Algebra Of Modular Forms Research Articles

On some free algebras of orthogonal modular forms

For 25 orthogonal groups of signature (2,n) related to the root lattices A1, 2A1, 3A1, 4A1, A2, A3, A4, A5, A6, A7, D4, D5, D6, D7, D8, E6, E7, we prove that the algebras of modular forms on symmetric domains of type IV are freely generated by the additive lifts of some special Jacobi forms. The proof is universal and elementary.

A simple method to extract zeros of certain Eisenstein series of small level

This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level N where $$N=2,3,5$$ and 7. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on SL(2, Z) and their zeros are ‘controlled’ by those of $$E_4$$ and $$E_6$$ in the fundamental domain of $$\Gamma _0(N)$$ .

Ramanujan-type partial theta identities and conjugate Bailey pairs, II. Multisums

In the first paper of this series, we described how to find conjugate Bailey pairs from residual identities of Ramanujan-type partial theta identities. Here we carry this out for four multisum residual identities of Warnaar and two more due to the authors. Applying known Bailey pairs gives expressions in the algebra of modular forms and indefinite theta functions.

Generators of graded rings of modular forms

We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A of C, and specifically the highest weight needed to generate these rings as A-algebras. In particular, we determine upper bounds, independent of N, for the highest needed weight that generates the C-algebras of modular forms over Γ1(N) and Γ0(N) with some conditions on N. For N⩾5, we prove that the Z[1/N]-algebra of modular forms over Γ1(N) with coefficients in Z[1/N] is generated in weight at most 3. We give an algorithm that computes the generators, and supply some computations that allow us to state two conjectures concerning the situation over Γ0(N).

Triangle groups, automorphic forms, and torus knots

This article is concerned with the relation between several classical and well-known objects: triangle Fuchsian groups, \mathbb{C}^\times -equivariant singularities of plane curves, torus knot complements in the 3-sphere. The prototypical example is the modular group PSL_2(\mathbb Z) : the quotient of the nonzero tangent bundle on the upper-half plane by the action of PSL_2(\mathbb Z) is biholomorphic to the complement of the plane curve z^3-27w^2=0 . This can be shown using the fact that the algebra of modular forms is doubly generated, by g_2,g_3 , and the cusp form \Delta=g_2^3-27g_3^2 does not vanish on the half-plane. As a byproduct, one finds a diffeomorphism between PSL_2(\mathbb R)/PSL_2(\mathbb Z) and the complement of the trefoil knot - the local knot of the singular curve. This construction is generalized to include all (p,q,\infty) -triangle groups and, respectively, curves of the form z^q+w^p=0 and (p,q) -torus knots, for p,q co-prime. The general case requires the use of automorphic forms on the simply connected group \widetilde{SL_2}(\mathbb R) . The proof uses ideas of Milnor and Dolgachev, which they introduced in their studies of the spectra of the algebras of automorphic forms of cocompact triangle groups (and, more generally, uniform lattices). It turns out that the same approach, with some modifications, allows one to handle the cuspidal case.

On the algebra of modular forms on a congruence subgroup

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.