Elliptic Modular Forms and Their Applications - Up To

12 D. Zagiermodular form f ∈ M k (Γ ) equals k Vol(Γ \H)/4π, where just as in the caseof Γ 1 we must count the zeros at elliptic fixed points or cusps of Γ withappropriate multiplicities. The same argument as for Corollary 1 of Proposition2 then tells us M k (Γ ) is finite dimensional and gives an explicit upperbound:Proposition 3. Let Γ be a discrete subgroup of SL(2, R) for which Γ \H hasfinite volume V .Then dim M k (Γ ) ≤ kV +1 for all k ∈ Z.4πIn particular, we have M k (Γ )={0} for k2 and the discriminant function Δ(z) of weight 12,whose definition is closely connected to the non-modular Eisenstein seriesE 2 (z).2.1 Eisenstein Series and the Ring Structure of M ∗ (Γ 1 )There are two natural ways to introduce the Eisenstein series. For the first,we observe that the characteristic transformation equation (2) of a modular