**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 $g$.

**Authors:**

**Knowl status:**

- Review status: beta
- Last edited by Jerry Shurman on 2016-03-29 10:18:56

**Referred to by:**

**History:**(expand/hide all)