Research Projects

There are two objectives in this proposal. We are interested in the algebra structure of Drinfeld modular forms, the Atkin-Lehner theory of oldforms and newforms for arbitrary level n, where n is any ideal of F_q[T]. The algebra of classical modular forms of level 1 is isomorphic to a polynomial ring in two variables, with generators Eisenstein series E_4 and E_6. Similarly, the algebra of classical modular forms of level \Gamma_0(2) is a polynomial ring in two variables but with a different set of generators. This explicit algebra structure enabled Serre and Swinnerton Dyer to study the properties of the weight filtration for level 1. The weight filtration is beneficial when we study congruences between classical modular forms. Though there are important works in the literature for higher levels, they are more concerned about the weight of the generators for the algebra of modular forms than the algebra structure of the space of modular forms of a given level. A similar study has been initiated to understand the C-algebra structure of the space of Drinfeld modular forms of level GL_2(A). The C-algebra of Drinfeld modular forms of level GL_2(A) is isomorphic to C[X, Y], with generators Eisenstein series g_1 and Poincare series h. Consequently, one can define and study the weight filtration for Drinfeld modular forms of weight k, type l, and level GL_2(A). Recently, in joint work with Dalal, we have shown that the ring of Drinfeld modular forms for \Gamma_0(T) is a quotient of the polynomial ring C[X,Y,Z]. Consequently, we could study the properties of weight filtration for Drinfeld modular forms of weight k, type l, and level \Gamma_0(T). The first objective of this proposal is to investigate the algebra structure of classical, Drinfeld modular forms for higher levels. We are mainly interested in the C-algebra structure of Drinfeld modular forms of any level \Gamma_0(n). The theory of oldforms and newforms is well-known for classical modular forms. However, the analogues theory of oldforms and newforms needs to be better understood for Drinfeld modular forms. In joint work with Dalal, we proposed a definition of oldforms and newforms for Drinfeld modular forms of square-free level and analyzed these spaces. Bandini and Valentino developed the theory of p-oldforms and p-newforms, and made a conjecture about their properties. The conjecture was shown to be true when the dimension of S_{k,l}(GL_2(A)) is less than or equal to 1. In joint work with Dalal, we obtained some partial results when the dimension of S_{k,l}(GL_2(A)) is less than or equal to 2. The second objective of this proposal is to investigate the theory of oldforms and newforms for arbitrary levels and check the validity of the Conjecture independent of the restriction on the dimension of S_{k,l}(GL_2(A)).