Fourier coefficients of half-integral weight modular forms modulo ell
Fourier coefficients of half-integral weight modular forms modulo ell
复制标题
半积分权模形式的傅立叶系数模 ell
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
C. Skinner
中科院分区：
文献类型：
--
作者：
K. Ono;C. Skinner
对于每个质数$\ell$，让$|\cdot|_\ell$作为$\Q$上通常的$\ell$ -adic绝对值的$\bar \Q$的扩展。假设$g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$是一个特征型，它的傅立叶系数是代数整数。在一个温和的条件下，除了有限个素数$\ell$之外，对于所有的素数，存在无限个无平方整数$m$，其中$|c(m)|_\ell = 1$。由此得到了模$L$ -函数中心临界值的“代数部分”和虚二次域的类数的不可分性结果。这些结果部分地回答了Kolyvagin关于模椭圆曲线的Tate-Shafarevich群的一个猜想。Jochnowitz早先用一种完全不同的方法得到了类似的结果。我们的方法使用关于伽罗瓦表示的标准事实附加到模形式，并愉快地揭示了$L$ -函数值的令人惊讶的kronecker风格同余。例如，如果$\Delta(z)$是拉马努金的尖点形式，$g(z)=\sum_{n=1}^{\infty}c(n)q^n$是尖点形式，$$L(\Delta_D,6)=\fracwithdelims(){\pi}{D}^6\frac{\sqrt{D}}{5!}\frac{<\Delta(z),\Delta(z)>} {}\cdot c(D)^2,$$是基本判别式$D>0,$则是 $N\geq 1$ $$\sum_{k=-\infty}^\infty c(N-k^2) \equiv \half \sum_{d|N}(\chi_{-1}(d)+\chi_{-1}(N/d))d^6 \pmod {61}. \tag{0}$$
For each prime $\ell$, let $|\cdot|_\ell$ be an extension to $\bar \Q$ of the usual $\ell$-adic absolute value on $\Q$. Suppose $g(z) = \sum_{n=0}^\infty c(n)q^n \in M_{k+\half}(N)$ is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes $\ell$ there are infinitely many square-free integers $m$ for which $|c(m)|_\ell = 1$. Consequently we obtain indivisibility results for ``algebraic parts'' of central critical values of modular $L$-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for $L$-function values. For example if $\Delta(z)$ is Ramanujan's cusp form and $g(z)=\sum_{n=1}^{\infty}c(n)q^n$ is the cusp form for which $$L(\Delta_D,6)=\fracwithdelims(){\pi}{D}^6\frac{\sqrt{D}}{5!}\frac{<\Delta(z),\Delta(z)>} {}\cdot c(D)^2,$$ for fundamental discriminants $D>0,$ then for $N\geq 1$ $$\sum_{k=-\infty}^\infty c(N-k^2) \equiv \half \sum_{d|N}(\chi_{-1}(d)+\chi_{-1}(N/d))d^6 \pmod {61}. \tag{0}$$