# Commutative Algebra by Hideyuki Matsumura

By Hideyuki Matsumura

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Additional info for Commutative Algebra

Example text

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