This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.

Abstract We determine the Bernstein–Sato polynomials for the ideal of maximal minors of a generic m × n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.

In analogy with the complex analytic case, Mustata constructed (a family of) Bernstein-Sato polynomials for the structure sheaf O-X and a hypersurface (f =3D 0) in X, where X is a regular variety over an F-finite field of positive characteristic (see Mustata, Bernstein-Sato polynomials in positive characteristic, J. Algebra 321(1) (2009), 128-151). He shows that the suitably interpreted zeros of his Bernstein-Sato polynomials correspond to the F-jumping numbers of the test ideal filtration tau(X; f(t)). In the present paper we generalize Mustata's construction replacing O-X by an arbitrary F-regular Cartier module M on X and show an analogous correspondence of the zeros of our Bernstein-Sato polynomials with the jumping numbers of the associated filtration of test modules tau(M; f(t)) provided that f is a nonzero divisor on M

In ([DS1], [DS2], [DS3]), Denker and Sato studied a Markov chain on the finite words space of the Sierpinski gasket (SG). They showed that the Martin boundary is homeomorphic to the SG. Recently, Lau and Wang (2015 Math. Z. 280 401–20) showed that the homeomorphism holds for an iterated function system with the open set condition provided that the transition probability on the finite words space is of DS-type. In this work, we continue studying this kind of transition probability on the unit interval. Using matrix expressions, we obtain a formula to calculate the Green function. By the ergodic arguments for non-negative matrices, we find that the Martin boundary is homeomorphic to the unit interval or the union of the unit interval and a countable set. This gives a good illustration for the results in Lau and Wang (2015 Math. Z. 280 401–20).

In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new q-deformed Toda hierarchy (QTH). Meanwhile the Block type additional symmetry and bi-Hamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the q-Toda hierarchy to an extended q-Toda hierarchy (EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov-Witten theory. The corresponding Sato theory including multi-fold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the q-Toda hierarchy and show the integrability including its bi-Hamiltonian structure, tau symmetry and conserved densities. (C) 2015 Elsevier Ltd. All rights reserved.

Abstract In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new q-deformed Toda hierarchy (QTH). Meanwhile the Block type additional symmetry and bi-Hamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the q-Toda hierarchy to an extended q-Toda hierarchy (EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov–Witten theory. The corresponding Sato theory including multi-fold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the q-Toda hierarchy and show the integrability including its bi-Hamiltonian structure, tau symmetry and conserved densities.

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum Sigma(S) contains a subset W-(S) over cap of points for which each fiber of the corresponding tautological subbundle TBWS is closed with respect to multiplication. Algebraically TBWS is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle TBWempty set represents the tower of families of normal rational (Veronese) curves of all degrees. For W-1 such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles TBW1,2,..., n represent families of plane (n+1, n+2) curves (trigonal curves at n = 2) and space curves of genus n. Two methods of regularization of singular curves contained in TBW(S) over cap, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.