论与θ级数全等的模数形式的模数p零点

IF 0.6 3区 数学 Q3 MATHEMATICS
Berend Ringeling
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引用次数: 0

摘要

对于大于 7 的素数 p,权重为 p-1 的爱森斯坦数列具有一些显著的同调性质,例如,这些性质意味着其零点(已知为区间 [0,1728] 中的实代数数)的 j 不变式在具有 p 元素的域上最多是二次,并且与某个截断超几何数列的零点同调。在本文中,我们引入了权重 k≥4 的全模态群的 "θ模态",即第一个 dim(Mk) 傅里叶系数与某些θ级数相同的模态。我们考虑了雅可比θ级数和六方格的θ级数的这些θ模形式。我们证明,雅可比θ级数的θ模形式零点的 j 不变性都是在具有 p 元素的基域中模数为 p 的。对于六边形网格的θ模形式,我们证明其零点在有 p 个元素的基域中最多是二次。此外,我们还证明了这两种情况下的这些零点都与某些截断超几何函数的零点相等。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the modulo p zeros of modular forms congruent to theta series

For a prime p larger than 7, the Eisenstein series of weight p1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight k4 for the full modular group as the modular forms for which the first dim(Mk) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

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来源期刊
Journal of Number Theory
Journal of Number Theory 数学-数学
CiteScore
1.30
自引率
14.30%
发文量
122
审稿时长
16 weeks
期刊介绍: The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field. The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory. Starting in May 2019, JNT will have a new format with 3 sections: JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access. JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions. Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.
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