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286 14. Elliptic Curves over Local Fields

see 9(4.1). In 10(1.4) other multiples of the basic Eisenstein series G k (L) = G 2k (τ)
for L τ = Zτ + Z were introduced where g 2 (τ) = 60G 4 (τ) and g 3 (τ) = 140G 6 (τ).
Now we use another normalization with q-expansion having 1 as constant term,
namely,
G 2k (τ) = 2ζ(2k)E 2k (τ).

Then the two relevant functions of weights 4 and 6 are respectively

n n
E 4 = 1 + 240 σ 3 (n)q and E 6 = 1 + 504 σ 5 (n)q
n≥1 n≥1
s
where σ s (n) d|n d . After suitable renormalization we have the equation of the
2
3
cubic as above y = x − 3E 4 (τ)x + 2E 6 (τ).
(4.4) Fundamental Domain. The fundamental domain D of SL(2, Z) in H is given
by the inequalities D : −1/2 ≤ Re(τ) ≤ 1/2and |τ|≥ 1,

Πm (τ) > 0

P
2

−1 +1

that is, the central strip outside the closed unit disc. When the strip −1/2 ≤ Re(τ) ≤
1/2 is mapped into the unit disc |q| < 1 by the exponential function q = q(t) =
e 2πiτ we see that q is real exactly for Re(τ) = 0, ±1/2. In terms of the equation of
3
2
the cubic curve E(τ): y = x − 3E 4 (τ)x + 2E 6 (τ), we know that E 4 and E 6 have
expansions in q(τ) with real coefficients. Furthermore the function j(τ) of 9(4.8)
3
becomes j(τ) = 12 J(τ) where
3 3 2
J(τ) = E 4 (τ) / (τ) and (τ) = E 4 (τ) − E 6 (τ) .
Recalling 10(1.7), we have thus four equivalent conditions for the equation
E"E 4 (τ), E 6 (τ)# to be defined with real coefficients: (1) q is real, (2) Re(τ) is a
half integer, (3) τ +¯τ is an integer, and (4) Zτ + Z is stable under complex conjuga-
tion.

(4.5) Notation. Let ζ n = exp(2πi/n), a special primitive nth root of unity. It satis-
fies the equation 1 + x +· · ·+ x n−1 = 0 like all nontrivial nth roots of unity.

The “corners” of the fundamental domain along the circle τ|= 1 are respectively
ζ 3 where Re(ζ 3 ) =−1/2, ζ 4 = i where Re(ζ 4 ) = 0, and ζ 6 where Re(ζ 6 ) = 1/2.
These are the intersections with the lines given real elliptic curves. We will use the