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${\mathit{P}}_{\mathit{k}}\mathrm{\left(}\mathit{j}\mathrm{\right)}\mathrm{\in }$ A $\mathrm{\left[}\mathit{j}\mathrm{\right]}\mathrm{[}\mathit{L}\mathrm{}\mathrm{:}\mathrm{}\mathit{K}\mathrm{]}\mathrm{<}\mathit{m}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{,}$ $\mathit{j}$-invariant $\mathit{j}$-invariants j-invariants $\mathrm{(}\mathit{k}\mathrm{,}\mathrm{}\mathit{K}\mathrm{)}$-lifting

$\mathrm{\setminus }\mathrm{\prime }\mathit{k}\mathrm{,}$ $\mathit{L}\mathrm{\left)}\mathrm{\text{-}}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{\right(}\mathit{k}\mathrm{,}\mathrm{}\mathit{L}\mathrm{)}\mathrm{\text{-}}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$. ${\mathit{P}}_{\mathit{k}}$ : ${\mathit{f}}_{\mathit{k}}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}{\mathit{P}}_{\mathit{k}}\mathrm{\left(}\mathit{j}\mathrm{\right(}\mathit{z}\mathrm{\left)}\mathrm{\right)}$. $\mathit{d}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathrm{=}\frac{{\mathit{q}}^{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{+}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{k}}}{\mathit{q}\mathrm{+}\mathrm{1}}$

4 $\mathit{A}\mathfrak{B}\mathfrak{B}\mathrm{\Delta }\mathit{g}\mathit{j}\mathit{L}\mathit{L}{O}\mathit{\phi }\mathrm{\Delta }\mathrm{,}$ j- $\mathit{k}$, L. $\mathrm{\Omega }\mathrm{,}$ $\mathrm{\Omega }$. ${\mathit{P}}_{\mathit{k}}{\mathit{P}}_{\mathit{k}}\mathit{q}$. $\mathit{j}\mathrm{=}\mathrm{0}\mathit{k}\mathrm{=}\mathrm{2}\mathit{k}\mathrm{\ge }\mathrm{2}\mathrm{:}{\mathit{q}}^{\mathrm{2}}\mathrm{+}\mathrm{1}$

$\mathit{j}\mathrm{\left(}\mathit{z}\mathrm{\right)}$. ${\mathit{q}}^{\mathrm{5}}\mathrm{/}\mathrm{4}$. ${\mathit{E}}_{{\mathit{q}}^{\mathit{k}}\mathrm{-}\mathrm{1}}\mathrm{,}$ $\mathit{j}\mathrm{=}{\mathit{T}}^{\mathit{q}}\mathrm{-}\mathit{T}\mathit{n}\mathrm{\le }\mathit{m}\mathrm{\left(}\mathit{k}\mathrm{\right)}\mathfrak{B}$-local $\mathit{G}\mathit{L}\mathrm{(}\mathrm{2}\mathrm{,}\mathrm{}\mathit{A}\mathrm{)}{\mathit{P}}_{\mathrm{0}}\mathrm{=}{\mathit{P}}_{\mathrm{1}}\mathrm{=}\mathrm{1}\mathrm{\{}{\mathit{j}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{j}}_{\mathit{n}}\mathrm{\}}$

$\mathrm{.}\mathrm{ċ}\mathit{K}\mathrm{(}{\mathit{j}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}{\mathit{j}}_{\mathit{n}}\mathrm{)}$ : $\mathit{K}$] $\mathrm{\ge }\mathit{n}\mathrm{!}\mathrm{/}\mathrm{2}$. ${\mathit{P}}_{\mathit{k}}\mathrm{\left(}\mathit{j}\mathrm{\right)}\mathrm{=}{\mathit{j}}^{\mathit{d}\mathrm{\left(}\mathit{k}\mathrm{\right)}}{\mathit{P}}_{\mathit{k}\mathrm{-}\mathrm{1}}\mathrm{-}\mathrm{⟨}\mathit{k}\mathrm{-}\mathrm{1}\mathrm{⟩}{\mathit{P}}_{\mathit{k}\mathrm{-}\mathrm{2}}$,

${\mathit{f}}_{\mathit{k}}\mathrm{\left(}\mathit{z}\mathrm{\right)}\mathrm{=}\mathrm{(}\mathrm{-}\mathrm{1}{\mathrm{)}}^{\mathit{k}\mathrm{-}\mathrm{1}}{{\displaystyle \prod }}_{\mathit{i}\mathrm{\le }\mathit{k}}\mathrm{⟨}\mathrm{i}\mathrm{⟩}\mathit{g}\mathrm{(}\mathit{z}{\mathrm{)}}^{\mathrm{-}\mathit{\chi }\mathrm{\left(}\mathit{k}\mathrm{\right)}}\mathrm{\Delta }\mathrm{\left(}\mathit{z}{\mathrm{)}}^{\mathrm{-}\mathit{m}\mathrm{\left(}\mathit{k}\mathrm{\right)}}{\mathit{E}}_{{\mathit{q}}^{\mathit{k}}\mathrm{-}\mathrm{1}}\mathrm{\right(}\mathit{z}\mathrm{)}$

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