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Fearless Symmetry by Robert Gross, Avner Ash

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108 CHAPTER 9
place according to the rules of the field R.) Let
x
3
= λ
2
x
1
x
2
.Lety
3
=−(λx
3
+ ν). Then P + Q has
coordinates (x
3
, y
3
).
5. If x
1
= x
2
and y
1
= y
2
= 0, then compute the number
λ =
3x
2
1
+A
2y
1
. Compute ν as before, and then use the same
formula for x
3
and y
3
as in the previous case.
A Much-Needed Example
This description is somewhat mysterious.
4
In fact, it is not obvious
(unless you looked at the footnote) that (x
3
, y
3
) even solves the
same equation y
2
= x
3
+ Ax + B that we started with. We take
some examples in E(Q) with the particular curve y
2
= x
3
+ 1, which
means that A = 0 and B = 1. Remember that (0, 1), (0, 1), and
(2, 3) are points on this curve. The first case of the group law
tells us that O + (0, 1) = (0, 1), and the second case tells us that
(0, 1) + O = (0, 1). The third case tells us that (0, 1) + (0, 1) = O.
Now things get messier. We compute (0, 1) + (2, 3). We start by
computing λ =
31
20
=
2
2
= 1. Next, ν = 1 1 · 0 = 1. Then x
3
= 1
2
0 2 =−1, and y
3
=−(1 ·−1 + 1) = 0. So we just worked out that
(0, 1) + (2, 3) = (0, 1).
Finally, we compute (2, 3) + (2, 3). Using the last case, we com-
pute that λ =
3·2
2
+0
2·3
=
12
6
= 2. Next, ν = 3 2 · 2 =−1. Now, x
3
= 4
2 2 = 0 and y
3
=−(2 · 0 + (1)) = 1. So (2, 3) + (2, 3) = (0, 1).
EXERCISE: Let E be the elliptic curve y
2
= x
3
+ 17. Compute
(1, 4) + (2, 5) and (2, 5) + (2, 5).
SOLUTION: To compute (1, 4) + (2, 5), we first compute that
λ =
1
3
, and then ν =
13
3
, and then (1, 4) + (2, 5) = (
8
9
,
109
27
).
To compute (2, 5) + (2, 5), we first compute that λ =
6
5
, and
then ν =
13
5
, and then (2, 5) + (2, 5) = (
64
25
,
59
125
).
4
Geometrically, what is going on is this: We take P and Q and connect them with a line.
This line will intersect the elliptic curve in exactly three points: P, Q, and a third point T.
We negate the y coordinate of T and the resulting point is P + Q. This is what underlies
formula (4). The other formulas have similar geometric interpretations. The fact that
this geometric construction defines a group law is amazing and not so easy to prove.

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