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Shinnosuke Izumi
Kyushu Journal of Mathematics, Volume 75, pp 235-247; https://doi.org/10.2206/kyushujm.75.235

Abstract:
Let X be a compact metric space and E be a Banach space. Then lip (X, E) denotes the Banach space of all E-valued little Lipschitz functions on X. We show that lip(X, E)∗∗ is isometrically isomorphic to the Banach space of E∗∗-valued Lipschitz functions Lip(X, E∗∗) under several conditions. Moreover, we describe the isometric isomorphism from lip(X, E)∗∗ to Lip(X, E∗∗).
Sohei Ashida
Kyushu Journal of Mathematics, Volume 75, pp 277-294; https://doi.org/10.2206/kyushujm.75.277

Abstract:
We study the Hartree-Fock equation and the Hartree-Fock energy functional universally used in many-electron problems. We prove that the set of all critical values of the Hartree-Fock energy functional less than a constant smaller than the first energy threshold is finite. Since the Hartree-Fock equation, which is the corresponding Euler-Lagrange equation, is a system of nonlinear eigenvalue problems, the spectral theory for linear operators is not applicable. The present result is obtained by establishing the finiteness of the critical values associated with orbital energies less than a negative constant and combining the result with Koopmans' well-known theorem. The main ingredients are the proof of convergence of the solutions and the analysis of the Fréchet second derivative of the functional at the limit point.
Densuke Shiraishi
Kyushu Journal of Mathematics, Volume 75, pp 95-113; https://doi.org/10.2206/kyushujm.75.95

Abstract:
The ℓ-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in ℓ-adic numbers, which arises from Galois actions on ℓ-adic étale paths on ℙ1\{0, 1, ∞}. In the present paper, we discuss a relationship between ℓ-adic Galois polylogarithms and triple ℓth power residue symbols in some special cases studied in a work of Hirano and Morishita [J. Number Theory 198 (2019), 211-238]. We show that a functional equation of ℓ-adic Galois polylogarithms by Nakamura and Wojtkowiak [Non-abelian Fundamental Groups and Iwasawa Theory. Cambridge University Press, 2012, pp. 258-310] implies a reciprocity law of triple ℓth power residue symbols.
Kento Fujita, Yasushi Komori
Kyushu Journal of Mathematics, Volume 75, pp 149-167; https://doi.org/10.2206/kyushujm.75.149

Abstract:
We prove a congruence between symmetric multiple zeta-star values and multiple zeta-star values. This congruence together with Aoki and Ohno's relation, the sum formula and the generalized height-one duality for multiple zeta-star values directly lead to those for the symmetric counterparts.
Hiroshi Kihara, Nobuyuki Oda
Kyushu Journal of Mathematics, Volume 75, pp 129-147; https://doi.org/10.2206/kyushujm.75.129

Abstract:
Given a set X = (X1, X2,..., Xm) of pointed spaces, we introduce a family {X(k,l)} of subquotients of X1 × X2 × ··· × Xm. This family extends the family of subspaces of X1 × X2 × ··· × Xm introduced by G. J. Porter and contains the product, the fat wedge, the wedge and the smash product. The (co)homology with field coefficients of X(k,l) is completely determined, which is used to study the group ℇ(X(k,l)) of self-homotopy equivalences of X(k,l). Especially, in the case of X1 = X2 = ··· = Xm = X, we construct a homomorphism Ψ(k,l) from the semi-direct product of the m-fold product ℇ(X)m and the symmetric group Sm to ℇ(X(k,l)) and give sufficient conditions for Ψ(k,l) to be injective. We apply this result to the case where X = Sn, ℂPn, or K(Ar, n) with A a subring of ℚ or a field ℤ/p, providing an important subgroup of ℇ(X(k,l)).
Tsudoi Kaminaka, Fumiharu Kato
Kyushu Journal of Mathematics, Volume 75, pp 351-364; https://doi.org/10.2206/kyushujm.75.351

