A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms
A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms
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模形式零点上 Rankin 和 Swinnerton-Dyer 定理的推广
DOI：
10.1090/s0002-9939-04-07478-7
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发表时间：
2004
期刊：
影响因子：
13.3
通讯作者：
Jayce R. Getz
中科院分区：
文献类型：
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作者：
Jayce R. Getz
兰金和Swinnerton-Dyer（1970）证明了Eisenstein级数Ek在标准基本域中的所有零点都位于A：= {ei θ：π/2 ≤ θ ≤ 2π/3）上。本文推广了他们的定理，给出了其它模形式的零点仅位于弧A上的条件。利用这一结果，我们证明了Ono的一个猜想，即M k中唯一间隙函数的零点，即在其q-展开式中具有最大数目的连续零系数的模形式，只在A上有零点.此外，我们还证明了j-不变量映射这些零到完全真实的代数整数的程度有界的一个简单的函数的重量k。
Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series E k in the standard fundamental domain for Γ lie on A:= {e iθ : π/2 ≤ θ ≤ 2π/3). In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique gap function in M k , the modular form with the maximal number of consecutive zero coefficients in its q-expansion following the constant 1, has zeros only on A. In addition, we show that the j-invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight k.