Construction of Siegel cusp forms of genus two in terms of cusp forms of genus one

One proves that the convolution of a modular form of semiintegral weight and genus one with the theta-series of an indefinite quadratic form of signature (3.2) is a Siegel modular form of genus two of an even weight. One establishes the relationship between the zeta-functions of these modular forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 186–191, 1980.     

L-functions for Jacobi forms à la Hecke

Hecke's method to associate anL-function to a modular form by a Mellin transform is applied here to Jacobi forms. One comes up with a functional equation and some connection to Shimura's theory for modular forms of half integral weight.     

Zeta function of degree six for Hermite modular forms of genus two

One defines the zeta function Z_F(s) corresponding to Hermite modular forms [the groups SU(2, 2) over the imaginary quadratic field]. The degree of the local pfactors of the introduced zeta-function is equal to six. It is established that Z_F(s) coincides with the Dirichlet series constructed from the Fourier coefficients of the form F, for sufficiently large Re s, and it is proved that Z_F(s) can be continued analytically into the entire complex plane and satisfies a certain functional equation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 46–66.     

Generalized modular forms

The theory of “generalized modular forms,” initiated here, grows naturally out of questions inherent in rational conformal field theory. The latter physical theory studies q-series arising as trace functions (or partition functions), which generate a finite-dimensional SL(2,Z)-module. It is a natural step to investigate whether these q-series are in fact modular forms in the classical sense. As it turns out, the existence of the module does not, of itself, guarantee that this is so. Indeed, our Theorem 1 shows that such q-series of necessity behave like modular forms in every respect, with the important exception that the multiplier system need not be of absolute value one. The Supplement to Theorem 1 shows that such q-series are classical modular forms exactly when the scalars relating the q-series generators of the module have absolute value one. That is, the SL(2,Z)-module in question is unitary. (There is the further restriction that the associated representation is monomial.) We prove as well that there exist generalized modular forms which are not classical modular forms. (Hence, as asserted above, the q-series need not be classical modular forms.)Beyond Theorem 1 and its Supplement, which serve to relate our generalized modular forms to classical modular forms (and thus justify the name), this work develops a number of their fundamental properties. Among these are a basic result relating generalized modular forms to classical modular forms of weight 2 and so, as well, to abelian integrals. Further, we prove two general existence results and a complete characterization of weight k generalized modular forms in terms of generalized modular forms of weight 0 and classical modular forms of weight k.     

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.     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

Algebraic modular forms

In this paper, we develop an algebraic theory of modular forms, for connected, reductive groupsG overQ with the property that every arithmetic subgroup Γ ofG(Q) is finite. This theory includes our previous work [15] on semi-simple groupsG withG(R) compact, as well as the theory of algebraic Hecke characters for Serre tori [20]. The theory of algebraic modular forms leads to a workable theory of modular forms (modp), which we hope can be used to parameterize odd modular Galois representations. The theory developed here was inspired by a letter of Serre to Tate in 1987, which has appeared recently [21]. I want to thank Serre for sending me a copy of this letter, and for many helpful discussions on the topic.     

The Eichler-Shimura cohomology in the case of Siegel modular forms

One generalizes the Eichler-Shimura result, connecting the space of parabolic modular forms relative to a group with the cohomologies of the group , to the case of the Siegel modular forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 128–135, 1985.     

Differential operators for Hermitian Jacobi forms and Hermitian modular forms

Kim (Arch Math (Basel) 79(3):208–215, 2002) constructs multilinear differential operators for Hermitian Jacobi forms and Hermitian modular forms. However, her work relies on incorrect actions of differential operators on spaces of Hermitian Jacobi forms and Hermitian modular forms. In particular, her results are incorrect if the underlying field is the Gaussian number field. We consider more general spaces of Hermitian Jacobi forms and Hermitian modular forms over \(\mathbb {Q}(i)\), which allow us to correct the corresponding results in Kim (2002).     

Modular forms arising from divisor functions

For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its application.     

Secord-order cusp forms and mixed mock modular forms

In this paper, we consider the space of second order cusp forms. We determine that this space is precisely the same as a certain subspace of mixed mock modular forms. Based upon Poincaré series of Diamantis and O’Sullivan (Trans. Am. Math. Soc. 360:5629–5666, 2008) which span the space of second order cusp forms, we construct Poincaré series which span a natural (more general) subspace of mixed mock modular forms.     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation r: G? GL(n,\Bbb C)\rho: \Gamma \rightarrow GL(n,{\Bbb C}) of a discrete subgroup G ì SL(2,\Bbb C)\Gamma \subset SL(2,{\Bbb C}) in \Bbb Cⁿ{\Bbb C}^n and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.     

Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its k-th power form the space of (algebraic) Drinfeld modular forms of weight k. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.     

A property of the Fourier coefficients of Siegel cusp forms of genus two

Let F be a Siegel cusp modular form of genus two and of even weight k > 0. One proves that if the zeta-function Z_F(s) of the form F has a pole at the point s=k, then the Fourier coefficients a(N) of this form depend only on the determinant and the divisor of the matrix N.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 93, pp. 159–163, 1980.     

A simple proof of Zagier duality for Hilbert modular forms

In this paper, we give a simple proof of an identity between the Fourier coefficients of the weakly holomorphic modular forms of weight 0 arising from Borcherds products of Hilbert modular forms and those of the weakly holomorphic modular forms of weight satisfying a certain property.

     

Ramanujan and coefficients of meromorphic modular forms

The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.     

Local constancy of dimensions of Hecke eigenspaces of automorphic forms

We use a method of Buzzard to study p-adic families of Hilbert modular forms and modular forms over imaginary quadratic fields. In the case of Hilbert modular forms, we get local constancy of dimensions of spaces of fixed slope and varying weight. For imaginary quadratic fields we obtain bounds independent of the weight on the dimensions of such spaces.     

On the zeros of weakly holomorphic modular forms

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen (Comment Math Univ St Pauli 52:55–57, 2003). Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.     

Zero loci of differential modular forms

The theory of p-adic modular forms initiated by Serre, Dwork, and Katz (p-Adic Properties of Modular Schemes and Modular Forms, Lecture Notes in Mathematics, Vol. 350, Springer, Berlin, 1973) “lives” on the complement (in the p-adic completion of the appropriate modular curve) of the zero locus of the Eisenstein form E_p−1. On the other hand, most of the interesting phenomena in the theory of differential modular forms (J. Reine Angew. Math. (520) (2000) 95) take place on the complement of the zero locus of a fundamental differential modular form called f_jet. We establish that the zero locus of the reduction modulo p for p not congruent to one modulo 12 of the Eisenstein form E_p−1 is not contained in the zero locus of the reduction modulo p of the differential modular form f_jet implying that the theory of differential modular forms is applicable in certain situations not covered by the theory of p-adic modular forms.     

Vector-valued Jacobi-like forms

We introduce vector-valued Jacobi-like forms associated to a representation of a discrete subgroup in and establish a correspondence between such vector-valued Jacobi-like forms and sequences of vector-valued modular forms of different weights with respect to ρ. We determine a lifting of vector-valued modular forms to vector-valued Jacobi-like forms as well as a lifting of scalar-valued Jacobi-like forms to vector-valued Jacobi-like forms. We also construct Rankin-Cohen brackets for vector-valued modular forms.