Zeros of certain modular functions and an application
Zeros of certain modular functions and an application
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某些模函数的零点及其应用
DOI：
10.1007/bf01209019
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发表时间：
1997
影响因子：
2.4
通讯作者：
Hirohito Ninomiya
中科院分区：
文献类型：
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作者：
T. Asai;M. Kaneko;Hirohito Ninomiya
根据F.K.C. Rankin和H.P.F. Swinnerton-Dyer[8]的研究，SL2(Z)上任意权值k的爱森斯坦级数Ek（τ）在H上的零点处j（τ）的值总是在区间[0,1728]内，或者等价地，标准基域上Ek（τ）的所有零点都在单位圆上。这一结果由R.A. Rankin[7]推广到某些庞加莱级数。此外，D. Zagier和Kaneko[5]在联合论文中研究的与超奇异椭圆曲线的j不变量密切相关的Atkin正交多项式的零和某种“超几何模形式”的零具有相同的性质。（这里我们提到爱森斯坦级数也与超奇异j不变量[9]有关。）我们的定理1提供了另一个例子，这个看似奇怪的，还没有完全理解的零的性质。
As has been known since the work of F.K.C. Rankin and H.P.F. Swinnerton-Dyer [8], the values of j(τ) at the zeros in H of the Eisenstein series Ek(τ) of any weight k on SL2(Z) always lie in the interval [0, 1728], or equivalently, all the zeros of Ek(τ) in the standard fundamental domain lie on the unit circle. This result was generalized by R.A. Rankin [7] to certain Poincare series. Furthermore, the zeros of Atkin’s orthogonal polynomials, as well as of certain “hypergeometric modular form”, both of which are studied in a joint paper by D. Zagier and Kaneko [5] and have an intimate connection to the j-invariants of supersingular elliptic curves, have the same property. (Here we mention that the Eisenstein series is also related to the supersingular j-invariants [9].) Our Theorem 1 supplies another example with this seemingly peculiar, and not yet fully understood property of zeros.