Modular forms and period polynomials
Modular forms and period polynomials
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DOI：
10.1112/plms/pdt003
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发表时间：
2013-10-01
影响因子：
1.8
通讯作者：
Popa, Alexandru A.
中科院分区：
文献类型：
--
作者：
Pasol, Vicentiu;Popa, Alexandru A.
研究了模群的有限指数子群的整权模形式所对应的周期多项式空间。对于模群，这个空间被赋予了一个配对，对应于模形式上的彼得森内积，通过哈伯兰公式，并具有由Zagier代数定义的Hecke算子作用。我们推广Haberland的公式（不一定尖）有限指数子群的模形式，我们发现，它隐藏了两个更强的公式。本文将Hecke算子的作用推广到模形式的周期多项式上，证明了Haberland公式中周期多项式的配对是非退化的，并确定了Hecke算子关于它的伴随，给出了Gamma（1）（N）的几个应用：Eichler-Shimura同构推广到整个模形式空间;确定与尖点形式相关的周期多项式的奇偶部分所满足的关系，这些关系与周期关系无关;以及Hecke本征形的Fourier系数的显式公式，推广了Manin的系数定理。
We study the space of period polynomials associated with modular forms of integral weight for finite-index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite-index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for Gamma(1)(N): an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficient theorem of Manin.