一种关于全实代数数域上Kudla-Millson提升的adele方法
An adelic approach to the Kudla-Millson lifts over totally real number fields
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- Institute of Advanced Studies（高等研究院）
- Technische Universität Darmstadt（达姆施塔特工业大学）
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中文总结 AI 辅助
本文通过adele方法计算全实域上正交Shimura簇的Kudla-Millson theta提升的Fourier展开，证明其单射性并给出上同调维数下界。
中文摘要 AI 辅助
我们计算了平行权重的向量值Hilbert尖形式在Kudla-Millson theta提升下的Fourier展开。这些展开是关于定义在任意全实代数数域上的非紧正交Shimura簇的0维尖点，并利用部分Fourier变换下的adele Weil表示混合模型进行计算。作为推论，在簇的某些附加假设下，我们获得了theta提升的单射性。此外，我们证明了theta提升是平方可积的调和微分形式，并推导了此类簇的上同调群维数的下界。
英文摘要
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.
