**Miyawaki Lift, Type II (conjectural):** Let $k$ be an even weight.
For each pair of elliptic Hecke cusp eigenforms
$$f\in{\mathcal S}_{2k−2}(\SL(2,{\mathbb Z}))
\quad\text{quad}\quad
g\in{\mathcal S}_{k−2}(\SL(2,{\mathbb Z}))$$
their Miyawaki lift of type II is a Siegel Hecke cusp eigenform
$$F\in{\mathcal S}_k(\Sp(6,{\mathbb Z}))$$
whose standard Euler factors and standard $L$-function factor as
$$Q_p^{\rm st}(F,X)=Q_p(f,p^{1−k}X)Q_p(f,p^{2−k}X)Q_p^{\rm st}(g,X)$$
and $$L^{\rm st}(F,s)=L(f,s+k−1)L(f,s+k−2)L^{\rm st}(g,s)$$
and whose spinor Euler factors and spinor L-function factor as
$$Q_p^{\rm spin}(F,X)=Q_p(g,p^{k−1}X)Q_p(g,p^{k−2}X)Q_p(f\otimes g,X)$$
and $$L^{\rm spin}(F,s)=L(g,s−k+1)L(g,s−k+2)L(f\otimes g,s).$$

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**Knowl status:**

- Review status: beta
- Last edited by Jerry Shurman on 2016-03-29 10:21:13

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