6. 62
LECTURE MODULAR FORMS
6.5 Show that
0
(z1 )
det @ (z2 )
r(z3 )
(z1 )
(z2 )
(z3 )
1
1
1A = 0
1
whenever z1 + z2 + z3 = 0. Deduce from this an explicit formula for the group law on
the projective cubic curve y 2 t = 4x3 g2 xt2 g3 t3 .
6.6 (Weierstrass -function) It is dened by
1
Z (z ; ) = +
z
X
\{0}
1
1
z
+ +2.
Let = Z1 + Z2 . Show that
(i) Z (z ) = (z );
(ii) Z (z + i ) = Z (z ) + i , i = 1, 2 where i = Z (i /2);
(iii) 1 2 2 1 = 2i;
(iv) Z (z ; ) = 1 Z (z ; ), where is any nonzero complex number.
6.7 Let (z ) be a holomorphic function satisfying
(z ) /(z ) = Z (z ),
(i) Show that (z ) = (z );
(ii) (z + i ) = ei (z+
i
2
(z );
(iii) (z ) = (z ), where (z ) is the Weierstrass -function.
6.8 Using the previous exercise show that the Weierstrass -function (z ) admits an
in nite product expansion of the form
Y
z z 1z2
(z ) = z
(1 )e + 2 ( )
\{0}
which converges absolutely, and uniformly each in disc |z | R.
6.9 Let E be an elliptic curve and y 2 = 4x3 g2 x g3 be its Weierstrass equation.
Show that any automorphism of E is obtained by a linear transformation of the
variables (x, y ) which transforms the Weierstrass equation to the form y 2 = 4x3
c4 g2 x c6 g3 for some c = 0. Show that E is harmonic (resp. anharmonic) if and only
if g3 = 0 (resp. g2 = 0).
6.10 Let k be an even integer and let L Rk be a lattice with a basis (e1 , . . . , ek ).
Assume that ||v ||2 is even for any v L. Let D be the determinant of the matrix
(ei ej ) and N be the smallest positive integer such that N ||v ||2 2Z for all v Rk
satisfying v w Z for all w L. Dene the theta series of the lattice L by
L ( ) =
X
#{v L : ||v ||2 = 2n}e2in .
n=0
(i) Show that L ( ) =
P
v L
2
ei ||v|| ;
(ii) Show that the functions (0, )k discussed in the beginning of Lecture 6 are
special cases of the function L .