p-adic Aspects of Jacobi Forms
p-adic Aspects of Jacobi Forms
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雅可比形式的 p-adic 方面
DOI：
10.1006/jnth.1997.2095
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发表时间：
1997
影响因子：
0.7
通讯作者：
A. Sofer
中科院分区：
文献类型：
--
作者：
A. Sofer
我们有兴趣理解和描述雅可比形式的 p-adic 属性。与模块化形式的情况相反，在这方面还没有做太多工作。文献包括[3,4,7]。在第一部分中，我们从 Serre 的 p 进模形式理论中遵循 Serre 的思想。我们研究雅可比形式，其傅里叶展开式具有积分系数，并研究它们之间的同余。雅可比-爱森斯坦级数给出了重要的例子。事实证明，两个雅可比形式需要具有相同的指数并满足权重条件才能全等。如果我们在这种情况下以自然的方式定义 p 进雅可比形式，并将自己限制在 SL2(Z) 的情况，我们就得到了给定权重 χ ∈ Z ′ p 和索引 m ∈ Z 的 SL2(Z) 的 p 进雅可比形式空间的结构定理。另一个特征是 Γ0(p) 的 p 进雅可比形式也是 SL2(Z) 的形式。这与模块化形式的类似结果相似，并且它很可能在定义一些不是直接由复杂运算符产生的 p 进运算符时发挥重要作用。在第二部分中，我们将每个具有积分系数的雅可比形式与 Zp 上的度量与卡茨广义模形式的 p 进环中的值相关联。这是一种注入，使我们能够将具有 p-adic 系数的雅可比形式解释为真正的 p-adic 对象，这表明在哪里为模块化 p-adic 理论寻找足够的“测试对象”。它还提供了模形式的 p-adic 分析族的示例。
We are interested in understanding and describing the p-adic properties of Jacobi forms. As opposed to the case of modular forms, not much work has been done in this area. The literature includes [3, 4, 7]. In the first section, we follow Serre’s ideas from his theory of p-adic modular forms. We study Jacobi forms whose Fourier expansions have integral coefficients and look at congruences between them. Non-trivial examples are given by Jacobi-Eisenstein series. It turns out that two Jacobi forms need to have the same index and satisfy a condition on the weights in order to be congruent. If we define p-adic Jacobi forms in the natural way in this context, and restrict ourselves to the case of SL2(Z), we obtain a structure theorem for the space of p-adic Jacobi forms for SL2(Z) of a given weight χ ∈ Z ′ p and index m ∈ Z. Another feature is that p-adic Jacobi forms for Γ0(p) are also forms for SL2(Z). This parallels the similar result for modular forms, and it will most probably play an important role in defining some p-adic operators that do not arise directly from complex operators. In the second section, we associate to every Jacobi form with integral coefficients a measure on Zp with values in the p-adic ring of Katz’s generalized modular forms. This is an injection that allows us to interpret Jacobi forms with p-adic coefficients as truly p-adic objects, and this suggests where to look for the adequate “test objects” for a modular p-adic theory. It also provides examples of p-adic analytic families of modular forms.