论文标题
尖端形式的傅立叶系数在规范形式参数下的符号变化
Sign Changes of Fourier Coefficients of Cusp Forms at Norm Form Arguments
论文作者
论文摘要
令$ f $为1级和固定重量的非CM Hecke eigencusp形式，让$ \ {λ_f（n）\} _ n $为其标准化的傅立叶系数的顺序。我们表明，如果$ k/ \ mathbb {q} $是任何数字字段，而$ \ m natercal {n} _k $表示代表的整数的集合，则表示为$ k $的整体理想规范，那么正整数的正整数$ n \ in \ nathcal in \ nathcal {n} _k $ n} _k $ n} _k $ sequence a seme a vardence a sem vardence a sequence a sequence a sember a sem vardence a sem sequence a sequence a sem vardence a sem vardence a n} _k $ n} $ \ {λ_f（n）\} _ {n \ in \ mathcal {n} _k} $。更准确地说，对于$ n \ in \ mathcal {n} _k \ cap [1，x] $的正比例，我们有$λ_f（n）λ_f（n）λ_f（n'）<0 $，其中$ n'$是$ \ Mathcal {n} _k $ n $ n $ n $ n $ ne $λ_f（n'q neq 0的$ \ natcal {n} _k $ n $ ne）的第一元素。 例如，对于$ k = \ mathbb {q}（i）$和$ \ mathcal {n} _k = \ {m^2+n^2：m，n \ in \ mathbb {z} \} $，我们获得了两个正方形的总和，我们获得了$ \ f gg_f x/pogright x/sign undign -wins undign congrion pog x/pogrognofe。隐含的常数）并改善了Banerjee和Pandey的工作。我们的证明依赖于Matomäki和Radziwiłłparss稀少的乘法功能的最新工作，以及由于作者而对其结果进行了一些技术改进。 在相关的静脉中，我们还考虑了两个正方形的符号变化的问题，为此，乘法技术不直接应用。使用偏换卷积总和和其他技术的估计值，我们确定对于任何固定的$ a \ neq 0 $，都有$ \ gg_ {f，ε} x^{1/2-} $符号更改$λ_f$的更改沿形式$ a + m a + m^2 + n^2 \ leq leq x $的整数序列。
Let $f$ be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{λ_f(n)\}_n$ be its sequence of normalized Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of $K$, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{λ_f(n)\}_{n \in \mathcal{N}_K}$. More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $λ_f(n)λ_f(n') < 0$ where $n'$ is the first element of $\mathcal{N}_K$ greater than $n$ for which $λ_f(n') \neq 0$. For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 : m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author. In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,ε} X^{1/2-ε}$ sign changes for $λ_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$.
