Download PDF by K. Mathiak, Steffen Supra: Algebra[Lecture notes]

By K. Mathiak, Steffen Supra

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33 - El - Eg is 504 . 504 = -1728 . 24. 2 Theta Functions and Weight Enumerators 50 So (E! - E~) (T) = 0 + 1728q + 1728(-24). q2 + .... g. [Ser73, pp. 95-96]): II (1- qr)24. 00 ~:= E! 6 The Algebra of Modular Forms Let k be an integer. The modular forms of weight k form a C-vector space which we denote by M k . We have Mk = 0 for k odd (by definition or extending the definition of a modular form to allow odd weights). In this section we want to determine the vector spaces Mk for k even. We shall first derive a formula about the orders of the zeros of a modular form.

Then ImT + nl 2 = m 2 TT + 2mnReT + n 2 This implies for keven, k ImT + nl- k ~ m2 - ~ mn + n 2 = Imry + nl 2 Imry + nl- k ::; 2. 3, the series 1 L 2 (m,n)EZ (m,n)¥(O,O) Imry+nlk converges for k > 2. This shows that the series Gk(T) converges uniformly on every compact subset of D. Applying the result to Gk(g-lT) for 9 E SL 2 (Z)/{±1} shows that the same holds for each of the transforms 9 D of D, which cover 1Hl. Thus Gk is the limit of a sequence of holomorphic functions, which converges uniformly on every compact subset of 1Hl.

E~. So i,j i,j where B = « bij )) is the inverse matrix of the matrix A = « ei . ej)) (cf. 1). 2. An: Consider a fundamental system of roots {el, ... -----+---- ....... ---.. Then _l_[(n - i + l)el + 2(n - i + 1)e2 + ... n+1 +i(n - i + l)ei + i(n - i)ei+l + ... + ienl e* • + (i - l)(n - i + 1)ei-l for i = 1, ... , n. Dn: Consider a fundamental system ofroots {el, ... , en} with Coxeter-Dynkin diagram ) 0 - - - - - - - 4 > - - - - - - - . . . . . - -.... el Then el + 2e2 + ... + (i - l)ei-l + i(ei + ei+l + ...

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