论文标题
高层次曲线和新形式系数
Hyperelliptic Curves and Newform Coefficients
论文作者
论文摘要
我们研究哪些整数可以作为整数重量新形式的傅立叶系数。在tau功能的具体情况下，我们表明，对于所有奇数$ \ ell <100 $和所有整数$ m \ geq 1 $，我们有$$τ（n）\ neq \ pm \ ell，\ pm 5^m。 $$对于一般新形式$ f $，甚至整数重量$ 2K $和整数系数，我们证明大多数整数$ j $ divine $ 2k-1 $，所有普通的普通素质$ p $ the $ a_f（p^2）$从来都不是$ j $ - the $ j $ -th power。我们证明了$ a_f（p^4）$的类似结果，这是在弗雷·马祖（Frey-Mazur）猜想上的。我们的主要方法涉及将有关新形式价值的问题与某些二进制复发序列中的完美能力的存在联系起来，并利用对数中线性形式理论的界限。该方法毫不费力地扩展到具有固定指数的大型Lebesgue-Nagell方程。为了证明一般新形式的结果，我们还利用模块化方法和Ribet的降级定理。
We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 100$ and all integers $m \geq 1$, we have $$ τ(n) \neq \pm \ell, \pm 5^m. $$ For general newforms $f$ with even integer weight $2k$ and integer coefficients, we prove for most integers $j$ dividing $2k-1$ and all ordinary primes $p$ that $a_f(p^2)$ is never a $j$-th power. We prove a similar result for $a_f(p^4)$, conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.
