Harder's conjecture and Hermitian automorphic forms

Hidenori Katsurada; Nobuki Takeda

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arXiv · 2606.30063

Harder's conjecture and Hermitian automorphic forms

Hidenori Katsurada·Nobuki Takeda

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Abstract

Let $k\geq 4$ and $j\geq 2$ 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.

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Hidenori Katsurada, Nobuki Takeda. 2026-06-29. Harder's conjecture and Hermitian automorphic forms. https://arxiv.org/abs/2606.30063

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