Some new modular equations and their applications

Ramanujan derived 23 beautiful eta-function identities, which are certain types of modular equations. We found more than 70 of certain types of modular equations by using Garvan's Maple q-series package. In this paper, we prove some new modular equations which we found by employing the theory of modular form and we give some applications for them.     

On some of Ramanujan's Schläfli-type “mixed” modular equations

We give elementary proofs of seven Schläfli-type “mixed” modular equations recorded by Ramanujan on p. 86 of his first notebook. Previously, these equations were proved by Berndt by using the theory of modular forms. In the process, we also found three new Schläfli-type mixed modular equations of the same nature.     

On certain identities for ratios of theta-functions and some new modular equations of mixed degree

In this paper, we derive certain identities for ratios of theta-functions. As applications of the identities, we establish certain new modular equations of mixed degree in the theory of signature 3, which are analogous to Ramanujan-Weber type modular equations and Ramanujan-Schläfli type mixed modular equations.     

On modular solutions of certain meromorphic modular differential equations

Kaneko and Koike gave the “extremal” quasimodular forms of depth 1 for PSL₂(ℤ) and modular differential equations they satisfy. In this paper, we study modular solutions of their modular differential equations.     

Arithmetic partial differential equations, II

We continue the study of arithmetic partial differential equations initiated in [7] by classifying “arithmetic convection equations” on modular curves, and by describing their space of solutions. Certain of these solutions involve the Fourier expansions of the Eisenstein modular forms of weight 4 and 6, while others involve the Serre-Tate expansions (Mori, 1995 [13], Buium, 2003 [4]) of the same modular forms; in this sense, our arithmetic convection equations can be seen as “unifying” the two types of expansions. The theory can be generalized to one of “arithmetic heat equations” on modular curves, but we prove that they do not carry “arithmetic wave equations.” Finally, we prove an instability result for families of arithmetic heat equations converging to an arithmetic convection equation.     

Theta function identities associated with Ramanujan’s modular equations of degree 15

We present alternative proofs of some of Ramanujan’s theta function identities associated with the modular equations of composite degree 15. Along the way we also find some new theta-function identities. We also give simple proofs of his modular equations of degree 15.     

Some new modular equations in Ramanujan’s alternate theory of signature 3

The Ramanujan Journal - In this paper, we prove some new modular equations in the theory of signature 3 or cubic modular equations by using theta-function identities. Particularly, we prove modular...     

Some modular relations for the Göllnitz-Gordon functions and Ramanujan’s modular equations

Huang used the methods of Rogers, Watson and Bressoud to derive some new modular relations involving the Göllnitz-Gordon functions. In this paper, using Ramanujan’s modular equations, we present a uniform method to prove these modular relations established by Huang.     

Hypergeometric series,modular linear differential equations and vector-valued modular forms

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in dimensions 2 and 3 leads naturally to close connections between modular forms and hypergeometric series.     

Ramanujan’s modular equations of degree 5

We provide alternative derivations of theta function identities associated with modular equations of degree 5. We then use the identities to derive the corresponding modular equations.     

Notes on Rankin–Cohen brackets

The Rankin–Cohen product of two modular forms is known to be a modular form. The same formula can be used to define the Rankin–Cohen product of two holomorphic functions f and g on the upper half-plane. Assuming that this product is a modular form, we prove that both f and g are modular forms if one of them is. We interpret this result in terms of solutions of linear ordinary differential equations.     

Towers of function fields over finite fields corresponding to elliptic modular curves

In this paper, we find several equations of recursive towers of function fields over finite fields corresponding to sequences of elliptic modular curves. This is a continuation of the work of Noam D. Elkies [8], [9] on modular equations of higher degrees.     

Certain new Ramanujan's Schläfli-type mixed modular equations

In this paper, we establish certain new Ramanujan's Schläfli-type mixed modular equations.     

Class invariants from a new kind of Weber-like modular equation

A technique is described for explicitly evaluating quotients of the Dedekind eta function at quadratic integers. These evaluations do not make use of complex approximations but are found by an entirely ‘algebraic’ method. They are obtained by means of specialising certain modular equations related to Weber’s modular equations of ‘irrational type’. The technique works for certain eta quotients evaluated at points in an imaginary quadratic field with discriminant \(d \equiv 1 \pmod {8}\).     

A simple example of modular forms as tau-functions for integrable equations

We show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakravarty, M. Ablowitz, and P. Clarkson [6]. We discuss possible consequences of this novel phenomenon in integrable systems which indicate deep connections between integrable equations, group representations, modular forms, and moduli spaces.In Memory of Prof. M. C. PolivanovDepartment of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3651. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 93, No. 2, pp. 330–341, November, 1992.     

On inequalities and linear relations for 7-core partitions

Recently, Ramanujan’s modular equations have been applied by N.D. Baruah and B.C. Berndt to obtain a linear relation for 5-core partitions and by A. Berkovich and H. Yesilyurt to obtain inequalities for 7-core partitions. In this paper, we generalize their results by using the theory of modular forms. In particular, we prove conjectures of Berkovich and Yesilyurt.     

Partition identities and Ramanujan's modular equations

We show that certain modular equations and theta function identities of Ramanujan imply elegant partition identities. Several of the identities are for t-cores.     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

A Few Theta-Function Identities and Some of Ramanujan's Modular Equations

We prove three modular equations of Ramanujan using theta-function identities. Proofs via methods known to Ramanujan were not available hitherto. One had previously been proved by classical methods, and two had been proved using the theory of modular forms.     

Non-hyperelliptic modular curves of genus 3

A curve C defined over Q is modular of level N if there exists a non-constant morphism from X₁(N) onto C defined over Q for some positive integer N. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve C of genus 3 and level N such that Jac C is Q-isogenous to a given three dimensional Q-quotient of J₁(N). Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus 3. Let C be a modular curve of level N, we say that C is new if the corresponding morphism between J₁(N) and Jac C factors through the new part of J₁(N). We compute equations of 44 non-hyperelliptic new modular curves of genus 3, that we conjecture to be the complete list of this kind of curves. Furthermore, we describe some aspects of non-new modular curves and we present some examples that show the ambiguity of the non-new modular case.