Modular forms of half integral weight,some explicit arithmetic

Mathematische Annalen -     

Modular forms

In this survey there are included results of recent years, concerning the theory of modular forms and representations connected with them of adele groups and Galois groups. There is discussed the hypothetical principle of functoriality of automorphic forms and other conjectures of Langlands concerning automorphic forms and the L-functions connected with them.     

Modular lattices and Hermitean forms

We give some application of the lattice method in the theory of infinite dimensional quadratic spaces over arbitrary division rings.Presented by Ralph Freese     

Modular forms and group representation

This paper considers the restriction of adjoint representations of simple Lie groups whose algebras are of even rank to finite subgroups in which each element has a rational characteristic polynomial. The nature of the virtual characters that appear is explained.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 25–31, July, 1992.     

Modular forms and ellipsoidal T-designs

The Ramanujan Journal - In recent work, Miezaki introduced the notion of a spherical T-design in $$mathbb {R}^2$$ , where T is a potentially infinite set. As an example, he offered the $$mathbb...     

Modular forms and differential operators

In 1956, Rankin described which polynomials in the derivatives of modular forms are again modular forms, and in 1977, H Cohen defined for eachn ≥ 0 a bilinear operation which assigns to two modular formsf andg of weightk andl a modular form [f, g]_n of weightk +l + 2n. In the present paper we study these “Rankin-Cohen brackets” from two points of view. On the one hand we give various explanations of their modularity and various algebraic relations among them by relating the modular form theory to the theories of theta series, of Jacobi forms, and of pseudodifferential operators. In a different direction, we study the abstract algebraic structure (“RC algebra”) consisting of a graded vector space together with a collection of bilinear operations [,]_n of degree + 2n satisfying all of the axioms of the Rankin-Cohen brackets. Under certain hypotheses, these turn out to be equivalent to commutative graded algebras together with a derivationS of degree 2 and an element Φ of degree 4, up to the equivalence relation (∂,Φ) ~ (∂ - ϕE, Φ - ϕ² + ∂(ϕ)) where ϕ is an element of degree 2 andE is the Fuler operator (= multiplication by the degree). Dedicated to the memory of Professor K G Ramanathan     

Modular forms and Eisenstein's continued fractions

Found in the collected works of Eisenstein are twenty continued fraction expansions. The expansions have since emerged in the literature in various forms, although a complete historical account and self-contained treatment has not been given. We provide one here, motivated by the fact that these expansions give continued fraction expansions for modular forms. Eisenstein himself did not record proofs for his expansions, and we employ only standard methods in the proofs provided here. Our methods illustrate the exact recurrence relations from which the expansions arise, and also methods likely similar to those originally used by Eisenstein to derive them.     

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.     

Modular forms and special cubic fourfolds

We study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ADE singularities. In this paper, we show that the generating series of the degrees of such divisors is a level three modular form.     

Modular forms and effective Diophantine approximation

After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey?s height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey?s height conjecture which applies to all elliptic curves over $\mathbb{Q}$, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of ${\mathbb{P}}^1?\{0,1,\infty \}$), with a completely explicit and effective bound, without using any variant of Baker?s theory or the Thue–Bombieri method.     

Modular forms and representations of symmetric groups

We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of the symmetric group S_n over a field of characteristic p for n . For p < 7 we give simple explicit formulas for the number of blocks of defect zero. The study of the modular forms leads to interesting identities involving the Dedekind -function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 74–85, 1982.     

Automorphically rigid group algebras. II. Modular algebras

We define classes of automorphically rigid modular group algebras KG over a finite field K and a finite group G.     

Modular forms arising from spherical polynomials and positive definite quadratic forms

The dimension over the complex numbers of the vector space of θ-series

$\text{Φ(τ;P,Q)=∑n?Z}{}^{\mathrm{r}}\text{P(n)e}{}^{\mathrm{2\pi iQ(n)\tau }}\text{,}$

where P(X) is a “spherical polynomial” with respect to the positive definite quadratic form Q(X) is computed for binary forms. A sufficient condition that θ(τ; P, Q) be identically zero is proved, and its necessity is also proved for binary forms.