Hecke Theory Research Articles

The Bogomolny–Carioli Twisted Transfer Operators and the Bogomolny–Gauss Mapping Class Group

The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized PSL (2,Z) group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized PSL (2,Z) domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy.

Hecke theory for SO+(2,n + 2)

We describe the foundations of a Hecke theory for the orthogonal group SO+(2,n+2). In particular we consider the Hermitian modular group of degree 2 as a special example of SO+(2,4). As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients.

Paramodular forms of level 8 and weights 10 and 12

We study degree 2 paramodular eigenforms of level 8 and weights 10 and 12, and determine all their local representations. We prove dimensions by the technique of Jacobi restriction. A level divisible by a cube permits a wide variety of local representations, but also complicates the Hecke theory by involving Fourier expansions at more than one zero-dimensional cusp. We overcome this difficulty by the technique of restriction to modular curves. An application of our determination of the local representations is that we obtain the Euler 2-factor of each newform.

Lattices in the cohomology of U(3) arithmetic manifolds

Under hypotheses required for the Taylor–Wiles method, we prove for forms of U(3) which are compact at infinity that the lattice structure on upper alcove algebraic vectors or on principal series types given by the $$\lambda $$ -isotypic part of completed cohomology is a local invariant of the Galois representation attached to $$\lambda $$ when this Galois representation is residually irreducible locally at places dividing p. As a crucial input, we establish corresponding mod p multiplicity one results. Our main innovation is the combination of integral Hecke theory and the Taylor–Wiles method.

The spaces of new forms for harmonic weak Maalss forms and their level raising properties modulo $$\ell $$ ℓ

Bringmann et al. (Math. Ann., 355(3):1085–1121, 2013) and Guerzhoy (Proc. Am. Math. Soc., 136:3051–3059, 2008) studied the multiplicity two Hecke theory for weakly holomorphic modular forms on the full modular group. In this note, we consider a generalization of these results to congruence subgroups \(\Gamma _1(N)\) with a new form theory. Using this, we define new forms for harmonic weak Maass forms and study a level raising property modulo \(\ell \) for them.

Rédei's Triple Symbols and Modular Forms

In 1939, L. Rédei introduced a certain triple symbol in order to generalize the Legendre symbol and Gauss' genus theory. Rédei's triple symbol $[a_1,a_2, p]$ describes the decomposition law of a prime number $p$ in a certain dihedral extension over $\mathbb{Q}$ of degree 8 determined by $a_1$ and $a_2$. In this paper, we show that the triple symbol $[-p_1,p_2, p_3]$ for certain prime numbers $p_1, p_2$ and $p_3$ can be expressed as a Fourier coefficient of a modular form of weight one. For this, we employ Hecke's theory on theta series associated to binary quadratic forms and realize an explicit version of the theorem by Weil-Langlands and Deligne-Serre for Rédei's dihedral extensions. A reciprocity law for the Rédei triple symbols yields certain reciprocal relations among Fourier coefficients.

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular \(L\)-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on \(M_{k}^{!}\) in terms of periods.

A remark on solutions of functional equations of Riemann's type

It is proved that a general functional equation of the Riemann type with multiple gamma factors has non-trivial solutions in the space of generalized Dirichlet series. Moreover, for a fixed functional equation, the space of such solutions has uncountable basis. The proof is based on Hecke's theory of Dirichlet series associated with modular forms for the groups $G(\lambda)$. This is in constrast with the situation in the extended Selberg class where there exist functional equations without non-trivial solutions. Presumably this holds for non-integer degrees $d$, but up to date was confirmed only for $0 \leqslant d \lt 5/3$. In the case of $d = 0$ or $d = 1$ the space of solutions belonging to the extended Selberg class, through non-trivial, is finite dimensional.

