Harder's conjecture and Hermitian automorphic forms

Abstract

Let $k\ge 4$ and $j\ge 2$ be integers with $j$ even. Harder's conjecture predicts a congruence between a primitive elliptic cusp form of weight $2k+j-2$ and a degree-two vector-valued Siegel Hecke cusp eigenform of weight ${\det }^k{\mathrm{Sym}}^j$. We prove the corresponding Harder-type congruence for the spinor $L$-polynomials under explicit arithmetic hypotheses on a congruence prime and an auxiliary imaginary quadratic field. Our approach uses Hermitian automorphic forms on the quasi-split unitary group ${\mathrm{U}}_{2,2}$. We construct Hermitian spin lifts of Siegel cusp forms and show that the Hermitian cusp form produced by a Hermitian Klingen-Eisenstein congruence lies in the image of this lift. The key step is to prove that the Galois representation attached to this Hermitian cusp form is conjugate invariant. For this, we combine the endoscopic classification for quasi-split unitary groups with Selmer-group vanishing results. This allows us to identify the Hermitian cusp form as the spin lift of a Siegel cusp eigenform and thereby obtain the Harder-type congruence. The argument applies uniformly to both even and odd $k$.

47 pages