Reduction mod p and Torsion Points

Abstract

Reduction modulo p defined Z → Z/p Z = F _p is a fundamental construction for studying equations in arithmetic. Another basic advantage of projective space over affine space is that the entire rational projective space can be reduced modulo p, P _h (Q) → P _n (F _p ), in such a way that rational curves (curves defined 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.