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.     

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.     

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...     

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.     

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.     

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.     

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.     

On the interlacing of the zeros of Poincaré series

The Ramanujan Journal - Rankin proved that the Poincaré series for $$mathbf{SL}(2,{{mathbb {Z}}})$$ that are not cusp forms have all their zeros on the unit circle in the standard...     

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.     

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.     

The radius of convergence of Poincaré series of loop spaces

LetR _S (resp.R _A) be the radius of convergence of the Poincaré series of a loop space (S) (resp. of the Betti-Poincaré series of a noetherian connected graded commutative algebraA over a field of characteristic zero).IfS is a finite 1-connected CW-complex, the rational homotopy Lie algebra ofS is finite dimensional if and only ifR _S-1. OtherwiseR _S<1.There is an easily computable upper bound (usually less than 1) forR _S ifS is formal or coformal.On the other handR _A=+ if and only ifA is a polynomial algebra andR _A=1 if and only ifA is a complete intersection (Golod and Gulliksen conjecture). OtherwiseR _A<1 and the sequence dim Tor _p ^H grows exponentially withp.     

Denominators for the Poincaré series of invariants of small matrices

We determine explicit denominators for the Poincaré series of (a) the invariants ofm genericN ×N matrices, and (b) the ring generated bym genericN ×N matrices and their traces, forN≤4. ForN≤3 we prove (and forN=4 we conjecture) that the denominators we obtain are of minimum degree. We also give explicit rational fractions for both series for small values ofm andN. Research supported by NSF grants DMS-9622062 and DMS-9700787.     

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.     

Denominators for the Poincaré series of invariants of matrices with involution

The goal of this paper is to study some Poincaré series associated to the invariants of the symplectic and odd orthogonal groups. These series turn out to be rational functions and our main results will describe the denominators. This work will generalize some known results on the invariants of the general linear groups. In addition to whatever intrinsic interest we hope our results may have, the subject involves an interesting interplay of invariant theory and complex variables. The first author gratefully acknowledges Support from DePaul University Research Council. The second author was supported in part by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities, and by an Internal Research Grant from Bar-Ilan University.     

Multi-variable Poincaré series of algebraic curve singularities over finite fields

Let be the local ring at a singular point of a geometrically integral algebraic curve defined over a finite field, and let m be the number of branches centered at the curve singularity. By encoding cardinalities of certain finite sets of ideals, we associate to each pair of ideal classes of a power series in m variables with integer coefficients, which can be represented by an integral within the framework of harmonic analysis. We prove that partial local zeta functions can be expressed in terms of these multi-variable power series. The main objective of this paper is to investigate the properties of these series, and to provide in this way a deeper insight into the nature of local zeta functions.      

Amenability,Poincaré series and quasiconformal maps

Summary Any coveringYX of a hyperbolic Riemann surfaceX of finite area determines an inclusion of Teichmüller spaces Teich(X)Teich(Y). We show this map is an isometry for the Teichmüller metric if the covering isamenable, and contracting otherwise. In particular, we establish <1 for classical Poincaré series (Kra's Theta conjecture).The appendix develops the theory of geometric limits of quadratic differentials, used in this paper and a sequel.Research partially supported by an NSF Postdoctoral Fellowship