Page 133 -

110 5. Reduction mod p and Torsion Points

2
(3.6) Example. If the minimal normal form of E over k is y = f (x), where f (x)
is a cubic polynomial, then E has bad reduction at all p where k(p) has characteristic
2 and at all irreducibles p which divide the discriminant D( f ) of the cubic f (x).

(3.7) Example. If the minimal normal form of E over k is
2
y = (x − α)(x − β)(x − γ),
2
then no p divides all roots α, β,and γ for any irreducible p. The elliptic curve E
has good reduction at p > 2 if and only if p does not divide any of the differences
α − β, β − γ ,and γ − α.

2
2
3
(3.8) Example. The curve E defined by y + y = x − x has good reduction at

2 since over F 2 it is the curve E of (4.2). Denoting the curve mod p by E (p) and
3
the reduction modulo p by r p : E(Q) → E (p) (F p ), we have in this case a cyclic
subgroup of order 5 mapping isomorphically onto E (2) (F 2 ). In order to determine

whether E (p) has a singularity for odd primes p, we calculate the derivative y , that
2

is, (2y+1)y = 3x −2x, and look for points where it is indeterminate, i.e., 0·y = 0.

Mod 3. The tangent slope (2y + 1)y =−2x is indeterminate for (0, 1), but the
point (0, 1) is not on the curve. Now reduction modulo 3 is defined r 3 : E(Q) →
E (3) (F 3 ) and is an isomorphism of the cyclic group of order 5 generated by (0, 0)
onto the group E (3) (F 3 ).
Mod 11. The point (−3, 5) is a point on the curve E (11) which is singular so that
E has bad reduction at 11. Substituting x − 3 for x and y + 5 for y,weobtainthe
2
2
3
equation y = x + x with a node at (0, 0) over F 11 . The reduction r 11 is defined
on the cyclic group of order 5 generated by (0, 0) mapping into E (11) (F 11 ) ns which
is a cyclic group of order 10.
Exercises
1. For which rational odd primes do the following curves have bad reduction?
3
2
a) y = x + ax.
2 3
b) y = x + a.
2 3
c) y = x − 43x + 166.
2 3
d) y = x − 16 · 27x + 19 · 16 · 27.
2. For which rational odd primes do the following curves have bad reduction?
2
3
2
a) y = x + x − x.
3
2
2
b) y = x − x + x.
2
2
3
c) y = x − 2x − x.
2
3
2
d) y = x − 2x − 15x.
3. For which rational primes do the following curves have bad reduction? If the curve has
good reduction at 2, then identify the curve reduced at 2 in the list 3(6.4), and if one
has the list of curves over F 3 as asked for in 3(6, Ex.6), then identify the curve reduced
modulo 3 in the list when it has good reduction at 3.
3
2
a) y + y = x − x.
2
3
2
b) y + y = x + x .
3
2
c) y + y = x + x.