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 Hecke cusp form of weight $2k+j-2$ and a degree $2$ vector-valued Siegel Hecke cusp form of weight ${\det }^k{\mathrm{Sym}}^j$. In this paper, we prove this conjecture under several explicit arithmetic hypotheses on a congruence prime and an auxiliary imaginary quadratic field. We construct Hermitian spin lifts from degree $2$ Siegel cusp forms to Hermitian cusp forms on ${\mathrm{U}}_{2,2}$, and use congruences between Hermitian Klingen-Eisenstein lifts and Hermitian cusp forms to obtain the congruence predicted by Harder's conjecture.

46 pages