= Good-reduction map respects elliptic addition
{title2=$\widetilde{P+Q}=\widetilde P+\widetilde Q$}
An integral <Weierstrass equation of an elliptic curve> with unit <elliptic-curve discriminant> defines a smooth proper elliptic curve over the <valuation ring>. Properness identifies points over the <local field> with integral sections. The group law, constructed by adding degree-zero divisor classes, is a morphism over the valuation ring, and base change to the <residue field> preserves it. Hence reducing primitive projective coordinates is a <group homomorphism>, including exceptional affine addition cases.
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