Miyawaki lift of type I, Ikeda-Miyawaki lift (awaiting review)

Genus 3 Miyawaki Lift, Type I: Let $k$ be an even weight. Each pair of elliptic Hecke cusp eigenforms $$f\in{\mathcal S}_{2k-4}(\SL(2,{\mathbb Z})) \quad\text{and}\quad g\in{\mathcal S}_k(\SL(2,{\mathbb Z}))$$ has an Ikeda-Miyawaki lift, or Miyawaki lift of type I, a Siegel Hecke cusp eigenform $$F\in{\mathcal S}_k(\Sp(6,{\mathbb Z}))$$ If $F$ is nonzero then its standard Euler factors and standard $L$-function factor as $$Q_p^{\rm st}(F,X)=Q_p(f,p^{2−k}X)Q_p(f,p^{3−k}X)Q_p^{\rm st}(g,X)$$ and $$L^{\rm st}(F,s)=L(f,s+k−2)L(f,s+k−3)L^{\rm st}(g,s)$$ and its spinor Euler factors and spinor $L$-function factor as $$Q_p^{\rm spin}(F,X)=Q_p(g,p^{k−2}X)Q_p(g,p^{k−3}X)Q_p(f\otimes g,X)$$ and $$L^{\rm spin}(F,s)=L(g,s−k+2)L(g,s−k+3)L(f\otimes g,s).$$

More generally, Ikeda proved the following.

Theorem (Ikeda-Miyawaki Lift, Type I) Let $k$, $n+r$, and $m=(n+r+k)/2$ be all even integers with $r\le n$. Given: an elliptic cusp eigenforms, $f$ of weight $k$ and a cusp eigenform $g$ in degree $r$ of weight $2m$, there exists an eigenform in degree $n$ of weight $m$ such that if nonzero it has an $L$-function that is a product of $L$-functions related to those of $f$ and $g$.