A Laurent series expansion formula and its applications

This paper gives a certain Laurent series expansion for a generalized Rodrigues type formula. The main result finds many applications which are enumerated briefly.     

Kuznietsov trace formula and weighted distribution of Hecke eigenvalues

Let (X,μ) be a measurable topological space. Let S₁,S₂,… be a family of finite subsets of X. Suppose each x∈S_i has a weight w_ix∈R⁺ assigned to it. We say {S_i} is {w_i}-distributed with respect to the measure μ if for any continuous function f on X, we have .Let S(N,k) be the space of modular cusp forms over Γ₀(N) of weight k and let be a basis which consists of Hecke eigenforms. Let a_r(h) be the rth Fourier coefficient of h. Let x_p^h be the eigenvalue of h relative to the normalized Hecke operator T′_p. Let ||·|| be the Petersson norm on S(N,k). In this paper we will show that for any even integer k?3, is -distributed with respect to a polynomial times the Sato-Tate measure when N→∞.     

An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials

In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ₀(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once.     

A Frobenius formula for the characters of the Hecke algebras

Summary This paper uses the theory of quantum groups and the quantum Yang-Baxter equation as a guide in order to produce a method of computing the irreducible characters of the Hecke algebra. This approach is motivated by an observation of M. Jimbo giving a representation of the Hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group . By rewriting the solutions of the quantum Yang-Baxter equation for in a different form one can avoid the quantum group completely and produce a Frobenius formula for the characters of the Hecke algebra by elementary methods. Using this formula we derive a combinatorial rule for computing the irreducible characters of the Hecke algebra. This combinatorial rule is aq-extension of the Murnaghan-Nakayama for computing the irreducible characters of the symmetric group. Along the way one finds connections, apparently unexplored, between the irreducible characters of the Hecke algebra and Hall-Littlewood symmetric functions and Kronecker products of symmetric groups.Work partially supported by an NSF grant at the University of California, San Diego     

A mixed hook-length formula for affine Hecke algebras

Let be the affine Hecke algebra corresponding to the group GL_l over a p-adic field with residue field of cardinality q. We will regard as an associative algebra over the field . Consider the -module W induced from the tensor product of the evaluation modules over the algebras and . The module W depends on two partitions λ of l and μ of m, and on two non-zero elements of the field . There is a canonical operator J acting on W; it corresponds to the trigonometric R-matrix. The algebra contains the finite dimensional Hecke algebra H_l+m as a subalgebra, and the operator J commutes with the action of this subalgebra on W. Under this action, W decomposes into irreducible subspaces according to the Littlewood–Richardson rule. We compute the eigenvalues of J, corresponding to certain multiplicity-free irreducible components of W. In particular, we give a formula for the ratio of two eigenvalues of J, corresponding to the “highest” and the “lowest” components. As an application, we derive the well known q-analogue of the hook-length formula for the number of standard tableaux of shape λ.     

A formula for the central value of certain Hecke L-functions

Let be the negative of a prime, and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to . Under these assumptions, there exists Hecke characters ψ_D of K with conductor (D) and infinite type (1,0). Their L-series L(ψ_D,s) are associated to a CM elliptic curve A(N,D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_D,s) of the form L(ψ_D,1)=Ω∑_[A],Ir(D,[A],I)m_[A],I([D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.     

A scalar product for copulas

We introduce a scalar product for n-dimensional copulas, based on the Sobolev scalar product for W1,2-functions. The corresponding norm has quite remarkable properties and provides a new, geometric framework for copulas. We show that, in the bivariate case, it measures invertibility properties of copulas with respect to the ∗-operation introduced by Darsow et al. (1992). The unique copula of minimal norm is the null element for the ∗-operation, whereas the copulas of maximal norm are precisely the invertible elements.     

Hypergeometric series with gamma product formula

We consider non-terminating Gauss hypergeometric series with one free parameter. Using various properties of hypergeometric functions we obtain some necessary conditions of arithmetic flavor for such series to admit gamma product formulas.     

A recursion formula for resistance distances and its applications

A recursion formula for resistance distances is obtained, and some of its applications are illustrated.     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

A trace formula for rigid varieties,and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When is the formal completion of f at a closed point x of the special fiber , we obtain the local motivic zeta function of f at x. The research for this article was partially supported by ANR-06-BLAN-0183.     

On a formula of Hecke

Hecke’s version of Kronecker’s limit formula for an algebraic number field is generalized using adelic analysis.     

A trace formula

To my Father, William F. Eberlein.     

A Fourier series formula for energy of measures with applications to Riesz products

In this paper we derive a formula relating the energy and the Fourier transform of a finite measure on the -dimensional torus which is similar to the well-known formula for measures on .

We apply the formula to obtain estimates on the Hausdorff dimension of Riesz product measures. These give improvements on the earlier, classical results which were based on completely different techniques.

     

A trace formula for Weyl transforms

A formula for the trace of a trace class Weyl transform associated to a symbol in L¹(R²ⁿ) is given.