Multiplikatorsysteme der symplektischen Thetagruppe

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

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

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).

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

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.

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].

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

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 classical generalizations (already investigated in the second half of last century) of the modular group SL(2, ℤ) are the groups ГK= SL(2,o)(othe principal order of a totally real number fieldK, [K:ℚ]=n), operating, originally, on a productofnupper half-planes or, forn=2, on the product1×−of an upper and a lower half-plane by(wherev(i), forv∈K, denotes the jth conjugate ofv), and Гn= Sp(n, ℤ), operating onn={Z∣Z=X+iY∈ℂ(n,n),tZ=Z, Y>0} byNowadays ГKis called Hilbert's modular group ofKand ГnSiegel's modular group of degree (or genus)n.Forn=1 we have Гℚ=Г1= SL(2, ℤ). The functions corresponding to modular forms and modular functions for SL(2, ℤ) and its subgroups are holomorphic (or meromorphic) functions with an invariance property of the formJ(L, t)for fixedL(orJ(M, Z)for fixedM) denoting a holomorphic function without zeros on) (or onn). A functionJ;, defined on ℤK×or ℤn×nto be able to appear in (1.3) withf≢0, has to satisfy certain functional equations (see below, (2.3)–(2.5) for ГK, (5.7)–(5.9) for Гn) and is called an automorphic factor (AF) then. In close analogy to the casen=1, mainly AFs of the following kind have been used:with a complex numberr, the weight ofJ, and complex numbersv(L), v(M). AFs of this kind are called classical automorphic factors (CAP) in the sequel. Ifr∉ℤ, the values of the functionvon ГK(or Гn) depend on the branch of (…)r. For a fixed choice of the branch (for eachL∈ГKorM∈Гn) the functional equations forJ, by (1.4), (1.5), correspond to functional equations forv.A functionvsatisfying those equations is called a multiplier system (MS) of weightrfor ГK(or Гn).

In doing p-adic analysis on spaces of classical modular functions and forms, it is convenient and traditional to broaden the notion of “modular form” to a class called “overconvergent p-adic modular forms.” Critical for the analysis of the p-adic Banach spaces composed of this wider class of forms is the “Atkin U -operator”, which is completely continuous and whose spectral theory (still not very well understood) seems to be the key to a good deal of arithmetic. The part of the spectrum of U corresponding to eigenvalues which are p-adic units is somewhat more understood, thanks to the work of Hida. As for the rest of the spectrum, it is surprising how fragmentary our information is (although recent work of Coleman, resolving in part some prior conjectures of ours, has improved the situation). We have begun an experimental search for nonclassical (but “overconvergent”) eigenfunctions in a fairly simple way. We take a number of classical modular functions which are p-adically overconvergent (e.g., j, 1/j,...), and try to find their p-adic “U -eigenfunction expansion.” There is a straightforward computational procedure to approximate such eigenfunction expansions, even though, on a theoretical level, we do not even know that the “expansions” that our algorithm produces converge in any sense, or even settle down numerically. Experimentally, they seem to, and they produce candidate Fourier expansions. In our computations, we specialize principally to p = 5. The same eigenfunctions (produced by our algorithm) seem to occur as terms in each of the eigenfunction expansions we have calculated. This leads us to suspect that we have in fact encountered all the 5-adic eigenfunctions

Representations by some quinary quadratic forms of level 8

Dedekind symbols generalize the classical Dedekind sums (symbols). These symbols are determined uniquely, up to additive constants, by their reciprocity laws. For k ≧ 2, there is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws of degree 2k − 2 and the space of modular forms of weight 2k for the full modular group \(PSL_2 (\mathbb{Z}).\) However, this is not the case when k = 1 as there is no modular form of weight two; nevertheless, there exists a unique (up to a scalar multiple) quasi-modular form (Eisenstein series) of weight two. The purpose of this note is to define the Dedekind symbol associated with this quasi-modular form, and to prove its reciprocity law. Furthermore we show that the odd part of this Dedekind symbol is nothing but a scalar multiple of the classical Dedekind sum. This gives yet another proof of the reciprocity law for the classical Dedekind sum in terms of the quasi-modular form.

Class fields and modular forms of weight one

Meromorphic scaling flow of [formula omitted] supersymmetric [formula omitted] Yang–Mills with matter

Modular forms of weight 1/2