Show 75
(i) that
rLC (2) = 48 + 16A4 ,
rLC (4) = 28 A8 + 640A4 + 1104.
(ii) Using (7.13) show that A8 = 759 4A4 .
7.9 Let A = An be a commutative graded algebra over a eld F . Assume A
n=
has no zero divisors, A0 = F 1 and dim AN > 1 for some nN > 0. Show that An = 0
for n < 0. Apply this to give another proof that Mk () = 0 for k < 0.
2
7.10 Find an explicit linear relation between the modular forms E16 , E8 and E4 E10 ,
where E2k denotes the Eisenstein series. Translate this relation into a relation between
the values of the functions d (m).
7.11 Let f ( ) be a parabolic modular form of weight with k respect to (1).
(i) Show that the function ( ) = |f ( )|(Im )k is invariant with respect to (1).
(ii) Show that ( ) is bounded on H (it is not true if f is not cuspidal).
P
(iii) Show that the coecient an in the Fourier expansion f ( ) =
an q n can be
computed as the integral
Z1
an =
f (x + iy )e2in(x+iy) dx.
0
(iv) Using (iii) prove that |an | = O(nk ) (Heckes Theorem).
7.12 Let L be an even unimodular lattice in R8k and rL (m) be dened as in Example
7.2. Using the previous exercise show that
rL (m) =
8k
4k1 (m) + O(m2k ).
B2k
7.13 Let L = E8 E8 E8 . Show that
L =
432000
1
E12 +
.
(12)
691