This proposal belongs to the area of modular forms and L-functions, and it consists of both algebraic and analytic problems related to the structure of the space of modular forms, of integral and half integral weight. The methods I use involve the theory of periods of modular forms, developed in the 1970s by Eichler-Shimura-Manin, and in the 1980s by Kohnen and Zagier. Part of the proposal is concerned with decompositions of modular forms of both integral and half integral weight, in terms of explicit generators with rational periods, or with rational Fourier coefficients. The coefficients of these decompositions can be explicitly expressed in terms of periods. Our results would contribute to the theory of periods of modular forms, an area of intense research due to connections with arithmetic algebraic geometry. Among the results we plan to obtain in this direction is a formula--in terms of periods--of the Petersson inner product between a Hecke eigenform for a congruence subgroup of SL(2,Z), and a Rankin-Cohen bracket of two Eisenstein series attached to arbitrary cusps of the congruence subgroup. This result would generalize to Rankin-Cohen brackets the classical Rankin-Selberg identity, in which the Rankin-Cohen bracket is simply a product of Eisenstein series. We plan to use adelic automorphic forms to prove the most general statement. This is a joint project with Ramin Takloo-Bighash In addition to these algebraic questions, I plan to study the rate of growth of certain arithmetic functions closely related to Fourier coefficients of Rankin-Cohen brackets of half integral weight. This has applications to proving bounds towards the Ramanujan conjecture for the coefficients of half integral weight forms, by a method different from the usual method of estimating Kloosterman sums appearing as coefficients of Poincare series.
Field of science
- /natural sciences/mathematics/pure mathematics/arithmetic
- /natural sciences/mathematics/pure mathematics/geometry
- /natural sciences/mathematics/pure mathematics/algebra/algebraic geometry
Call for proposal
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