Finite Weil representations and associated Hecke algebras

An algebra H(Gm) of double cosets is constructed for every finite Weil representation Gm. For the Clifford-Weil groups Gm=Cm(ρ) associated to some classical Type ρ of selfdual codes over a finite field, this algebra is shown to be commutative. Then the eigenspace decomposition of H(Cm(ρ)) acting on the space of degree N invariants of Cm(ρ) may be obtained from the kernels of powers of the coding theory analogue of the Siegel Φ-operator.     

Hecke operators on Drinfeld cusp forms

In this paper, we study the Drinfeld cusp forms for Γ₁(T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ₁(T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for Γ₁(T) of large weights, and not for Γ(T) even of small weights. The Hecke eigenvalues on cusp forms for Γ(T) with small weights are determined and the eigenspaces characterized.     

Rationality theorems for Hecke operators on GLn

We define n families of Hecke operators for GL_n whose generating series are rational functions of the form q_k(u)⁻¹ where q_k is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially .By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL₃ and GL₄.     

Lattice polytopes,Hecke operators,and the Ehrhart polynomial

Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.      

On a new circular summation of theta functions

In this paper, we prove a new formula for circular summation of theta functions, which greatly extends Ramanujan's circular summation of theta functions and a very recent result of Zeng. Some applications of this circular summation formula are given. Also, an imaginary transformation for multiple theta functions is derived.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.     

Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.Received: 10 September 2004     

Sixth order mock theta functions

Two new mock theta functions of the sixth order are defined. The main theorem in this paper (Theorem 1.1) provides four transformation formulas relating the new mock theta functions with Ramanujan's mock theta functions of the sixth order. Two further representations of the new mock theta functions are established. Lastly, a hitherto unproved entry from Ramanujan's lost notebook related to sixth order mock theta functions is proved.     

A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

Twisted Hecke L-values and period polynomials

Let f₁,…,fd be an orthogonal basis for the space of cusp forms of even weight 2k on Γ₀(N). Let L(fi,s) and L(fi,χ,s) denote the L-function of fi and its twist by a Dirichlet character χ, respectively. In this note, we obtain a “trace formula” for the values at integers m and n with 0<m,n<2k and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisely the value of the ratio L(f,χ,m)/L(f,n) for a Hecke eigenform f.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

A Hecke correspondence theorem for automorphic integrals with symmetric rational period functions on the Hecke groups

Marvin Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or ∞. For other automorphic integrals the corresponding Dirichlet series has a functional equation with a remainder term that arises from the nonzero poles of the rational period function. In this paper we prove a Hecke correspondence theorem for a class of automorphic integrals with rational period functions on the Hecke groups. We restrict our attention to automorphic integrals of weight that is twice an odd integer and to rational period functions that satisfy a symmetry property we call “Hecke-symmetry.” Each remainder term satisfies two relations (the second of which is new in this paper) corresponding to the two relations for the rational period function.     

An explicit formula of the Koecher-Maass series associated with Hermitian modular forms belonging to the Maass space

We give an explicit form of the Koecher-Maass series for Hermitian modular forms belonging to the Maass space. We express the Koecher-Maass series as a finite sum of products of two L-functions associated with automorphic forms of one variable. In particular the Koecher-Maass series associated with the Hermitian-Eisenstein series of degree two can be described by a finite sum of products of four shifted Dirichlet L-functions associated with some quadratic characters under the assumption that the class number of imaginary quadratic fields is one.     

关于次可分解算子的不变子空间

In this paper,we prove the Mohebi-Radjabalipour Conjecture under an ad-ditional condition,and obtain an invariant subspace theorem on subdecomposableoperators.     

An addition formula for the Jacobian theta function and its applications

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.     

On a dual pair of spaces of smooth and generalized random variables

A dual pairG andG ^* of smooth and generalized random variables, respectively, over the white noise probability space is studied.G is constructed by norms involving exponentials of the Ornstein-Uhlenbeck operator,G ^* is its dual. Sufficient criteria are proved for when a function onL(ℝ) is theL-transform of an element inG orG ^*.     

On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL ²[0, 1] is equal to {B(a):a[0, 1]}, whereB(a)={fL ²[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofL ^p -spaces. Our main result characterizesL-closed subspaces of a Banach function spaceL that are invariant underV, whereL denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.This work was supported by the Research Ministry of Slovenia.     

On certain properties of the spaces of order-preserving functionals

In the present paper it is proved that the functor Oτ of τ-smooth order preserving functionals and the functor OR of Radon order preserving functionals, do not change the weight of infinite Tychonoff spaces. It is shown that the density and the weak density of infinite Tychonoff spaces do not increase under these functors. Moreover, if X is a metric space with the second axiom of countability then the spaces Oτ(X) and OR(X) are also metrizable.