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5

Reduction mod p and Torsion Points

The reduction modulo p morphism Z → Z/pZ = F p is a fundamental construc-
tion for studying equations in arithmetic. A basic advantage of projective space over
affine space is that the entire rational projective space can be reduced modulo p,
yielding a map P n (Q) → P n (F p ), in such a way that rational curves (curves de-
fined over Q) and their intersection points also reduce modulo p. The first task is to
study when the reduced curve is again smooth and when intersection multiplicities
are preserved. This is an extension of the ideas in Chapter 2 to arithmetic.
Turning to the special case of an elliptic curve E, we need to choose a “mini-
mal” cubic equation for E. These have the best possible reduction properties, and
reduction is a group homomorphism when the curve E reduces to a nonsingular E p
modulo p.Wesay E has a good reduction at p in this case, and we look for criterions
for good reduction.
Finally we turn to the global arithmetic of elliptic curves, and prove the Nagell–
Lutz theorem which says that for the reduction at p homomorphism E(Q) →
E p (F p ) the restriction to

Tors(E(Q)) → E p (F p )

is injective for p odd and has kernel at most of order 2 at 2. Combining this arithmetic
study with the structure of the group of real points E(R), we deduce that the torsion
subgroup Tors(E(Q)) of an elliptic curve E over the rational numbers is either finite
cyclic or finite cyclic direct sum with the group of order 2. Moreover, it is effectively
computable.

§1. Reduction mod p of Projective Space and Curves

(1.1) Notations. We will use the following notation in the next three sections. Let R
be a factorial ring with field of fractions k. For each irreducible p in R we form the
quotient ring R/p = R/Rp and denote its field of fractions by k(p). Each element
a in k can be decomposed as a quotient