Abstract:
We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3 and 4, and partly classifies them, where the classification is complete for r = 2, 3 and 4. In fact, we show that there exists no normalized extremal quasimodular forms of depth four with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.
Winfried Kohnen
Kyushu Journal of Mathematics, Volume 75, pp 125-128; https://doi.org/10.2206/kyushujm.75.125

Abstract:
We give a characterization of cusp forms of half-integral weight of level four in the plus space in terms of a functional equation of attached L-series.
Mitsuo Kato
Kyushu Journal of Mathematics, Volume 75, pp 1-21; https://doi.org/10.2206/kyushujm.75.1

Abstract:
From Lauricella's hypergeometric functions in n variables, we construct a special type of Pfaffian equations of rank n + 1 called Okubo type differential equations. For each homogenized Okubo type differential equation, we construct special n + 1 coordinate functions called flat coordinate functions and an (n + 1)-tuple of homogeneous functions called a potential vector.
Matthew Randall
Kyushu Journal of Mathematics, Volume 75, pp 77-94; https://doi.org/10.2206/kyushujm.75.77

Abstract:
In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also establish the relationship to solutions given by Lamé's equation and investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to the first-order system of differential equations associated to the generalized Chazy equation with parameter k = 3/2.
Masanari Kida, Ryota Okano, Ken Yokoyama
Kyushu Journal of Mathematics, Volume 75, pp 41-54; https://doi.org/10.2206/kyushujm.75.41

Abstract:
We study modular properties of theta functions of binary quadratic forms with congruence condition and compute their values at arbitrary cusps.
Ken-Ichi Iwamoto
Kyushu Journal of Mathematics, Volume 75, pp 23-40; https://doi.org/10.2206/kyushujm.75.23

Abstract:
In this paper, we study non-singular extensions of Morse functions on closed orientable surfaces. By a non-singular extension of such a Morse function, we mean an extension to a function without critical points on some compact orientable 3-manifold having as boundary the given surface. In 1977, Curley characterized the existence of non-singular extensions of non-singular boundary germs in terms of combinatorics on associated labeled Reeb graphs. We apply Curley's result to show that every Morse function on a closed orientable (possibly disconnected) surface has a non-singular extension to a 3-manifold that is connected.
Katsusuke Nabeshima, Shinichi Tajima
Kyushu Journal of Mathematics, Volume 75, pp 55-76; https://doi.org/10.2206/kyushujm.75.55

Abstract:
New methods for computing parametric local b-functions are introduced for µ-constant deformations of semi-weighted homogeneous singularities. The keys of the methods are comprehensive Gröbner systems in Poincaré-Birkhoff-Witt algebra and holonomic D-modules. It is shown that the use of semi-weighted homogeneity reduces the computational complexity of b-functions associated with µ-constant deformations. In the case of inner modality two, local b-functions associated with µ-constant deformations are obtained by the resulting method and given the list of parametric local b-functions.
Minoru Hirose, Kohtaro Imatomi, Hideki Murahara, Shingo Saito
Kyushu Journal of Mathematics, Volume 75, pp 115-124; https://doi.org/10.2206/kyushujm.75.115

Abstract:
Ohno's relation is a generalization of both the sum formula and the duality formula for multiple zeta values. Oyama gave a similar relation for finite multiple zeta values, defined by Kaneko and Zagier. In this paper, we prove relations of similar nature for both multiple zeta-star values and finite multiple zeta-star values. Our proof for multiple zeta-star values uses the linear part of Kawashima's relation.
Tetsuya Ito
Kyushu Journal of Mathematics, Volume 75, pp 273-276; https://doi.org/10.2206/kyushujm.75.273

Abstract:
We show that a knot whose minimum crossing number c(K) is even and greater than 30 is not fertile; there exists a knot K′ with crossing number less than c such that K′ is not obtained from a minimum crossing number diagram of K by suitably changing the over-under information.
Yoshiaki Fukuma
Kyushu Journal of Mathematics, Volume 75, pp 211-233; https://doi.org/10.2206/kyushujm.75.211

