Kurokawa–Mizumoto congruence for Klingen–Eisenstein series

Let $k$ and $\nu$ be even integers with $k>4$. Let

$$f(z)=\sum_{n>0}a(n,f)\mathbf{e}(nz)\in S_{k+\nu}(\Gamma_1)$$

be a normalized Hecke eigenform, meaning that $a(1,f)=1$. Let $L(k-1,f,\mathrm{St})$ denote the relevant standard $L$-function, and suppose that a sufficiently large prime $\mathfrak{p}$ of $\mathbb{Q}$ divides this special value. Let $[f]^{(k+\nu,k)}$ be the Klingen–Eisenstein series associated with $f$, and let $\equiv_{ev}$ denote congruence of Hecke eigenvalues. Kurokawa–Mizumoto congruence. There exist a Hecke eigenform $F\in S_{(k+\nu,k)}(\Gamma_2)$ and a prime ideal $\mathfrak{p}'\mid\mathfrak{p}$ in $\mathbb{Q}(F)$ such that

$$F\equiv_{ev}[f]^{(k+\nu,k)}\pmod{\mathfrak{p}'}.$$

In particular, for every prime $p$,

$$\lambda_F(p)\equiv_{ev}(1+p^{k-2})\lambda_f(p)\pmod{\mathfrak{p}'}.$$

The conjecture predicts that divisibility of the relevant special $L$-value produces a cuspidal Siegel eigenform congruent to the Klingen–Eisenstein series, with the displayed relation between Hecke eigenvalues. The surrounding discussion records the corresponding eigenvalue formulas for the Klingen–Eisenstein component, but does not establish this congruence in general.