By Chowdhury K.C.
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Additional info for A first course in theory of numbers
Any lattice can be written as Λ = αΛz with Λz = Z + Zz where z in the upper half plane is unique up to a transformation from Γ1 . Then the assignment f (z) = F (Λz ) yields a bijection from functions F on lattices, homogeneous of degree −k, to functions f on the upper half plane satisfying the transformation law (1) in the deﬁnition of modular forms in Sect. 4 (for Γ = Γ1 , k integral, v = 1). The action of the mth Hecke operator on degree −k functions F on lattices is simply given by Tm F (Λ) = Λ F (Λ ) where Λ runs over all sublattices of index m in Λ.
The ﬁrst non-vanishing term in (−cz+a)−1/2 × fm (A−1 z) is a constant multiple of e(g 2 z/24m). Thus, by our deﬁnition of the order, we obtain ord(fm , r) = g 2 /24m, which is the second assertion. We note an immediate consequence of the second assertion: 36 2. 1), and let r = − dc ∈ Q, gcd(c, d) = 1. Then the order of f at the cusp r is ord(f, r) = 1 24 m|N (gcd(c, m))2 am . m An eta product f will be called a holomorphic eta product if its orders at all cusps are non-negative, ord(f, r) ≥ 0 for all r ∈ Q ∪ ∞.
They satisfy √ 1 = i( 3z)3 E3,3,i (z), E3,3,i − 3z √ 1 E3,3,−i − 3z = −i( 3z)3 E3,3,−i (z). The signs in these transformation formulae have been the reason for the choice of signs in the notation Ek,P, δi (z). We will meet the functions E3,3, δi (z) in Sect. 2. There are many more types of Eisenstein series which will not be presented here. We refer to , Chap. 4, , Chap. 7, and , Chap. 7 for a thorough discussion, including the delicate cases of small weights 1 and 2. We will meet several examples in Part II.
A first course in theory of numbers by Chowdhury K.C.