(f\times f\times g, s)$.

  • 2024-09-01

    Different characterizations of curvature in the context of Lie algebroids

    Rabah Djabri

    Abstract : We consider  a vector bundle map $F\colon E_{1}\longrightarrow E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$  over arbitrary bases $M_{1}$ and  $M_{2}$. We associate to it different notions of curvature which we call  A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten  structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and  we will prove that these curvatures are equivalent, in the  sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary  that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten   supermanifolds).

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  • 2024-09-01

    Eigenvalues and congruences for the weight 3 paramodular nonlifts of levels 61, 73, and 79

    Cris Poor , Jerry Shurman , David S. Yuen

    Abstract : We use Borcherds products to give a new construction of the weight~$3$paramodular nonlift eigenform~$f_N$ for levels~$N=61,73,79$.We classify the congruences of~$f_N$ to Gritsenko lifts.We provide techniques that compute eigenvalues to supportfuture modularity applications.Our method does not compute Hecke eigenvalues from Fouriercoefficients but instead uses elliptic modular forms, specificallythe restrictions of Gritsenko lifts and their images under the slashoperator to modular curves.

  • 2024-11-01

    Representation numbers by quadratic forms of certain levels

    Ick Sun Eum

    Abstract : Let $Q$ be an integral positive definite quadratic form of level $N$ in $2k(\geq4)$ variables. We assume that $(-1)^kN$ is a fundamental discriminant and the associated character $\chi$ of $Q$ is primitive of conductor $N$. Under our assumption, we find the pairs $(k,N)$ such that the dimension of spaces of cusp forms of weight $k$ and level $N$ with Nebentypus $\chi$ is one. Furthermore, we explicitly construct their bases by using Eisenstein series of lower weights. For the above pairs $(k,N)$, we use these cusp forms to provide closed formulas for the representation numbers by quadratic forms of level $N$ in $2k$ variables, which are expressed in terms of divisor functions and their convolution sums.

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  • 2024-11-01

    Associated primes and nilpotent associated primes of skew inverse Laurent series rings

    Azadeh Hajaliakbari, Ahmad Moussavi

    Abstract : Let $R$ be an associative ring with an identity, $ \sigma $ be an automorphism, and $\delta$ be a $\sigma$-derivation of $R$. In this article, we describe all (nilpotent) associated primes of the skew inverse Laurent series ring $ R((x^{-1}$; $\sigma,\delta)) $ in terms of the (nilpotent) associated primes of $R.$

  • 2024-11-01

    Timelike general rotational surfaces in Minkowski 4-space with density

    Mustafa Altın, Ahmet Kazan, Dae Won Yoon

    Abstract : In this study, we give weighted mean and weighted Gaussian curvatures of two types of timelike general rotational surfaces with non-null plane meridian curves in four-dimensional Minkowski space $\mathbb{E}_{1}^{4}$ with density $e^{\lambda_{1}x^{2}+\lambda_{2}y^{2}+\lambda_{3}z^{2}+\lambda_{4}t^{2}},$ where $\lambda_{i}$ ($i=1,2,3,4$) are not all zero. We give some results about weighted minimal and weighted flat timelike general rotational surfaces in $\mathbb{E}_{1}^{4}$ with density. Also, we construct some examples for these surfaces.

  • 2024-11-01

    On the unresolved conjecture for the algebraic transfers over the binary field

    Đặng Võ Phúc

    Abstract : The objective which drives the writing of this article is to study the behavior of the algebraic transfer for ranks $h\in \{6,\, 7,\, 8\}$ across various internal degrees. More precisely, we prove that the algebraic transfer is an isomorphism in certain bidegrees. A noteworthy aspect of our research is the rectification of the results outlined by M. Moetele and M. F. Mothebe in [East-West J. of Mathematics \textbf{18} (2016), 151--170]. This correction focuses on the $\mathcal A$-generators for the polynomial algebra $\mathbb Z/2[t_1, t_2, \ldots, t_h]$ in degree thirteen and the ranks $h$ mentioned above. As direct consequences, we are able to confirm the Singer conjecture for the algebraic transfer in the cases under consideration. Especially, we affirm that \textit{the decomposable element $h_6Ph_2 \in {\rm Ext}_{\mathcal A}^{6, 80}(\mathbb Z/2, \mathbb Z/2)$ does not reside within the image of the sixth algebraic transfer}. This event carries significance as it enables us to either strengthen or refute the Singer conjecture, which is relevant to the behavior of the algebraic transfer. Additionally, we also show that the indecomposable element $$q\in {\rm Ext}_{\mathcal A}^{6, 38}(\mathbb Z/2, \mathbb Z/2)$$ is not detected by the sixth algebraic transfer. \textit{Prior to this research, no other authors had delved into the Singer conjecture for these cases. The significant and remarkable advancement made in this paper regarding the investigation of Singer's conjecture for ranks $h,\, 6\leq h\leq 8,$ highlights a deeper understanding of the enigmatic nature of ${\rm Ext}_{\mathcal A}^{h, h+\bullet}(\mathbb Z/2, \mathbb Z/2)$}.

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  • 2024-11-01

    The convergence rate for Schr\"{o}dinger operators with complex time

    Tie Li, Yaoming Niu, Ying Xue

    Abstract : In this paper, in one spatial dimension, we study the convergence rate for Schr\"{o}dinger operators with complex time $P_{a,\gamma}^{t}$, which is defined by $$P_{a,\gamma}^{t}f(x)=S_{a}^{t+it^{\gamma}}f(x) =\int_{\mathbb{R}} e^{ix\xi}e^{it|\xi|^{a}}e^{-t^{\gamma}|\xi|^{a}} \hat{f}(\xi)d\xi,$$ where $\gamma>0$ and $a>0.$ The convergence rate for Schr\"{o}dinger operators with complex time is different from that of classical Schr\"{o}dinger operators in Cao-Fan-Wang (Illinois J. Math. 62: 365--380, 2018).

  • 2024-11-01

    A study on holomorphic isometries of weighted Bergman metrics

    Aakanksha Jain , Kaushal Verma

    Abstract : For a domain $D \subset \mathbb C^n$ and an admissible weight $\mu$ on it, we consider the weighted Bergman kernel $K_{D, \mu}$ and the corresponding weighted Bergman metric on $D$. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \ge 0$. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.

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  • 2024-11-01

    Tits Pentagons and the Root System $GH_2$

    Bernhard Mühlherr, Richard M. Weiss

    Abstract : We show that under a mild thickness condition, every Tits pentagon arises through a folding process from a free module of rank~$5$ over a unitary associative ring of stable range~$1$.

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November, 2024
Vol.61 No.6

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