By P. Deligne, N. Katz
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Additional info for SGA 7 II: Groupes de monodromie en geometrie algebrique
Sample text
Hence degree of T2 is 1, i. e . , f is an isomorphism. (In fact, it can be checked with the formulae we have at hand that the cusps co, 0, {, 1, 2 & 3 are respectively mapped to the points (1,1,0), ( 1 , 0 , 1 ) , ( 1 , - 1 , 0 ) , ( 0 , 1 , i), ( 1 , 0 , - 1 ) and ( 0 , 1 , - i ) . 52 An important consequence of this theorem is: (4) Corollary 1 0 . 2 . The ring Mod' of modular forms of level 4 is naturally isomorphic to CC *dl ( 0 - T)» *01(0'T)'*12C/0. rtH**oo-4l- *10> 2 i . e . , it is generated by A.
B: a factor exp (TTicd) a p p e a r s , s o w e u s e H o w e v e r , the p e r i o d i c behaviour of 0 for cd even in the v e r i f i c a t i o n ) . z |—> z+ T g i v e s a 2nd q u a s i - p e r i o d for Y, n a m e l y , cT+d c T +a We give s o m e of the c a l c u l a t i o n s this t i m e : f o r m a l l y writing we have by definition: T( ^ajL±L, T) 2 2 exp[TTic(cT+d)y + 2TTicy(a T+b) +TTic r J ^((cT+d)y+aT+b,T) (a T *b] cT + d But *((cT+d)y+aT+b,T) Y(y, T) a exp[-TTia 2 T - 2 f f i a y ( c T + d ) ] * ( ( c T+d)y, T) exp(TTic ( c T + d ) y 2 ) i>((cT+d) y, T) 2 2 = e x p ( - n i a T-2TTiay(c T+d) -TTic(cT+d)y ).
I(z,T) =Zexp(TTi(n+i) 2 T + 2TTi(n+|)(z + i)) = expfai T/4+TTi(z+J))*(z+l(l+T), T) f. 2 # 2 F o r simplicity, we write these a s £ **oi' *10 and *11 # ** i s immediatelv verified that * oo (-z, T) V-z-T) s = * oo (z, T) %i ( z - T ) *io(-z'T)=Vz'T) showing that * *11^°' T^ = ° ' i s different from the others, and confirming the fact that wnile tne other 3 are not z e r o at z - 0 (cf. Lemma 4. 1). Riemann's formula g i v e s us: