Elliptische Kurven/Morphismen/Summe/Explizit/Fakt/Beweis

Beweis

Nach Fakt ist

${}{\begin{aligned}\theta _{1}(x,y)+\theta _{2}(x,y)&=\left(f_{1}+g_{1}y,\,p_{1}+q_{1}y\right)+\left(f_{2}+g_{2}y,\,p_{2}+q_{2}y\right)\\&=\left(\alpha ^{2}-(f_{1}+f_{2})-(g_{1}+g_{2})y,\,-\alpha ^{3}+\alpha (2f_{1}+f_{2}+(2g_{1}+g_{2})y)-p_{1}-q_{1}y\right)\\&=\left(f+gy,\,p+qy\right)\end{aligned}}$

mit

${}{\begin{aligned}\alpha &={\frac {p_{2}-p_{1}+(q_{2}-q_{1})y}{f_{2}-f_{1}+(g_{2}-g_{1})y}}\\&={\frac {{\left(p_{2}-p_{1}+(q_{2}-q_{1})y\right)}{\left(f_{2}-f_{1}-(g_{2}-g_{1})y\right)}}{{\left(f_{2}-f_{1}+(g_{2}-g_{1})y\right)}{\left(f_{2}-f_{1}-(g_{2}-g_{1})y\right)}}}\\&={\frac {(p_{2}-p_{1})(f_{2}-f_{1})+(q_{2}-q_{1})(g_{2}-g_{1})(x^{3}+ax+b)+((p_{2}-p_{1})(g_{2}-g_{1})+(q_{2}-q_{1})(f_{2}-f_{1}))y}{(f_{2}-f_{1})^{2}-(g_{2}-g_{1})^{2}(x^{3}+ax+b)}}.\end{aligned}}$

Somit ist

${}{\begin{aligned}f&={\frac {((p_{2}-p_{1})(f_{2}-f_{1})+(q_{2}-q_{1})(g_{2}-g_{1})(x^{3}+ax+b))^{2}+((p_{2}-p_{1})(g_{2}-g_{1})+(q_{2}-q_{1})(f_{2}-f_{1}))^{2}(x^{3}+ax+b)}{((f_{2}-f_{1})^{2}-(g_{2}-g_{1})^{2}(x^{3}+ax+b))^{2}}}-(f_{1}+f_{2})\\\end{aligned}}$

Zur bewiesenen Aussage