Math 428, Assignment 12: due December 5
1)Fix `∈Nand let a, b be rational numbers so that `a, `b ∈Z. Consider
theta functions with rational characteristics
ϑa,b(v, τ ) =
n=∞
X
n=−∞
exp[πi((a+n)2τ+ 2(n+a)(v+b))].
a)Show that ϑ0,0=θ3,ϑ0,1/2=θ2,ϑ1/2,0=θ1,and ϑ1/2,1/2=−θ0.
b)Show that
ϑa,b(v, τ ) = exp[πi(a2τ+ 2a(v+b))]ϑ0,0(v+aτ +b, τ ).
c)Show that ϑa,b =ϑa+N,b and
ϑa,b+N(v, τ ) = ϑa,b(v+N, τ ) = exp[2πiaN ]ϑa,b(v, τ ).
Conclude that for fixed `, the theta functions with rational characteristics
span a vector space of complex dimension ≤`2.
d)Describe the set of all zeros of ϑa,b(v , τ) (as a function of vfor fixed τ).
2)Recall that complex projective space CP3is defined as the set of equivalence
classes [z0, z1, z2, z3], zj∈C, where
[z0, z1, z2, z3]∼[w0, w1, w2, w3] if for some λ6= 0, wj=λzj, j = 0,1,2,3.
Consider the complex torus E=C/Λ, where Λ is a lattice with basis ω1=
2, ω2= 2τ. Show that
Θτ(v) := [θ(v, τ ), θ1(v, τ ), θ2(v, τ ), θ3(v, τ )]
yields a well-defined map Θτ:E→CP3. Does it yield a well-defined map
C/(Z+Zτ)→CP3?
3)Verify the identities
θ3(v, τ )2θ3(0, τ )2=θ2(v, τ)2θ2(0, τ )2+θ1(v, τ )2θ1(0, τ )2
θ(v, τ )2θ3(0, τ )2=θ2(v, τ)2θ1(0, τ )2−θ1(v, τ )2θ2(0, τ )2.
Conclude that the image Θτ(E) satisfies the quadratic equations
z2
3θ3(0, τ )2=z2
2θ2(0, τ )2+z2
1θ1(0, τ )2z2
0θ3(0, τ )2=z2
2θ1(0, τ )2−z2
1θ2(0, τ )2.
(1)
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