Hecke theory of series formed with modular symbols and relations among convolution L -functions

In [G1], [G2], the distribution of modular symbols is studied and a new class of functions which satisfy a transformation law involving these objects is introduced. The goal of Goldfeld’s program is to prove Szpiro’s conjecture which states that for elliptic curves with minimal discriminant D and conductor N there is an absolute constant κ such that D ? N. To do this, an equivalent conjecture involving modular symbols is established in [G4] using the, now proven, conjecture of Shimura, Taniyama and Weil. It seems that a sufficiently good understanding of the new series proposed by Goldfeld should yield a resolution of these conjectures. We repeat here their definition in a somewhat more general form to include period polynomials rather than modular symbols only. For positive integers M,N such that M |N, let m, k be non-negative integers such that m ≥ k − 2 ≥ 0 and let χ be a Dirichlet character modulo N. First, for each f ∈ Sk(M) = {holomorphic cusp forms of weight k and level M} we denote by rf the map which sends γ ∈ Γ0(M), the Hecke congruence group of level M , to the polynomial function:

Hecke theory and equidistribution for the quantization of linear maps of the torus

We study semi-classical limits of eigenfunctions of a quantized linear hyperbolic automorphism of the torus ("cat map"). For some values of Planck's constant, the spectrum of the quantized map has large degeneracies. Our first goal in this paper is to show that these degeneracies are coupled to the existence of quantum symmetries. There is a commutative group of unitary operators on the state-space which commute with the quantized map and therefore act on its eigenspaces. We call these "Hecke operators", in analogy with the setting of the modular surface. We call the eigenstates of both the quantized map and of all the Hecke operators "Hecke eigenfunctions". Our second goal is to study the semiclassical limit of the Hecke eigenfunctions. We will show that they become equidistributed with respect to Liouville measure, that is the expectation values of quantum observables in these eigenstates converge to the classical phase-space average of the observable.

Jacobi-Hecke algebras and a rationality theorem for a formal Hecke series

It is well known that the L-function associated to a Siegel eigenform f is equal to a Rankin-Selberg type zeta-integral involving f and a restricted Eisenstein series ([3], [14]). At some point in the proof one has to show the equality of a certain Dirichlet series and the L-function, which follows from a rationality theorem for a certain formal power series over the Hecke algebra. The main purpose of this paper is to develop a Hecke theory for the Jacobi group and to prove such a rationality theorem.

Equivalence of factorizability theories

A framework is given for defining and comparing factorizability theories (as used, for example, to relate modules over the group rings of finite groups). The strict theory of Fröhlich [4] and the Hecke theory of [7] as well as several other related theories are shown to be equivalent. Of particular interest among the several equivalent theories examined is that of normaliser factorizability. This is perhaps the most user friendly theory and follows most closely the framework of the classical theory. Among other tools, the theory of permutation projectives and their species is used.

Hecke theory over arbitrary number fields

In 1954, Hermann [6] developed an explicit Hecke theory over totally real fields by using ideal numbers. For a field of class number h, he introduced modular forms which were actually vectors of h different functions. Recently, persons have studied modular forms over totally complex fields (Goldfeld et al. [2]) or over arbitrary number fields (Stark [8]) with h = 1. When h > 1, however, complications arise. This paper will develop the tools of Hecke theory for study of modular forms over arbitrary number fields when h> 1. Section 2 introduces ideal numbers following Hecke [S]. Section 3 defines the appropriate upper half space and the action of matrix operators. In Section 4, we define the notion of a vector modular form, with the crucial point being the introduction of h-length vectors, with h-length vectors of matrix operators. These ideas follow Hermann [6]. Section 5 describes Hecke operators, and their effect on the Fourier expansion. Section 6 outlines the theory of Dirichlet series, their Euler products, and functional equations. In Section 7, we describe the Petersson inner product and prove the self adjointness of the Hecke operators. Section 8 contains the main theorem. Intuitively, one might suspect that ideal numbers are unnecessary-the principal elements should suffice. In Section 8, we show the principal component of a vector modular form and its action under “principal” Hecke operators do determine the entire form. Throughout the paper, we illustrate the general theory with examples of Eisenstein series over Q(a). These concrete examples demonstrate the value of this approach to obtain results analogous to the classical Eisenstein series. It is well known that an adelic version of Hecke theory exists over any number field. For instance, Weil outlines an adelic approach, saying “technical difficulties arose when the number of ideal classes is greater than

