Maass operators and Eisenstein series

Given a totally real fieldL of degreeg, we constructg Hasse invariants on Hilbert modular varieties in characteristicp and characterize their divisors. We show that these divisors give the type stratification defined by the action of\(\mathcal{O}_L \) on theαp-elementary subgroup. Under certain conditions, involving special values of zeta functions, the product of these Hasse invariants is the reduction of an Eisenstein series of weightp−1

If M is a marked compact Riemann surface of genus g>0 and ξ is a factor of automorphy with characteristic class c(ξ)=r then by the Riemann-Roch theorem γ(ξ)=γ(κξ-1)+r+1-g, where κ is the canonical factor of automorphy; and if r≥2g-1 then c(κξ)=2g-2-r<0 so that γ(κξ-1)=0 and γ(ξ)=r+1-g. Thus all factors of automorphy with characteristic class r admit equally many complex analytic relatively automorphic functions whenever r≥2g-1. Now the set of factors of automorphy with characteristic class r can be parametrized by the complex manifold ℂ g , by associating to any point t∊- g the factor of automorphy ρtζ r where ρ: ℂ g →Hom(Г, ℂ*) is the canonical homomorphism considered previously and ζ is a factor of automorphy representing the point bundle associated to the divisor 1·p0 for the base point p0 of the marked surface M; and it can be asked whether the set of all complex analytic relatively automorphic functions for these factors of automorphy can be parametrized accordingly by the complex manifold ℂg× ℂr+1-g if r≥2g-1. The answer is provided by the following result.

On the Northcott property of zeta functions over function fields

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

In this paper we exhibit a new type of relation between 0 series and Eisenstein series. In Sect. 1 we give a new construction for a sequence of nonlinear differential operators, introduced by Rankin [6], which connect the various representations of the double cover of SL2(1R ) on differential forms of quarter-integral weight on the upper half plane. In Sect. 2 (resp. Sect. 3) we fix notation for 0 functions (resp. Eisenstein series). The main results are in Sect. 4. They are exemplified by: The Rankin operators map the Dedekind t/ function to the Eisenstein series for the modular group. Similar results are proved for the level-2 congruence subgroup and for the 0-group. The proofs are ad hoc and uninteresting but raise the general question (which will require more insightful methods): For what discrete groups can one associate to each cusp a form like Dedekind's t/which will be mapped to the corresponding Eisenstein series by the Rankin operators? The referee has pointed out that the first theorem of this paper is equivalent to one of Cohen [1], and that the Bessel function expansion following it can also be found in Cohen's thesis and in Kuznecov [5]. Most of this paper is taken from the author 's 1974 Brandeis thesis, written with the invaluable help of A. L. Mayer, whom we now thank.

We construct differential operators which act on holomorphic functions defined on the Hermitian half space of degree n and are equivariant under the action of Sp(n,R), thereby increasing the weight by ν. For this, we construct and use as a tool the νth Rankin–Cohen bracket for holomorphic functions on the Hermitian half space of degree n, which is equivariant under the action of SU(n,n). The explicit combinatorial description of the differential operators is given for n=2, an arbitrary ν; and for an arbitrary n, ν=1. We also treat the vector-valued case, where a twisted automorphy factor shows up naturally for equivariance under the group U(n,n). Furthermore, our differential operators are unique upto scalars after restriction of variables to the Siegel half space.

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

A generating function for the Euler characteristic of out( F n)

We consider complex-valued modular forms on finite upper half planesHqand obtain Fourier expansions of Eisenstein series invariant under the groupsΓ=SL(2,Fp)andGL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincaré upper half plane —theK-Bessel functions being replaced by Kloosterman sums.

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus [Formula: see text] must be integral or half integral. Actually he proved that for a system [Formula: see text] of complex numbers of absolute value 1 [Formula: see text] can be an automorphy factor only if [Formula: see text] is integral. We give a different proof for this. It uses Mennicke’s result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for [Formula: see text] and [Formula: see text], Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre.

Limit values of Eisenstein series and multiple cotangent functions

3.1 Description of the Shimura differential operators 3.2 Nearly holomorphic Siegel modular forms 3.2.1 Algebraic nearly holomorphic Siegel modular forms 3.2.2 Formal expansions of nearly holomorphic forms. 3.2.3 Action of the U-operator. 3.3 Algebraic differential operators of Maass and Shimura 3.3.1 Differential operators on ℍ m . 3.3.2 The polynomial R m (z; r,β ). 3.3.3 Proof of Lemma 3.7 3.3.4 Action of the Shimura differential operator on formal Fourier expansions 3.3.5 Commutation of the Shimura operator with Hecke operators 3.4 Arithmetical variables of nearly holomorphic forms 3.4.1 Arithmetical nearly holomorphic Siegel modular forms 3.4.2 Action on Fourier expansion 3.4.3 Differentiation on monomials

The generic possibility of full surplus extraction in models with large type spaces

The Eisenstein series of heigher weight has at least one transcendental zero point on upper half plane.