On the canonical decomposition of generalized modular functions
On the canonical decomposition of generalized modular functions
复制标题
关于广义模函数的正则分解
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
G. Mason
中科院分区：
文献类型：
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作者：
W. Kohnen;G. Mason
作者已经证明（引用{KoM}），如果定义在同余子群$Gamma$上的归一化广义模函数（GMF）$f$具有整数傅立叶系数，则$f$是经典的，因为某些幂$f^m$是$Gamma$上的模函数。证明了这个猜想的一个加强形式（同上）的情况下，$f$的除数是<${空}。本文研究了一个正规抛物型广义矩函数f = f_1f_0$的正则分解为一个正规抛物型广义矩函数f_1，f_0$的乘积，使得f_1 $有n个酉特征，f_0 $有n个空因子。我们表明，加强形式的猜想成立，如果第一个“少数”傅立叶系数的f_1 $是代数。我们推导证明了几个新的情况下的猜想，特别是如果要么f_0 =1$或如果因子的f$是集中在尖点的伽玛$。
The authors have conjectured (cite{KoM}) that if a normalized generalized modular function (GMF) $f$, defined on a congruence subgroup $Gamma$, has integral Fourier coefficients, then $f$ is classical in the sense that some power $f^m$ is a modular function on $Gamma$. A strengthened form of this conjecture was proved (loc cit) in case the divisor of $f$ is emph{empty}. In the present paper we study the canonical decomposition of a normalized parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1, f_0$ such that $f_1$ has emph{unitary character} and $f_0$ has emph{empty divisor}. We show that the strengthened form of the conjecture holds if the first "few" Fourier coefficients of $f_1$ are algebraic. We deduce proofs of several new cases of the conjecture, in particular if either $f_0=1$ or if the divisor of $f$ is concentrated at the cusps of $Gamma$.