Junior Ndayikengurukiye, Temple University
In Section 2 of "The 1-2-3 of Modular Forms", Zagier introduces Eisenstein series of weight $k\geq 2$ and the Discriminant function as first examples of modular and quasi-modular forms on well-behaved subgroups of $SL(2,R)$ (in particular for the full modular group $\Gamma_1 = SL(2,Z)$). In this talk, I will discuss Zagier's two natural ways of defining Eisenstein series on $\Gamma_1 = SL(2,Z)$ and prove their modularity. I will then describe the consequences of their existence on the spaces $M_k(\Gamma_1)$, and how these vector spaces, in concert with the Fourier expansion of the Eisenstein series, result in non-trivial identities for modular forms and their Fourier coefficients. We conclude by discussing the modularity of the
discriminant function and important congruences that arise involving the coefficients of its Fourier expansion.