= Reduction exponent obstruction to integral multiples
{title2=$N\widetilde E(k)=0\ \Longrightarrow\ NP\in E_1(K)$}
At a prime of <good reduction of an elliptic curve>, the reduction map is a group homomorphism on all local points. If $N$ annihilates the reduced finite group, every $NP$ reduces to the identity at infinity. A finite point with integral affine coordinates reduces to an affine point, so a nonidentity $NP$ cannot have integral affine coordinates. The identity has no affine coordinates and must be treated separately.
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