Executive Summary : | Modular forms are complex-valued holomorphic functions defined on the complex upper half-plane which transform in a certain way under the action of a discrete subgroup of SL(2, R). The set of modular forms of a fixed weight forms a finite dimensional vector space over the set of complex numbers. It is well-known that the space of modular forms can be decomposed as a sum of the space of Eisenstein series and the space of cusp forms. Eisenstein series are well-known objects as their Fourier coefficients are explicitly known. However, the Fourier coefficients of cusp forms cannot be determined explicitly in general. An important problem in this direction is to construct a basis for the space of cusp forms. The Dedekind eta function (a cusp form of weight 1/2) is a theta series which is an important building block for constructing modular forms. It is therefore natural to ask if one can express a cusp form in terms of powers of Dedekind eta function. More generally, we ask whether there exists an explicit basis for the space of cusp forms involving products or quotients of powers of Dedekind eta function. The aim of this research proposal is to construct such a basis for certain spaces of cusp forms. From applications point of view, such constructions are significant for problems in the theory of q-series and partitions. A partition of a natural number n is a representation of n as a sum of smaller numbers (which add up to n) in a prescribed order (generally, non-increasing). The partition function which counts the number of partitions of n, is denoted by p(n). p(n) satisfies remarkable divisibility properties modulo 5,7 and 11, viz. for all natural number n, 5|p(5n+ 4),7|p(7n+ 5) and 11|p(11n+ 6). Ramanujan proved certain identities between q-series, called witness identities which imply the divisibility 5|p(5n+ 4) and 7|p(7n+ 5). Ramanujan did not give any hint of a witness identity in modulus 11. Recently, in a joint work with Goswami and Singh, we obtain a new witness identity in modulus 11 case using the theory of modular forms. Our approach was motivated from Ramanujan’s work . In order to obtained a witness identity modulo 11, we constructed a good basis (appeared in Garvan's work on p-core partitions) for certain space of cusp form involving eta quotients. We also obtained several identities satisfied by p(n) modulo 13,17,19 and 23. Another application of such construction is an explicit evaluations of Fp(q), p = 11,17,19 and 23 where Fp is the function (appearing in the Ramanujan’s circular summation formula) given by Fn(q) := 1 + 2nq^{n(n−1)/2} +.... For primes p bigger than 23, a good basis is not explicitly known for the corresponding spaces of cusp forms and therefore we could not obtain similar identities involving p(n) in these cases. The aim of this project is to first construct such basis for certain spaces of cusp forms and then use this basis to obtain identities satisfied by p(n) and explicit expression for Fn in these cases. |