Abstract:
Let (X, L) denote a polarized manifold of dimension five. This study considers the dimension of the global sections of KX + mL with m ≥ 6. In particular, we prove that h0(KX + mL) ≥ (m−15) for any polarized 5-fold (X, L) with h0(L) > 0. Furthermore, we also consider (X, L) with h0 (KX + mL) = (m−15) for some m ≥ 6 with h0(L) > 0.
Makoto Nakamura, Yuya Sato
Kyushu Journal of Mathematics, Volume 75, pp 169-209; https://doi.org/10.2206/kyushujm.75.169

Abstract:
The Cauchy problem for the semilinear complex Ginzburg-Landau type equation is considered in homogeneous and isotropic spacetime. Global solutions and their asymptotic behaviours for small initial data are obtained. The non-existence of non-trivial global solutions is also shown. The effects of spatial expansion and contraction are studied through the problem.
Yuta Watanabe
Kyushu Journal of Mathematics, Volume 75, pp 323-349; https://doi.org/10.2206/kyushujm.75.323

Abstract:
We study the cohomology groups of vector bundles on neighborhoods of non-pluriharmonic loci in q-complete Kähler manifolds and in compact Kähler manifolds. Applying our results, we show variants of the Lefschetz hyperplane theorem.
Trevor Griffin, Nathan Kenshur, Abigail Price, Bradshaw Vandenberg-Daves, Hui Xue, Daozhou Zhu
Kyushu Journal of Mathematics, Volume 75, pp 249-272; https://doi.org/10.2206/kyushujm.75.249

Abstract:
Let Ek (z) be the normalized Eisenstein series of weight k for the full modular group SL(2, ℤ). Let a > 0 be an even integer. In this paper we completely determine when the zeros of Ek interlace with the zeros of Ek+a. This generalizes a result of Nozaki on the interlacing of zeros of Ek and Ek+12.
Ce Xu
Kyushu Journal of Mathematics, Volume 75, pp 295-322; https://doi.org/10.2206/kyushujm.75.295

Abstract:
We define a new kind of classical digamma function, and establish some of its fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler-type sums. The main results of Flajolet and Salvy's paper (Expo. Math. 7(1) (1998), 15-35) are immediate corollaries of the main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
Kalyan Chakraborty, , Mohit Mishra
Kyushu Journal of Mathematics, Volume 74, pp 201-210; https://doi.org/10.2206/kyushujm.74.201

Abstract:
We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be three. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be two and three.
Katsuhisa Mimachi
Kyushu Journal of Mathematics, Volume 74, pp 15-42; https://doi.org/10.2206/kyushujm.74.15

Abstract:
Some of the connection problems associated with the system of differential equations E2, which is satisfied by Appell's F2 function, are solved by using integrals of Euler type. The present results give another proof of connection formulas related with Appell's F2, Horn's H2 and Olsson's FP functions, which are obtained by Olsson.
Ryota Umezawa
Kyushu Journal of Mathematics, Volume 74, pp 233-254; https://doi.org/10.2206/kyushujm.74.233

Abstract:
We introduce an iterated integral version of (generalized) log-sine integrals (iterated log-sine integrals) and prove a relation between a multiple polylogarithm and iterated log-sine integrals. We also give a new method for obtaining relations among multiple zeta values, which uses iterated log-sine integrals, and give alternative proofs of several known results related to multiple zeta values and log-sine integrals.
Hidekazu Tanaka
Kyushu Journal of Mathematics, Volume 74, pp 441-449; https://doi.org/10.2206/kyushujm.74.441

Abstract:
We study the rationality of gamma factors associated to certain Hasse zeta functions. We show many explicit examples of rational gamma factors coming from products of GL(n).
Shunsuke Tamura
Kyushu Journal of Mathematics, Volume 74, pp 255-264; https://doi.org/10.2206/kyushujm.74.255

