In this article, we estimate the density of the set of primes p such that the p-th Hecke eigenvalue of an Ikeda lift is divisible by a fixed positive integer. One of the main ingredients involves the study of abelian subfields of fixed fields of the kernel of Galois representations attached to elliptic Hecke eigenforms. Further, we study the distribution of Fourier coefficients of elliptic Hecke eigenforms in arithmetic progressions.

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We present a formula for computing the Milnor number of certain weighted Lê-Yomdin hypersurface singularities. This formula partially extends to hypersurfaces of arbitrary dimension a similar result previously established for surface singularities by Artal-Bartolo et al (Int. Math. Res. Not. IMRN 2010(22) (2010), 4301–4318).

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In this paper, we consider subordinate symmetric Markov processes that correspond to non-killing Dirichlet forms enjoying heat kernel estimates on a metric measure space with the volume doubling property. We obtain estimates of the jump kernel of the subordinate process and establish equivalent conditions for the jump kernel following the work of Liu and Murugan (ALEA Lat. Am. J. Probab. Math. Stat. 20(2) (2023), 1271–1281). In particular, we clarify the scale of the jump kernel, which is different from the diffusion type. This result is applicable to non-subordinate processes by the transferring method, which uses the stability of Dirichlet forms.

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Our aim in this paper is to establish some weighted second main theorems in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which generalize and improve upon the previous results of Yan (J. Math. Anal. Appl. 422 (2015), 456–477) and Thoan et al (Kyushu J. Math. 77 (2023), 203–220) by removing the condition that the codimension of the intersection of inverse images of two moving targets by the mappings is at least two. An algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity is also given as an application of the main results.

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We study the geometry of curves passing through the Whitney umbrella singularities by using a Darboux frame. We define three functions using a Frenet–Serret type formula related to the geodesic curvature, the normal curvature, and the geodesic torsion. We investigate the degrees of divergence, the top terms of these functions, and their geometric meanings. We also consider a developable surface along the curve.

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We study the eigenvalues and eigenfunctions of the Folland–Stein operator L_α on some examples of three-dimensional Heisenberg Bieberbach manifolds, that is, compact quotients Γ\H of the Heisenberg group H by a discrete torsion-free subgroup Γ of H⋊U(1).

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It will be shown that every adapted transformation of order one on a Wiener space gives rise to an exponentially integrable quadratic form through a change-of-variables formula, and conversely each exponentially integrable quadratic form has an adapted transformation of order one realizing the form in such a manner. This bidirectional relationship is applied to adapted linear transformations to see the equivalence between the exponential integrability of quadratic forms and the solvability of the Riccati equation. Furthermore, thanks to this equivalence, an explicit representation of the Feynman–Kac density function will be achieved.

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In this paper, we compute the second integral homology of even Artin groups using Hopf's formula. We then apply our results to the computation of cup products and Pontryagin products on even Artin groups, as well as to the second integral homology of even Coxeter groups.

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We construct solutions of the two-dimensional Toda–Hirota equation (2dTHE) expressed by the Gelfand hypergeometric function (HGF) on the Grassmannian GM(2, N) of confluent or non-confluent type, which is labeled by a partition of N. A system of hyperbolic equations in N complex variables is obtained from the differential equations which form the main body of the Gelfand hypergeometric system and characterize the image of the Radon transform. We use the Laplace sequence of the system of hyperbolic operators to find an elementary seed solution of the 2dTHE and then use the Bäcklund transformation to obtain higher solutions expressed in terms of the Gelfand HGF. In constructing the Laplace sequence, the contiguity relations (operators) for the Gelfand HGF play an important role.

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