论文标题
在确定的四元组代数上的模块化形式的曲折周期的分布
Distribution of toric periods of modular forms on definite quaternion algebras
论文作者
论文摘要
令$ d $是$ \ mathbb {q} $的确定的四元素代数，而$ \ mathcal {o} $在$ d $ d $的无方面级别中的eichler订单。我们研究代数模块化级别$ \ MATHCAL {O} $的分布。我们专注于两个问题：非拼接和签名变化。首先，在$ \ Mathcal {o} $的某些条件下，我们证明了对假想的二次字段的积极比例的福利时期的不变。在某些情况下，这改善了对戈德菲尔德的猜想的已知下限，并为扭曲自动形态$ l $ functions的中心价值的类似非施加猜想提供了证据。其次，我们表明，复曲周期的序列具有无限的许多符号变化。这证明了傅立叶系数的符号更改$ \ {a（n）\} _ n $ 3/2模块化表单，其中$ n $的范围比基本歧视剂范围。在最后一部分中，我们在某些情况下介绍了数值实验，并根据它们制定了几种猜想。
Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.
