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.
We prove a Liouville property for any f-harmonic function with polynomial growth on a complete non-compact smooth metric measure space (M, g, e -f dv) when the Bakry-Émery Ricci curvature is non-negative and the diameter of its geodesic sphere has sublinear growth.
In this paper, we first give a new simple proof for the elimination theorem of definite folds by homotopy for generic smooth maps of manifolds of dimension strictly greater than two into the 2-sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but also algorithmic: the procedure enables us to construct various explicit examples. We also study simple stable maps of 3-manifolds into the 2-sphere without definite folds. Furthermore, we prove the non-existence of singular Legendre fibrations on 3-manifolds, answering negatively a question posed in our previous paper.
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.
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.
In the present paper, we discuss the hyperbolic ordinariness of hyperelliptic curves in characteristic 3. In particular, we prove that every hyperelliptic projective hyperbolic curve of genus less than or equal to 5 in characteristic 3 is hyperbolically ordinary.
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.
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.
In this paper, we shall introduce the notion of a KSM-manifold, which has the structure of a fiber bundle over an Einstein-Kähler Fano manifold whose fiber is a toric Fano manifold, and prove that every KSM-manifold admits a Kähler-Ricci soliton.
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isomorphism between Jacobi forms of integral weight and half-integral weight is also obtained.
The existence and the continuity of the transition density function of the diffusion process generated by the Baouendi-Grushin operator is shown with the help of the partial Malliavin calculus. For this purpose, the partial Malliavin calculus is reformulated in terms of Watanabe's distribution theory on Wiener spaces.