On The Fourier Expansion Of Eisenstein Series And Its Applications

As we know, the main reason for the study of modular forms is that the arithmetic information encoded in their Fourier coefficients, and the Eisenstein series are a class of important modular forms in number theory, thus the study of the Fourier expansions of the Eisenstein series is very important. In the textbooks on classical modular forms, authors discussed the Fourier expansions of the Eisenstein series at oo. However, they did not discuss the Fourier expansions of the Eisenstein series at cusps other than oo. This thesis solved this problem with respect to the congruence subgroups Γ(N), Γ(N) and Γ0(N) of SL2(Z), respectively. On the other hand, Srinivasa Ramanujan recorded the following interesting formula in his lost notebook:Note that the inte-grand on the right hand side of this formula can be expressed as the Fourier expansion of an Eisenstein series, and C3 is a multiple of the value of certain Dirichlet L-function. This thesis also discussed this kind of formulas based on [ABYZ02].In chapter 1, we give an brief introduction to the main results of this thesis. In chapter 2, we recall some basic definitions and results about classical modular forms. Particularly, when 1≤N≤12, we compute the set of representative elements of cusps of the congruence subgroups Γ(N), Γ1(N) and Γ0(N), respectively. In chapter 3, we study the Fourier expansions of Eisenstein series at various cusps. Particularly, we study the explicit formulas for Fourier expansions at general cusps, and compute some examples at the end of this chapter. In chapter 4, we study one class of Ramanujan formulas. Applying one result in [Liu03],we obtain more such formulas.