Harder's conjecture and Hermitian automorphic forms

Let $k\ge4$ and $j\ge2$ be integers with $j$ even, and let $f$ be a primitive elliptic cusp form of weight $2k+j-2$ for $\mathrm{SL}_2(\mathbb{Z})$.

We study congruences between a Hermitian Klingen--Eisenstein lift associated with $f$ and Hermitian cusp forms on the quasi-split unitary group $\mathrm{U}_{2,2}$.

Under explicit arithmetic hypotheses on a congruence prime, we prove that the Hermitian cusp eigenform appearing in such a congruence is the Hermitian spin lift of a Siegel cusp eigenform of weight ${\det}^{k}\mathrm{Sym}^{j}$.

As a consequence, we obtain the spinor $L$-polynomial congruence predicted by Harder's conjecture.

The proof combines Mok's endoscopic classification, Skinner's Galois representations for unitary groups, and Selmer-group vanishing arguments.