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An adelic approach to the Kudla-Millson lifts over totally real number fields
Abstract
We compute the Fourier expansions of the Kudla-Millson theta lifts of vector-valued Hilbert cusp forms of parallel weight. These expansions are with respect to 0-dimensional cusps of non-compact orthogonal Shimura varieties defined over any totally real number field, and computed using a mixed model of the adelic Weil representation under a partial Fourier transform. As a consequence, we obtain the injectivity of the theta lifts under some additional hypothesis on the variety. Furthermore, we show that the theta lifts are square-integrable harmonic differential forms, and deduce lower bounds for dimensions of cohomology groups of such varieties.
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BibTeX
@misc{li2026adelic,
title = {{An adelic approach to the Kudla-Millson lifts over totally real number fields}},
author = {Yingkun Li and Mingkuan Zhang and Riccardo Zuffetti},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12148},
eprint = {2609.12148},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12148},
}
RIS
TY - UNPB
TI - An adelic approach to the Kudla-Millson lifts over totally real number fields
AU - Yingkun Li
AU - Mingkuan Zhang
AU - Riccardo Zuffetti
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12148
UR - https://arxiv.org/abs/2609.12148
AB - We compute the Fourier expansions of the Kudla-Millson theta lifts of vector-valued Hilbert cusp forms of parallel weight. These expansions are with respect to 0-dimensional cusps of non-compact orthogonal Shimura varieties defined over any totally real number field, and computed using a mixed model of the adelic Weil representation under a partial Fourier transform. As a consequence, we obtain the injectivity of the theta lifts under some additional hypothesis on the variety. Furthermore, we show that the theta lifts are square-integrable harmonic differential forms, and deduce lower bounds for dimensions of cohomology groups of such varieties.
ER -