The Theory of Vector-Valued Modular Forms for the Modular Group
The Theory of Vector-Valued Modular Forms for the Modular Group
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模群的向量值模形式理论
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
T. Gannon
中科院分区：
文献类型：
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作者：
T. Gannon
我们解释的基本思想，描述与证明的主要结果，并证明其有效性，一个不断发展的理论向量值模形式（vvmf）。为了使论述具体化，我们在这里仅限于模群的特殊情况。除此之外，我们构造vvmf的任意乘数，解决了Mittag-Leffler问题，建立塞尔对偶，并找到一个尺寸公式全纯vvmf，所有在更大的普遍性比其他地方已经做了。更重要的是，所涉及的新思想是足够简单和强大的，这整个理论可以直接扩展到任何亏格0的Fuchsian群。
We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things, we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.