论文标题
Lehmer对Ramanujan tau功能的猜想的变化
Variations of Lehmer's Conjecture for Ramanujan's tau-function
论文作者
论文摘要
我们认为Lehmer尚未解决的猜想的自然变体，Ramanujan的Tau功能永不消失。也就是说，对于$ n> 1 $，我们证明$τ（n）\ in \ {\ pm 1，\ pm 3，\ pm 5，\ pm 5，\ pm 7，\ pm 691 \}。$ $ $ $ $ $这个结果是与trivial mod 2居住的新形式的典型示例，该示例具有琐碎的mod 2居住式持续的代表，并将与以下的工作一起出现在wee中。 Ramanujan著名的一致性$τ（N）$允许在这些特殊情况下简化证明。我们利用卢卡斯序列理论，用于纤维化曲线的chabauty-coleman方法以及有关某些thue方程的事实。
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for $n>1$ we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \pm 7, \pm 691\}.$$ This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for $τ(n)$ allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.
