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288 14. Elliptic Curves over Local Fields

The two curves C(ev) and C(od) are treated as disjoint spaces with the com-
pactifying points also distinct. Returning to the diagram in (4.2), we describe J :

C(ev) ∪ C(od) = E (R) → P 1 (R).

(4.9) Theorem. The function J : E (R) → P 1 (R) is 2:1 and the separate restric-
tions of J to C(ev) and C(od) have the form:
(1) J ev = J|C(ev) : C(ev) → [1, ∞] ⊂ P 1 (R) and corresponds to (τ) > 0.
Over (∞, 1) the function J ev is a smooth 2: 1 mapping with exactly on one point in
each of the inverse images of 1 and ∞.
(2) J od = J|C(od) : C(od) → [∞, 1] ⊂ P 1 (R) and corresponds to (τ) < 0.
Over (∞, 1) the function J od is a smooth 2: 1 mapping except over the point 0 ∈
[∞, 1]. There is exactly one point in each of the inverse images of ∞ and 1.

2
Proof. The 2:1 character of the restrictions of J follow from (4.7) applied to y =
3
4
x − 3E 4 (τ)x + 2E 6 (τ). In the case (1) we have the relations E 4 (i/t) = t E 4 (it)
6
and E 6 (i/t) =−t E 6 (it) for τ = it which up to rescaling give the two signs for
E 6 , and except for the two special points give a 2:1 mapping.
In the case (2) we have the two signs at ζ 6 = T (ζ 6 ) and T (ζ 3 ) for τ = (1/2)+ti.

By continuity this sign difference extends over pairs of points τ = T (β ) and τ =

T (β ) where β and β are two boundary points on the fundamental domain giving

the same elliptic curve over C. since they are nonisomorphic at {β ,β }={ζ 3 ,ζ 6 },

they are nonisomorphic in general. This proves the theorem.
(4.10) Remark. At the two points over 0 where J od is not smooth, we have vertical
tangents.
2
3
y = x + x 2
3
2
y = x − x 2
c (od) c (od)
c (ev)
∆ < 0
∆ < 0
J
∆ > 0
( )
A
0 1 ∞
2
y = x + 1 y = x + x ∆ > 0
3
3
2
y = x − 1 y = x − x ∆ < 0
2
3
3
2