Eisenstein series over complex quadratic fields when the class number exceeds one

INTRODUCTION In 1954, Hermann [IS] developed an explicit Hecke theory over totally real fields by using the ideal numbers of Hecke. For a field of class number h, he introduced modular forms which were actually vectors of Ir different functions. The author [7] developed the analogous theory for complex fields with h > 1. In 1982, Goldfeld et

Lattices with theta functions for G(√2) and linear codes

Modular hermitian lattices over Z [ i]and, in particular, unimodular lattices over Z [ e πi 4 ] give rise to modular forms for Hecke's group G( 2 )=<( 1 2 0 1 ), ( 0 1 −1 0 )> .Two general constructions of such lattices are performed, using codes over F 2 and F 9. Lattices with an extremal theta-function (i.e., with the largest minimum that Hecke's theory allows) are obtained in C 2n for all n < 12, including the densest known sphere-packings of R 4n for n = 1, 4, and 8.

Vector valued Siegel's modular forms of degree two and the associated Andrianov L-functions

In [1], [2], Andrianov constructed a remarkable Hecke theory for Siegel's modular forms of degree two. In this article we extend some of his results to the case of vector valued Siegel's modular forms of degree two.

Whittaker models for real groups

Introduction. Whittaker functions were first introduced for the principal series representations of Chevalley groups by H. Jacquet [3]. Later, they were pursued by G. Schiffmann for algebraic groups of real rank one [12]. They played a very important role in the development of the Hecke theory for GLn through the work of H. Jacquet, R. P. Langlands, I. I. Piatetski-Shapiro, and J. A. Shalika [4, 5]. More precisely, they were the main tools for the definitions of local and global L-functions and e-factors. They also appeared quite useful in the development of the Hecke theory for other groups (cf. [10]), as well as in the definition of the local y-factors of certain functional equations [13, 14], particularly in their factorization. There seems to be other evidence of interest, especially in the work of W. Casselman, B. Kostant [7], and G. Zuckerman. The analytic behavior of these functions is much simpler when the ground field is non-archimedean; a good account of their analytic properties and some interesting formulas for certain class of such functions may be found in a recent paper of W. Casselman and J. A. Shalika [2]. But when the ground field is archimedean, these functions were believed to behave in a rather complicated manner. In fact, this has been one of the main obstacles in the development of the Hecke theory for number fields. To make a more precise statement of the problem, we let G be a split reductive algebraic group over R. We fix a maximal torus T of G and we let B be a fixed Borel subgroup of G containing T. We write B = M0AU, the Langlands decomposition of B with T=M0A, and fix a non-degenerate (unitary) character x of U (see section 1). Now, let IT be a continuous representation of G on a Frechet space V. Denote by ( oo > ^oo) e corresponding differentiable representation. Topologize V^ with the relative topology inherited from C°°(G, V). Let VK be the subspace of ^-finite vectors of F, where K is a fixed maximal compact subgroup of G with G = KB. We say that the representation (7r, V) is non-degenerate, if there exists a continuous linear functional A on V^, called a Whittaker functional, such that

On the ζ-Functions of a Division Algebra

In this paper, we will develop a theory of C-functions with characters in a division algebra. The ordinary C-function of a division algebra was introduced by K. Hey [4], and generalized by M. Eichler [1] to L-functions with abelian characters. The first attempt to generalize these theories to C-functions with non-abelian characters is due to H. Maass [5]. Later, R. Godement [2] gave a method to get the most general formulations on these matters. In this note, we will define a type of C-functions of a division algebra over an algebraic number field which are included in Godement's work as a special case, and for which one can develop the theory of Euler products. The latter theory has its own meaning as an application of the theory of spherical functions on p-adic algebraic groups. Here we have a generalization of Hecke's theory of so-called Heckeoperators. (Theorem 1-6). One can expect that there exist similar theories for other simple algebraic groups defined over p-adic number fields, and that there will be applications of these theories to non-commutative number theory. The author wishes to express his thanks to Professor Shimura for some valuable suggestions about the first part of this paper.