Harder's conjecture and Hermitian automorphic forms

Let $k\ge 4$ and $j\ge 2$ be integers with $j$ even, and let $f$ be a primitive elliptic cusp form of weight $2k+j-2$ for ${\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$. 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 ${\det }^{}{\mathrm{S}\mathrm{y}\mathrm{m}}^j$. 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.