Number Theory: An Introduction to Mathematics

520 XII Elliptic Functions

One important property of the theta function is almost already known to us:

Proposition 3Fo r a l lv∈Candτ∈H,

θ(v;− 1 /τ)=(τ/i)^1 /^2 eπiτv

2

θ(τv;τ), (32)

where the square root is chosen to have positive real part.

Proof Suppose first thatτ=iy,wherey>0. We wish to show that

∑∞

n=−∞

e−n

(^2) π/y
e^2 nπiv=y^1 /^2

∑∞

n=−∞

e−(v+n)

(^2) πy
.
But this was already proved in Proposition IX.10.
Thus (32) holds whenτis pure imaginary. Since, with the stated choice of square
root, both sides of (32) are holomorphic functions forv∈Candτ∈H, the relation
continues to hold throughout this extended domain, by analytic continuation.

Following Hermite (1858), for any integersα,βwe now put
θα,β(v)=θα,β(v;τ)=

∑∞

n=−∞

(− 1 )βneπiτ(n+α/^2 )

2

e^2 πiv(n+α/^2 ).

(The factor(− 1 )βnmay be made less conspicuous by writing it aseπiβn.) Since

θα+ 2 ,β(v)=(− 1 )βθα,β(v), θα,β+ 2 (v)=θα,β(v),

there are only four essentially distinct functions, namely

θ 00 (v)=

∑∞

n=−∞

eπiτn

2

e^2 πivn,

θ 01 (v)=

∑∞

n=−∞

(− 1 )neπiτn

2

e^2 πivn,

θ 10 (v)=

∑∞

n=−∞

eπiτ(n+^1 /^2 )

2

eπiv(^2 n+^1 ),

θ 11 (v)=

∑∞

n=−∞

(− 1 )neπiτ(n+^1 /^2 )

2

eπiv(^2 n+^1 ).

(33)

Moreover,

θ 00 (v;τ)=θ(v;τ), θ 01 (v;τ)=θ(v+ 1 / 2 ;τ),

θ 10 (v;τ)=eπi(v+τ/^4 )θ(v+τ/ 2 ;τ), θ 11 (v;τ)=eπi(v+τ/^4 )θ(v+ 1 / 2 +τ/ 2 ;τ).

In fact, for all integersm,n,

θα,β(v+mτ/ 2 +n/ 2 )=θα+m,β+n(v)e−πi(mv+m

(^2) τ/ 4 −αn/ 2 )

. (34)