Tata Lectures on Theta I (Modern Birkhauser Classics) by David Mumford, C. Musili, M. Nori, E. Previato, M. Stillman

By David Mumford, C. Musili, M. Nori, E. Previato, M. Stillman

This quantity is the 1st of 3 in a sequence surveying the speculation of theta capabilities. in response to lectures given by means of the writer on the Tata Institute of primary examine in Bombay, those volumes represent a scientific exposition of theta capabilities, starting with their old roots as analytic capabilities in a single variable (Volume I), referring to the various appealing methods they are often used to explain moduli areas (Volume II), and culminating in a methodical comparability of theta features in research, algebraic geometry, and illustration conception (Volume III).

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Example 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: : *oo< x ) *oo ( y>*o° ( u ) *oo< v > + * U | W * 0 l W *01 * O l W ^10w*10(y)*10

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