Harder's conjecture and Hermitian automorphic forms
Abstract
Let k≥4 and j≥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 detkSymj. 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 U2,2. 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.
Related
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- Construction of vector-valued adelic automorphic forms
- Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
- K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
- Conformal field theory from automorphic forms on a quantum group
BibTeX
@misc{katsurada2026harders,
title = {{Harder's conjecture and Hermitian automorphic forms}},
author = {Hidenori Katsurada and Nobuki Takeda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.30063},
eprint = {2606.30063},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.30063},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Harder's conjecture and Hermitian automorphic forms
AU - Hidenori Katsurada
AU - Nobuki Takeda
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2606.30063
UR - https://arxiv.org/abs/2606.30063
AB - Let k≥4 and j≥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^kSym^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 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.
ER -