By Dolgachev I.
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Example text
A weak meromorphic modular form) of weight k with respect to Γ. We shall later add one more condition to get rid of the adjective ”weak”. 1. Some authors prefer to call 2k the weight of a weak modular form admitting k to be equal 1/2. Since jg has a meaning for any group Γ acting discretely on a complex manifold M , our definition can be easily extended to a more general situation leading to the notion of an automorphic form of weight k. 2 Suppose we have n + 1 linearly independent functions f0 , .
Since we can decrease |δ ± 2γ| in this way, the assertion will follow by induction. Note that we used that |δ ± 2γ| is not equal to |δ| or |γ| because (γ, δ) = 1 and γδ is even. Now, if |δ| < |γ|, we use the substitution τ → −1/τ . 13) we see that the asssertion for M follows from the assertion fo M ′ = This reduces again to the case |δ| > |γ|. β −α δ −γ . 16) is equal to i δ−1 2 γ ( |δ| ) when γ is even and δ is odd. If γ is odd and δ is even, it is equal to e−πiγ/4 ( γδ ). Here ( xy ) is the Jacobi-Legendre symbol, where we also set ( 10 ) = 1.
As we shall see a little later, ℘1 (z) = d℘ . dz Consider the map Eτ = C /Λ → P2, z → (T (z)3 , T (z)X(z), Y (z)). Since T (z)3 , T (z)X(z), Y (z) all belong to the same space Th(3, Λτ ) 1 , 1 this map is 2 2 well-defined and holomorphic. 2 only by a composition with a translation on Eτ and a linear change of the projective coordinates coordinates. This is because, for any f ∈ Th(k; Λ) we have eπi[a 2 τ +2(z+a)b] f (z + b + aτ ) ∈ Th(k; Λ)ab k (see Lecture 3). So it is an isomorphism onto its image.