Poincaré series

Let N_α denote the number of solutions to the congruence F(x_i,..., x_m) ≡ 0 (mod p^α) for a polynomial F(x_i,..., x_m) with integral p-adic coefficients. We examine the series \(\varphi (t) = \sum\nolimits_{\alpha = 0}^\infty {N_{\alpha ^{t^\alpha } } } \) . called the Poincaré series for the polynomial F. In this work we prove the rationality of the series ?(t) for a class of isometrically equivalent polynomials of m variables, m ≥ 2, containing the sum of two forms ?_n(x, y) + ?_n+1(x, y) respectively of degrees n and n+1, n ≥ 2. In particular the Poincaré series for any third degree polynomial F₃(x, y) (over the set of unknowns) with integral p-adic coefficients is a rational function of t.     

Szegő kernels and Poincaré series

Let \(M = {{\widetilde M} \mathord{\left/ {\vphantom {{\widetilde M} \Gamma }} \right. \kern-\nulldelimiterspace} \Gamma }\) be a Kähler manifold, where Γ ~ π₁(M) and \(\widetilde M\) is the universal Kähler cover. Let (L, h) → M be a positive hermitian holomorphic line bundle. We first prove that the L² Szeg? projector \({\widetilde \Pi _N}\) for L²-holomorphic sections on the lifted bundle \({\widetilde L^N}\) is related to the Szeg? projector for H⁰(M, L^N) by \({\widehat \Pi _N}\left( {x,y} \right) = \sum\nolimits_{\gamma \in \Gamma } {{{\widetilde {\widehat \Pi }}_N}} \left( {\gamma \cdot x,y} \right)\). We then apply this result to give a simple proof of Napier’s theorem on the holomorphic convexity of \(\widetilde M\) with respect to \({\widetilde L^N}\) and to surjectivity of Poincaré series.     

Non-vanishing of Taylor coefficients and Poincaré series

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan’s Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincaré series. At a “generic” point, all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.     

Equivariant Poincaré series of filtrations and topology

Earlier, for an action of a finite group G on a germ of an analytic variety, an equivariant G-Poincaré series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of G-sets with an additional structure. We discuss to which extent the G-Poincaré series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors.     

Generalised Poincaré series and embedded resolution of curves

The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein–Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.     

Generalised Poincaré series and embedded resolution of curves

The purpose of this paper is to extend the notions of generalised Poincaré series and divisorial generalised Poincaré series (of motivic nature) introduced by Campillo, Delgado and Gusein–Zade for complex curve singularities to curves defined over perfect fields, as well as to express them in terms of an embedded resolution of curves.     

Hyperbolic dimension and Poincaré critical exponent of rational maps

We study the Poincaré series of rational maps. By investigating the property of conical Julia set and dissipative measure, we prove that the Poincaré critical exponents are equal to the hyperbolic dimensions for a large class of rational maps.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Nonvanishing of Siegel Poincaré series

We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.     

Hilbert function,generalized Poincaré series and topology of plane valuations

To a multi-index filtration (say, on the ring of germs of functions on a germ of a complex analytic variety) one associates several invariants: the Hilbert function, the Poincaré series, the generalized Poincaré series, and the generalized semigroup Poincaré series. The Hilbert function and the generalized Poincaré series are equivalent in the sense that each of them determines the other one. We show that for a filtration on the ring of germs of holomorphic functions in two variables defined by a collection of plane valuations both of them are equivalent to the generalized semigroup Poincaré series and determine the topology of the collection of valuations, i.e. the topology of its minimal resolution.     

A weighted Poincaré inequality and removable singularities for harmonic maps

We give a sufficient condition on a closed subset R ⁿ for the weighted Poincaré inequality (1.5) below to be valid. As an application, we prove that, for any 2p<n and any such closed subset R ⁿ , if uC ¹( , N) W ^1,p (, N) is a stationary p-harmonic map such that |Du| ^p (x) dx is sufficiently small, then uC ¹(, N). This extends previously known removal singularity theorems for p-harmonic maps. Mathematics Subject Classification (2000):58E20, 58J05, 35J60This revised version was published online in September 2003 with a corrected date of receipt of the article.     

Alexander polynomials and Poincaré series of sets of ideals

Earlier the authors considered and, in some cases, computed Poincaré series for two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular, on the complex plane). A filtration of the first class was defined by a curve (with several branches) on the surface singularity. A filtration of the second class (called divisorial) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define and compute in some cases the Poincaré series corresponding to a set of ideals in the ring of germs of functions on a surface singularity. For the complex plane, this notion unites the two classes of filtrations described above.     

Hyperbolic Fourier coefficients of Poincaré series

The Ramanujan Journal - Poincaré (Ann Fac Sci Toulouse Sci Math Sci Phys 3:125–149, 1912) and Petersson (Acta Math 58(1):169–215, 1932) gave the now classical expression for the...     

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V.I. Arnold and V.I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.     

Generators of Jacobi forms are Poincaré series

The ring of Jacobi forms of even weights is generated by the weak Jacobi forms \(\phi _{-2,1}\) and \(\phi _{0,1}\). Bringmann and the first author expressed \(\phi _{-2,1}\) as a specialization of a Maass–Jacobi–Poincaré series. In this paper, we extend the domain of absolute convergence of Maass–Jacobi–Poincaré series which allows us to show that \(\phi _{0,1}\) is also a Poincaré series.     

Fourier coefficients of logarithmic vector-valued Poincaré series

We give an exact expression (Theorem 3.2) for the Fourier coefficients of logarithmic vector-valued Poincaré series associated to representations where \(\rho (T)\) is a single Jordan block.     

Poincaré series of Klein groups,coxeter polynomials,the Burau representation,and Milnor invariants

We obtain several formulas for the Poincaré series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincaré series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincaré series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincaré series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincaré series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.     

McKay correspondence for the Poincaré series of Kleinian and Fuchsian singularities

We give a uniform and, to a large extent, geometrical proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using 2-Calabi-Yau triangulated categories and spherical twist functors.