Miyawaki lift (awaiting review)

A Miyawaki Lift is a lift from elliptic modular forms to Siegel modular forms of degree 3 over the full symplectic group.

Miyawaki conjectured two types of lift. Type I, also called the Ikeda–Miyawaki lift, was established by Ikeda using a Duke-Imamoglu-Ikeda lift provided that the construction does not produce 0. Ikeda's construction is valid for degrees other than 3. Type II remains conjectural.

On the right sides of the Miyawaki lift Euler factor and $L$-function descriptions, the elliptic Euler factor is $$Q_p(f,X)=1-a_p(f)X+p^{k-1}X^2$$ with $a_p(f)$ the $p$-th Fourier coefficient of the normalized weight $k$ eigenform $f$. Let $Q_p(f,X)$ factor as $(1-\gamma_+X)(1-\gamma_-X)$; the standard elliptic Euler factor of $f$ is $Q_p^{\rm st}(f,X)=(1-X)(1-(\gamma_+/\gamma_-)X)(1-(\gamma_-/\gamma_+)X)$, and this is $$Q_p^{\rm st}(f,X)=(1-X)(1-(a_p(f)^2/p^{k-1}-2)X+X^2).$$ Also, if $g$ has weight $\ell$ and $Q_p(g,X)=(1-\delta_+X)(1-\delta_-X)$ then $Q_p(f\otimes g,X)=(1−\gamma_+\delta_+X)(1−\gamma_+\delta_−X)(1−\gamma_−\delta_+X)(1−\gamma_−\delta_−X)$, and this is $$Q_p(f\otimes g,X)=1−a_p(f)a_p(g)X+(a_p(f)^2p^{\ell−1}+a_p(g)^2p^{k−1}−2p^{k+\ell−2})X^2−a_p(f)a_p(g)p^{k+\ell−2}X^3+p^{2k+2\ell−4}X^4.$$