导数周期多项式的奇数部分与对数过渡尺度
Odd Parts of Derivative Period Polynomials: Zero Geometry and a Logarithmic Transition
AI总结：
研究偶数权重$k$的归一化一级赫克本征形式$f$的周期多项式奇数部分$Q^-_{f,m}$，证明其单位圆外非零零点若存在则成实倒数四重奏，确定临界尺度$m_c(k)$，给出不同$m/m_c(k)$情形下零点分布及相关公式，结合多种方法完成证明。
AI中文摘要：
设$f$为偶数权重$k$的归一化一级赫克本征形式，$Q_{f,m}$为由其完备$L$函数的$m$阶导数的临界值形成的周期多项式。我们研究奇数部分$Q^-_{f,m}(z)=(Q_{f,m}(z)-Q_{f,m}(-z))/2$，保留因奇数性而在原点处的零点。全多项式的单位圆定理无法解决此问题：即使原多项式的所有零点都在单位圆上，取奇数部分也可能产生单位圆外的零点。我们证明存在一个绝对的$K_0$，使得对于每个偶数$k\geq K_0$，每个权重为$k$的归一化一级赫克本征形式$f$，以及每个整数$m\geq0$，$Q^-_{f,m}$在单位圆外的非零零点（若有）形成单个实倒数四重奏$\{\pm b,\pm b^{-1}\}$，其中$0\lt b\lt1$。对于每个固定权重，一旦$m$足够大，所有非零零点都是简单的且位于单位圆上。因此，实圆或单位圆包含性的任何失败都局限于有限多个权重 - 导数对。我们还确定了可能的四重奏的大权重过渡。其临界尺度为$m_c(k)=(k - 1)\log((k - 1)/\pi)$。如果$m/m_c(k)\to\theta\in(0,1)$，恰好出现一个四重奏且其内部正零点趋于$(1 + \theta)/2$；如果$\theta\gt1$，每个非零零点都是简单的且位于单位圆上。在临界比率$\theta = 1$时，相同的实圆或单位圆包含性仍然有效。更准确地说，如果$|m - m_c(k)|/\log k\to\infty$，$m - m_c(k)$的符号决定相位。我们还获得了在解析预临界侧和正导数阶数$m = O(\log k)$时四重奏的一阶公式。证明结合了精确的奇数自反完备、边界敏感的缠绕数计数以及对分裂梅林积分的一致鞍点估计。
英文摘要：
Let $f$ be a normalized level-one Hecke eigenform of even weight $k$, and let $Q_{f,m}$ be the derivative period polynomial formed from the critical values of the $m$-th derivative of its completed $L$-function. We study its odd part $Q^-_{f,m}(z)=(Q_{f,m}(z)-Q_{f,m}(-z))/2$. We prove that there is an absolute $K_0$ such that, for every even $k\ge K_0$, every normalized level-one Hecke eigenform $f$ of weight $k$, and every integer $m\ge0$, the nonzero zeros of $Q^-_{f,m}$ off the unit circle, if any, consist of four simple zeros forming a single real reciprocal quartet $\{\pm b,\pm b^{-1}\}$, where $0<b<1$. The occurrence and location of this possible quartet are governed by the critical derivative order $m_c(k)=(k-1)\log((k-1)/π)$. If $m/m_c(k)\toθ\in(0,1)$, exactly one quartet occurs and $b\to(1+θ)/2$; if $θ>1$, every nonzero zero is eventually simple and lies on the unit circle. At $θ=1$ the same real-or-unit-circle containment remains valid, and any quartet that is present consists of four simple zeros. For each fixed weight, all nonzero zeros are eventually simple and lie on the unit circle as $m\to\infty$. Consequently, the Diamantis--Rolen containment conjecture holds outside finitely many weight--derivative pairs. The proof combines an exact signed-reciprocal completion, uniform split-Mellin saddle estimates yielding a moving-sine model, and a winding count that transfers disk-zero information to the unit circle.
