An extension of Hecke's converse theorem
An extension of Hecke's converse theorem
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赫克逆定理的推广
DOI：
10.1155/s1073792895000328
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
D. Farmer
中科院分区：
文献类型：
--
作者：
J. Conrey;D. Farmer
与新形式$f(z)$相关联的是具有泛函方程和欧拉积的Dirichlet级数$L_f(s)$。Hecke证明了如果Dirichlet级数$F(s)$具有适当形式的泛函方程，那么对于$\Gamma(1)$上的全纯新形式$F(z)$，则$F(s)=L_f(s)$。Weil将这一结果推广到$\Gamma_0(N)$，这是基于狄利克雷字符对$F(s)$的扭曲的假设。我们证明，至少对于小的$N$，关于扭转的假设可以用关于$F(s)$欧拉积的局部因子的假设来代替。
Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of the appropriate form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $\Gamma(1)$. Weil extended this result to $\Gamma_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. We show that, at least for small $N$, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of $F(s)$.