Representations by some quinary quadratic forms of level 8

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.

Representations of loop groups, Dirac operators on loop space, and modular forms

Introduction. Let E/F be a totally real quadratic extension of a totally real algebraic number field. To a suitably defined automorphic form h defined with respect to a quaternion algebra BE over E, we can associate a Hilbert modular form I(z, h) defined with respect to the field F . Such a lifting has been explicitly considered in the author’s recent paper [D99], via a convolution with a theta function, and the Fourier coefficients of the theta lift have been computed in terms of certain periods of the original form. The purpose of the current paper is to establish an explicit formula relating the actions of the Hecke operators on the original automorphic form and its theta lift. This result is analogous to theorems of Shimura in, for example, [Sh82] and [Sh88], where he has shown commutativity properties with respect to these Hecke operators. In addition to their inherent interest, these commutativity results are necessary in obtaining algebraicity results concerning Shimura’s period invariants as well as certain special L-values which naturally arise. The reader can consult Shimura’s [Sh88] for further motivation in this regard. See also [Y]. It is the author’s hope to discuss applications in this direction in the near future. Since it is essential for our purposes to consider everything explicitly, the quantity of technical details is considerable. In the interest of space, and also in order not to obscure essential ideas, a large amount of computation has been suppressed. Instead we have endeavored to point out key ingredients in the computations and to give precise references. It is then hoped that

The basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series, in order that this question makes sense one has of course to specify which type of theta series should be taken into account. We want here to follow (in a special case) the approach taken e.g. in [B61, W a l l which (in its general form) asks" Which (cuspidal) modular forms F of degree n and weight k for the group FoC")(N) can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m = 2 k and the same level N? Moreover, one wants to derive an explicit expression for such a linear combination in terms of the Petersson inner products of F with the theta series involved. The representation theoretic version of this question is to ask: Which (cuspidat) automorphic representations of the adelic symplectic (or metaplectic) group

Finding elementary formulas for theta functions associated to even sums of squares

Let Γθ be the subgroup of Siegel modular groupSp(n, ℤ) consisting of all matrices\(M = \left( {\begin{array}{*{20}c} {A B} \\ {C D} \\ \end{array} } \right)\), such that the diagonal elements ofAtC andBtD are even. A multiplier system of weightr(∈ℂ) is a system of complex numbers ν (M)≠0,M∈Γθ, such thatJ (M, Z)=ν(M) det(CZ+D)r is an automorphy factor (that isJ (M N, Z)=J (M, N Z) J (N, Z) forM, N ∈Sp(n,ℤ) and $$Z \in S_n = \left\{ {Z = X + i Y \in M^{(n,n)} (\mathbb{C}); X = X^t , Y = Y^t > 0} \right\})$$ . We show that in casen≥2 such a multiplier system exists if and only if 2r∈ℤ. A corollary of this fact is the following. From the cohomology theory of Siegel modular group we derive that in casen≥8 any Γθ-invariant divisor is the exact zero divisor of a modular form for Γθ. Therefore the zero divisor of classical theta function\(\theta (Z) = \sum\limits_{g \in \mathbb{Z}^n } {e^{\pi iZ[g]} } \), a modular form of weight 1/2 is irreducible. In the second part of this paper we calculate the commutator factor group of Γn, θ forn≥2.

We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.

Entire modular forms of weights and for the congruence group Γ0(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.

The Apery-like numbers J2(n) associated to the special value ζQ(2) of the spectral zeta function ζQ(s) for the non-commutative harmonic oscillator Q have remarkable modular form interpretation. In fact, we show that the differential equation satisfied by the generating function w2(t) of J2(n) is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4-torsion. The parameter t of this family can be regarded as a modular function for the congruent subgroup Γ0(4). Further, we see that the function w2(t) is regarded as a Γ0(4)-modular form of weight 1 in the variable τ by taking t as the classical Legendre modular function λ(τ).

An elliptic modular form of positive integral weight k is a linear combination of a cusp form and Eisenstein series. This well-known fact was proved by Hecke and was generalized by Kloostermann to the Hilbert modular case (Hecke [5] and Kloostermann [8]). The case k = 1, which was left unsettled in [-8], was solved by Gundlach under certain conditions; a Hilbert modular form for the full modular group over a real quadratic field is a linear combination of a cusp form and Eisenstein series (Gundlach [3, 4]). The aim of the present paper is to show that the same holds in general for any principal congruence subgroup over a totally real number field. As one expects by [3], the number of linearly independent Eisenstein series of weight 1 is less than (roughly equal to one half of) the number of cusps. It is enough to prove that the number of linearly independent non-cuspidal forms equals that of linearly independent Eisenstein series. To do this, we shall apply the theory of automorphic representations (the equivalence condition, the uniqueness of Whittaker models, etc .... ) as is developed in Jacquet and Langlands [6].

Recently, Harvey and Wu proposed a suitable Hecke operator for vector-valued SL(2, ℤ) modular forms to connect the characters of different 2d rational conformal field theories (RCFTs). We generalize such an operator to the 2d fermionic RCFTs and call it fermionic Hecke operator. The new Hecke operator naturally maps the Neveu-Schwarz (NS) characters of a fermionic theory to the NS characters of another fermionic theory. Mathematically, it is the natural Hecke operator on vector-valued Γθ modular forms of weight zero. We find it can also be extended to NS~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overset{\\sim }{\ extrm{NS}} $$\\end{document} and Ramond (R) sectors by combining the characters of the two sectors together. We systematically study the fermionic Hecke relations among 2d fermionic RCFTs with up to five NS characters and find that almost all known supersymmetric RCFTs can be realized as fermionic Hecke images of some simple theories such as supersymmetric minimal models. We also study the coset relations between fermionic Hecke images with respect to c = 12k holomorphic SCFTs.

In the previous chapter we examined representations of Galois groups of global fields into GL(n, ℂ), and their L-functions. Such representations are necessarily of finite image. We saw at the end that an important example is supplied by modular forms of weight 1.

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construction of infinite sequences of eigenfunctions for all regular Hecke operators on spaces of Maass theta series; in the case of Maass theta series of genus $2$, all the eigenfunctions are constructed and the corresponding Andrianov zeta functions are expressed in the form of products of two $L$-functions of the relevant imaginary quadratic rings. The proofs are based on a combination of explicit formulas for the action of Hecke operators on theta series with Gauss composition of binary quadratic forms.

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.