Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c ( n ) denote the n th Fourier coefficient of g , normalized so that c ( n ) is real for all $$n \ge 1$$ n ≥ 1 . A theorem of Waldspurger determines the magnitude of c ( n ) at fundamental discriminants n by establishing that the square of c ( n ) is proportional to the central value of a certain L -function. The signs of the sequence c ( n ) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that $$c(n) < 0$$ c ( n ) < 0 and respectively $$c(n) > 0$$ c ( n ) > 0 holds for a positive proportion of fundamental discriminants n . Moreover we show that the sequence $$\{c(n)\}$$ { c ( n ) } where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c ( n ) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4 N with N odd, square-free.

中文翻译：

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半积分权模形式的傅里叶系数符号

设 g 是半积分权的 Hecke 尖点形式，水平 4，属于 Kohnen 的加子空间。让 c ( n ) 表示 g 的第 n 个傅立叶系数，归一化使得 c ( n ) 对于所有 $$n \ge 1$$ n ≥ 1 都是实数。Waldspurger 的定理通过确定 c ( n ) 的平方与某个 L 函数的中心值成正比来确定 c ( n ) 在基本判别式 n 处的大小。然而，序列 c ( n ) 的符号仍然是神秘的。以广义黎曼假设为条件，我们证明 $$c(n) < 0$$ c ( n ) < 0 和 $$c(n) > 0$$ c ( n ) > 0 分别适用于正比例的基本判别式 n . 此外，我们表明序列 $$\{c(n)\}$$ { c ( n ) } 其中 n 范围在基本判别式上的变化表示时间的正比例。无条件，不知道这些系数的正比例是非零的，我们证明了关于 c ( n ) 符号的结果，这些结果与最著名的非消失结果具有相同的质量。最后，我们讨论了将我们的结果扩展到具有 N 奇数、无平方的 4 N 级的一般半积分权重形式 g。