Abstract:
For each positive integer m, an arbitrary finite non-solvable group acts smoothly on infinitely many standard spheres with exactly m fixed points. However, for a given finite non-solvable group G and a given positive integer m, all standard spheres do not admit smooth actions of G with exactly m fixed points. In this paper, for each of the alternating group A6 on six letters, the symmetric group S6 on six letters, the projective general linear group PGL(2, 9) of order 720, the Mathieu group M10 of order 720, the automorphism group Aut(A6) of A6 and the special linear group SL(2, 9) of order 720, we will give the dimensions of homology spheres whose fixed point sets of smooth actions of the group do not consist of odd numbers of points.
Steven Charlton
Kyushu Journal of Mathematics, Volume 74, pp 337-352; https://doi.org/10.2206/kyushujm.74.337

Abstract:
We prove an identity for multiple zeta star values, which generalizes some identities due to Imatomi, Tanaka, Tasaka and Wakabayashi. This identity gives an analogue of cyclic insertion-type identities, for multiple zeta star values, and connects the block decomposition with Zhao's generalized 2-1 formula.
Tamio Koyama
Kyushu Journal of Mathematics, Volume 74, pp 415-427; https://doi.org/10.2206/kyushujm.74.415

Abstract:
The Fisher integral is the normalizing constant of a statistical model on the special orthogonal group. In this paper, we discuss a system of differential equations for the Fisher integral. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we show that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. The Fisher integral for diagonal matrices is related to the hypergeometric function of matrix arguments. We also give a new approach to get differential operators annihilating the Fisher integral for the diagonal matrix.
Taiki Yamada
Kyushu Journal of Mathematics, Volume 74, pp 291-311; https://doi.org/10.2206/kyushujm.74.291

Abstract:
Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.
Xiaohan Wang, Jay Mehta, Shigeru Kanemitsu
Kyushu Journal of Mathematics, Volume 74, pp 313-335; https://doi.org/10.2206/kyushujm.74.313

Abstract:
As has been pointed out by Chakraborty et al (Seeing the invisible: around generalized Kubert functions. Ann. Univ. Sci. Budapest. Sect. Comput. 47 (2018), 185-195), there have appeared many instances in which only the imaginary part—the odd part—of the Lerch zeta-function was considered by eliminating the real part. In this paper we shall make full use of (the boundary function aspect of) the q-expansion for the Lerch zeta-function, the boundary function being in the sense of Wintner (On Riemann's fragment concerning elliptic modular functions. Amer. J. Math. 63 (1941), 628-634). We may thus refer to this as the ‘Fourier series-boundary q-series', and we shall show that the decisive result of Yamamoto (Dirichlet series with periodic coefficients. Algebraic Number Theory. Japan Society for the Promotion of Science, Tokyo, 1977, pp. 275-289) on short character sums is its natural consequence. We shall also elucidate the aspect of generalized Euler constants as Laurent coefficients after a brief introduction of the discrete Fourier transform. These are rather remote consequences of the modular relation, i.e. the functional equation for the Lerch zeta-function or the polylogarithm function. That such a remote-looking subject as short character sums is, in the long run, also a consequence of the functional equation indicates the ubiquity and omnipotence of the Lerch zeta-function—and, a fortiori, the modular relation (S. Kanemitsu and H. Tsukada. Contributions to the Theory of Zeta-Functions: the Modular Relation Supremacy. World Scientific, Singapore, 2014).
Takahiro Sudo
Kyushu Journal of Mathematics, Volume 74, pp 223-232; https://doi.org/10.2206/kyushujm.74.223

Abstract:
We study some finite discrete groups such as semi-direct products of finite cyclic groups by their automorphisms, the corresponding group and subgroup C* -algebras, and their K-theory. Consequently, we obtain several isomorphism classification theorems of such groups by their group C* -algebras and K-theory.
Toshie Takata, Rika Tanaka
Kyushu Journal of Mathematics, Volume 74, pp 265-289; https://doi.org/10.2206/kyushujm.74.265

