Page 143 -

120 5. Reduction mod p and Torsion Points

Exercises

3
2
2
1. Find all torsion points on E: y = x −x +x by the method using (6.1) and (6.2). Show
that the curve has good reduction modulo 5 and compare with E(F 5 ).
2
3
2
2. Find all torsion points on E: y = x − 2x − x by the method using (6.1) and (6.2).
Show that the curve has good reduction both modulo 3 and modulo 5, and compare with
E(F 3 ) and E(F 5 ).
3
2
2
3. Show that E: y = x +x −x has good reduction modulo 3 and compare E(Q) tors with
E(F 3 ).
4. Let (x, y) be a point on E(Q) defined by a normal cubic over the integers. If the slope
of the tangent line of E at (x, y) is not an integer, then show that (x, y) is not a torsion
point.
§7. Bad Reduction and Potentially Good Reduction
We continue with the notations of (1.1) and recall that an elliptic curve E over k with
discriminant has good (resp. bad) reduction at p if and only if ord p ( ) = 0 (resp.
ord p ( ) > 0). Bad reduction divides into two cases using the description of singular
cubic curves in Chapter 3, §7.

(7.1) Definition. An elliptic curve E over k has:
(1) multiplicative reduction (or semistable reduction) at p provided the reduction
E (p) has a double point (or node), or
(2) additive reduction (or unstable reduction) at p provided the reduction E (p) has a
cusp.
Multiplicative reduction is divided into split or nonsplit depending on whether or not
E (p) (k(p)) is isomorphic to the multiplicative group of k(p) or to the elements of
norm one in a quadratic extension of k(p).
(7.2) Remark. Let E be an elliptic curve E over k with discriminant and having
bad reduction at p, that is, ord p ( ) > 0or ¯ = 0. The reduction is:
(1) multiplicative reduction if and only if ord p (c 4 ) = 0 or, equivalently, ord p (b 2 ) =
0, or
(2) additive reduction if and only if ord p (c 4 )> 0 or, equivalently, ord p (b 2 )> 0.

Observe that ord p ( j(E)) is positive if E has good reduction and can be either
positive or negative if E has bad reduction since it is a quotient of elements in R.

(7.3) Remark. Let E be an elliptic curve over k with good reduction at p.Thenthe
reduction modulo p of j(E) is given by r p ( j(E)) = j(E (p) ) and ord p ( j(E)) ≥ 0.
We also have two congruences:
3
ord p ( j(E)) ≡ 0 (mod 3) and ord p ( j(E) − 12 ) ≡ 0(mod2)