Commutative Algebra by Hideyuki Matsumura

By Hideyuki Matsumura

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Let us stand back from our calculation now and consider what we have got so far. In the first place, the substitutions in the variables z, T for which £ is quasi-periodic form a group: in fact, SL(2, 2Z) acts on C x H b y (z, T) I—» (z/cT+d, (aT+b)/(cT+d)) because z/(cT+d) a'((aT+b)/(cT+d))+b' » V((aT+b)/(cT+d))+d»' C '((aT*b)/(cT+d))+d' ' . / I (a'a+b'c)T+(a'b+b'd) v " '(c'a+d'cjT+fc'b+dd1)' ( C 'a+d f c) T+(c'b+d'd) Moreover, this action normalises the lattice action on z, i . e . , we have an action of a semi-direct product SL(2, ZZ)X ZZ2 a b on ((z+mT+n)/(cT+d),(aT+b)/(cT+d)).

Homogeneous coordinates, zeros at z = b^/a^ and poles 25 Method III. Second logarithmic derivatives: Note that log & (z) is periodic upto addition of a linear function. Thus the (doubly) periodic function ^ 5 - log *(*) dz 2 is meromorphic. This is essentially Weierestrass » (p -function. To be precise, $>(z) = - - £ - log * (z) + (constant), ll dz 2 the constant being adjusted so that the Laurent expansion of {p(z) at z = 0 has no constant term. Method IV: Sums of first logarithmic derivatives: Choose a .

Replacing x , y , u and v by x+ a , y + p,u+Y , v +6 where a , p, Y , & e | A and a + 0 + Y + $ e A . listed below in an abbreviated form all the r e s u l t s . We have 19 First we make a table containing the fundamental transformation relations between the d' 's that are needed for a quick verification of Riemann's theta formulae. Table 0 (zi >-z) (z i *OO(-Z'T>=*OO(Z'T) * o o ( z + >z+i) - V"Z'T) = V Z ' T ) VZ+*'T) ^o(-Z,T)=^10(z, *10(Z+i' T) *ll(z+*' * n ( - » . 1) • - * n ( z , T) (zi *oo ( z + * T ' T ) = (exp ( T ) = = T ) = T) = > » + £T) " TTi T / 4 " T T i z ) ) *10

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