Abstract:
We give a formula for the quantum SU(2) invariant at ζ = e4πi/r for Lens space L(p, q), and we prove that the asymptotic expansion is represented by a sum of contributions from SL2C flat connections whose coefficients are square roots of the Reidemeister torsions.
Norio Iwase, Mitsunobu Tsutaya
Kyushu Journal of Mathematics, Volume 74, pp 197-200; https://doi.org/10.2206/kyushujm.74.197

Abstract:
We show that tcM(M) ≤ 2 cat(M) for a finite simplicial complex M. For example,we have tcM(Sn ∨ Sm) = 2 for any positive integers n and m.
Henrik Bachmann, Tatsushi Tanaka
Kyushu Journal of Mathematics, Volume 74, pp 169-176; https://doi.org/10.2206/kyushujm.74.169

Abstract:
Recently, inspired by the Connes-Kreimer Hopf algebra of rooted trees, the second named author introduced rooted tree maps as a family of linear maps on the non-commutative polynomial algebra in two letters. These give a class of relations among multiple zeta values,which are known to be a subclass of the so-called linear part of the Kawashima relations. In this paper we show the opposite implication, that is, the linear part of the Kawashima relations is implied by the relations coming from rooted tree maps.
Nobuaki Yagita
Kyushu Journal of Mathematics, Volume 74, pp 43-62; https://doi.org/10.2206/kyushujm.74.43

Abstract:
In this paper, we study the mod(p) motivic cohomology of twisted complete flag varieties X over some restricted fields k. Here we take fields k such that the Milnor K-theory KMn+2 (k) / p = 0 for some n ≥ 2. For these fields, we compute the mod(p) motivic cohomologies of the Rost motives Rn and the flag variety X containing R2.
Saiei-Jaeyeong Matsubara-Heo
Kyushu Journal of Mathematics, Volume 74, pp 109-125; https://doi.org/10.2206/kyushujm.74.109

Abstract:
We consider Mellin-Barnes integral representations of GKZ hypergeometric equations. We construct integration contours in an explicit way and show that suitable analytic continuations give rise to a basis of solutions.
Akio Kodama
Kyushu Journal of Mathematics, Volume 74, pp 149-167; https://doi.org/10.2206/kyushujm.74.149

Abstract:
We introduce a new class of domains Dn,m(μ, p), called FBH-type domains, in ℂn × ℂm, where 0 < μ ∈ ℝ and p ∈ ℕ. In the special case of p = 1, these domains are just the Fock-Bargmann-Hartogs domains Dn,m(μ) in ℂn × ℂm introduced by Yamamori. In this paper we obtain a complete description of an arbitrarily given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group Aut(Dn,m(μ, p)) of any FBH-type domain Dn,m(μ, p) with p ≠ 1 is a Lie group isomorphic to the compact connected Lie group U(n) × U(m). This tells us that the structure of Aut(Dn,m(μ, p)) with p ≠ 1 is essentially different from that of Aut(Dn,m(μ)).
Shigeo Mukai
Kyushu Journal of Mathematics, Volume 74, pp 63-104; https://doi.org/10.2206/kyushujm.74.63

Abstract:
We study Pfaffian systems of confluent hypergeometric functions of two variables with rank three, by using rational twisted cohomology groups associated with Euler-type integral representations of them. We give bases of the cohomology groups, whose intersection matrices depend only on parameters. Each connection matrix of our Pfaffian systems admits a decomposition into five parts, each of which is the product of a constant matrix and a rational 1-form on the space of variables.
Shuji Watanabe
Kyushu Journal of Mathematics, Volume 74, pp 177-196; https://doi.org/10.2206/kyushujm.74.177

Abstract:
We show that the transition from a normal conducting state to a superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity from the viewpoint of operator theory. Here we have no magnetic field. Moreover we obtain the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature. To this end, we have to differentiate the thermodynamic potential with respect to the temperature twice. Since there is a solution to the BCS-Bogoliubov gap equation in the form of the thermodynamic potential, we have to differentiate the solution with respect to the temperature twice. Therefore, we need to show that the solution to the BCS-Bogoliubov gap equation is differentiable with respect to the temperature twice, as well as its existence and uniqueness. We carry out its proof on the basis of fixed point theorems.
Junko Inoue, Jean Ludwig
Kyushu Journal of Mathematics, Volume 74, pp 127-148; https://doi.org/10.2206/kyushujm.74.127

