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§4. Elliptic Curves over the Real Numbers 287

following basic modular transformations S(τ) =−1/τ and T (τ) =−1/(τ − 1).
These two transformations have the following values on the corners.
(1) For S it preserves the line Re(τ) = 0and S(ζ 4 ) = ζ 4 .Also S(ζ 3 ) = ζ 6 and
S(ζ 6 ) = ζ 3 interchanging the lines Re(τ) =±1.
(2) For T we have T (ζ 6 ) = ζ 6 , T (ζ 4 ) = (1/2) 1/2 ζ 8 ,and T (ζ 3 ) = (1/3) 1/2 ζ 12
carries the circle |τ|= 1tothe line Re(τ) = 1/2 in the upper half plane.
(4.6) The Modular Forms E 4 and E 6 . The transformation law

aτ + b 2k ab

E 2k = (cτ + d) E 2k (τ) for ∈ SL(2Z)
cτ + d cd
specializes for the two transformations S(τ) and T (τ) to the following: E 2k (S(τ)) =
2k
τ 2k E 2k (τ) and E 2k (T (τ)) = (τ − 1) E 2k (τ).Wehavetwo casesfor thelines
Re(τ) = 0 and 1/2.
(1) For Re(τ) = 0: E 6 (ζ 4 ) = 0and J(ζ 4 ) = 1. Moreover, τ = it, t > 0, and q ≥
n
0 in this case, and the expansion E 4 = 1+240 σ 3 (n)q shows that E 4 (it) ≥ 0
n≥1
on this line. In particular J(q) ≥ 1 on this line. The relation S(it) = i/t shows that
the two parts of the positive imaginary axis i[1, ∞) in the fundamental domain and
i(0, 1] are interchanged. The coefficients of the curve (it) are transformed by the
modular relations to give

i 4 i 6
E 4 = t E 4 (it) and E 6 =−t E 6 (it).
t t
(2) For Re(τ) = 1/2: We have three distinguished values:

τ = T (ζ 6 ) = ζ 6 T (ζ 4 ) = (1/2) 1/2 ζ 8 T (ζ 3 ) = (1/3) 1/2 ζ 12
E 4 (τ) = 0 E 4 (i)> 0 0
E 6 (τ) = E 6 (ζ 6 ) 0 −27 · E 6 (ζ 6 )
J(t) = 0 1 0

6
Here we use that E 6 (T (ζ 3 )) = (ζ 3 − 1) E 6 (ζ 6 ) =−27 · E 6 (ζ 6 ) for the last entry
in the table. For τ = 1/2 + it, t ≥ 0, we have −1 < q ≤ 0and J(q) take values
(−∞, 1], that is, the values not coming from Re(τ) = 0.

(4.7) Two Real Forms. The two equations

2 3 4 6 2 3 4 6
E"α, β#: y = x − 3αx + 2β and E"t α, −t β#: y = x − 3αt x − 2t β
over the real numbers are nonisomorphic, but over the complex numbers become
2
3
isomorphic under the substitution which carries (x, y) to (−t x, it y) followed by
6
dividing by −t . We can see how this can be carried out with the modular property
of the coefficients.
(4.8) Notation. Let C(ev) denote iR + compactified with a point, denoted 0 = i∞,
and C(od) denote (1/2)+iR + compactified with a point, denoted 1/2 = 1/2+i∞.