Elliptische Kurve/Fq/Zeta-Funktion/Funktionalgleichung/Fakt/Beweis

Beweis

Dies ergibt sich aus Fakt durch die Umformungen

{}{\begin{aligned}Z(E;{\frac {1}{qt}})&={\frac {1+(N_{1}-q-1){\frac {1}{qt}}+q\left({\frac {1}{qt}}\right)^{2}}{{\left(1-{\frac {1}{qt}}\right)}{\left(1-q{\frac {1}{qt}}\right)}}}\\&={\frac {{\left(qt\right)}^{2}+(N_{1}-q-1)qt+q}{{\left(qt-1\right)}{\left(qt-q\right)}}}\\&={\frac {q{\left(1+(N_{1}-q-1)t+qt^{2}\right)}}{q{\left(1-t\right)}{\left(1-qt\right)}}}\\&={\frac {1+(N_{1}-q-1)t+qt^{2}}{(1-t)(1-qt)}}\\&=Z(E;t).\end{aligned}}