Abstract:
Let G = exp(g) be an exponential solvable Lie group and Ad(G) ⊂ D an exponential solvable Lie group of automorphisms of G. Assume that for every non-∗-regular orbit D · q, q ∈ g∗, of D = exp(∂) in g∗, there exists a nilpotent ideal n of g containing ∂ · g such that D · qǀn is closed in n∗. We then show that for every D-orbit Ω in g∗ the kernel kerC∗(Ω) of Ω in the C∗-algebra of G is L1-determined, which means that kerC∗(Ω) is the closure of the kernel kerL1(Ω) of Ω in the group algebra L1(G). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group D = Ad(G). We finally give an example of a non-closed non-∗-regular orbit of an exponential solvable group G and of a coadjoint orbit O ⊂ g∗, for which the corresponding kernel kerC∗(πO) in C∗(G) is not L1-determined.
Katsuhisa Mimachi
Kyushu Journal of Mathematics, Volume 74, pp 1-13; https://doi.org/10.2206/kyushujm.74.1

Abstract:
We give integral representations of Euler type for Appell's hypergeometric functions F2, F3, Horn's hypergeometric function H2 and Olsson's hypergeometric function FP. Their integrands are the same (up to a constant factor), and only the regions of integration vary.
Hiroshi Goda
Kyushu Journal of Mathematics, Volume 74, pp 211-221; https://doi.org/10.2206/kyushujm.74.211

Abstract:
The twisted Alexander polynomial is an invariant of the pair of a knot and its group representation. Herein, we introduce a digraph obtained from an oriented knot diagram,which is used to study the twisted Alexander polynomial of knots. In this context, we show that the inverse of the twisted Alexander polynomial of a knot may be regarded as the matrix-weighted zeta function that is a generalization of the Ihara-Selberg zeta function of a directed weighted graph.
Souheyb Dehimi, Mohammed Hichem Mortad
Kyushu Journal of Mathematics, Volume 74, pp 105-108; https://doi.org/10.2206/kyushujm.74.105

Abstract:
In this paper, we give an example of a closed unbounded operator whose square domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.
Nakao Hayashi, Elena I. Kaikina
Kyushu Journal of Mathematics, Volume 74, pp 375-400; https://doi.org/10.2206/kyushujm.74.375

Abstract:
We consider the inhomogeneous Dirichlet-boundary value problem for the quadratic nonlinear Schrödinger equations, which is considered as a critical case for the large-time asymptotics of solutions. We present sufficient conditions on the initial and boundary data which ensure asymptotic behavior of small solutions to the equations by using the classical energy method and factorization techniques of the free Schrödinger group.
Yutaro Aoki
Kyushu Journal of Mathematics, Volume 74, pp 353-373; https://doi.org/10.2206/kyushujm.74.353

Abstract:
In this paper, we study the mean square of the logarithmic derivative of the Selberg zeta function for cocompact discrete subgroups. Our results are analogues of the results on the mean square of the logarithmic derivative of the Riemann zeta function by Goldston, Gonek,and Montgomery (J. Reine Angew. Math. 537 (2001), 105-126). We obtain an asymptotic formula for the mean square of the logarithmic derivative of the Selberg zeta function,including a term on the pair correlation of the zeros of the Selberg zeta function. In addition,we introduce an integral related to the prime geodesic theorem in short intervals and prove that the integral is bounded by the mean square of the logarithmic derivative of the Selberg zeta function. The upper bound for the integral is improved in the case of the Selberg zeta function for arithmetic cocompact groups by proving an asymptotic formula for the mean square near the left side of the vertical line whose real part is one.
Yoshinori Hamahata
Kyushu Journal of Mathematics, Volume 74, pp 429-439; https://doi.org/10.2206/kyushujm.74.429

Abstract:
Let k1, . . . , kr be positive integers. Let q1, . . . ,qr be pairwise coprime positive integers with qi > 2 (i = 1, . . . , r), and set q = q1 . . . qr. For each i = 1, . . . , r, let Ti be a set of φ(qi)/2 representatives mod qi such that the union Ti ∪ (-Ti) is a complete set of coprime residues mod qi. Let K be an algebraic number field over which the qth cyclotomic polynomial Φq is irreducible. Then, φ(q)/2 r numbers Πri=1d ki-1/dz ki-1i (cotπzi)|zi=ai/qi (ai ∈ Ti , i = 1, . . . , r) are linearly independent over K. As an application, a generalization of the Baker-Birch-Wirsing theorem on the non-vanishing of the multiple Dirichlet series L(s1, . . . , sr; f) with periodic coefficients at (s1, . . . , sr) = (k1, . . . , kr) is proven under a parity condition.
Federico Pellarin, Gabriele Nebe
Kyushu Journal of Mathematics, Volume 74, pp 401-413; https://doi.org/10.2206/kyushujm.74.401

Abstract:
Kaneko and Koike introduced the notion of extremal quasi-modular forms and proposed conjectures on their arithmetic properties. The aim of this paper is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The paper ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalized extremal quasi-modular form of weight 14 and depth one.
Yosuke Irie
Kyushu Journal of Mathematics, Volume 73, pp 357-377; https://doi.org/10.2206/kyushujm.73.357

Abstract:
The loxodromic Eisenstein series is defined for a loxodromic element of cofinite Kleinian groups. It is the analogue of the ordinary Eisenstein series associated to cusps. We study the asymptotic behavior of the loxodromic Eisenstein series for degenerating sequences of three-dimensional hyperbolic manifolds of finite volume. In particular, we prove that if the loxodromic element corresponds to the degenerating geodesic, then the associated loxodromic Eisenstein series converges to the ordinary Eisenstein series associated to the newly developing cusp on the limit manifold.
Nils Matthes, Koji Tasaka
Kyushu Journal of Mathematics, Volume 73, pp 337-356; https://doi.org/10.2206/kyushujm.73.337

Abstract:
Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown, respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.
Katsunori Iwasaki
Kyushu Journal of Mathematics, Volume 73, pp 251-294; https://doi.org/10.2206/kyushujm.73.251

Abstract:
Following a previous article we continue our study on non-terminating hypergeometric series with one free parameter, which aims to find arithmetical constraints for a given hypergeometric series to admit a gamma product formula. In this article we exploit the concepts of duality and reciprocity not only to extend already obtained results to a larger region but also to strengthen themselves substantially. Among other things we are able to settle the rationality and finiteness conjectures posed in the previous article.
Yoshiaki Okumura
Kyushu Journal of Mathematics, Volume 73, pp 295-316; https://doi.org/10.2206/kyushujm.73.295

Abstract:
In the arithmetic of function fields, Drinfeld modules play the role that elliptic curves play in the arithmetic of number fields. The aim of this paper is to study a non-existence problem of Drinfeld modules with constraints on torsion points at places with large degree. This is motivated by a conjecture of Christopher Rasmussen and Akio Tamagawa on the non-existence of abelian varieties over number fields with some arithmetic constraints. We prove the non-existence of Drinfeld modules satisfying Rasmussen-Tamagawa type conditions in the case where the inseparable degree of the base field is not divisible by the rank of Drinfeld modules. Conversely if the rank divides the inseparable degree, then we prove the existence of a Drinfeld module satisfying Rasmussen-Tamagawa type conditions.
Yann Bugeaud, Hajime Kaneko
Kyushu Journal of Mathematics, Volume 73, pp 221-227; https://doi.org/10.2206/kyushujm.73.221

Abstract:
We prove that there are only finitely many perfect powers in any linear recurrence sequence of integers of order at least two and whose characteristic polynomial is irreducible and has a dominant root.
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