Odd Parts of Derivative Period Polynomials: Zero Geometry and a Logarithmic Transition

Abstract

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 $m\ge 0$, 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 $m_c(k)=(k-1)\log ((k-1)/\pi )$. If $m/m_c(k)\to \theta \in (0,1)$, exactly one quartet occurs and $b\to (1+\theta )/2$; if $\theta >1$, every nonzero zero is eventually simple and lies on the unit circle. At $\theta =1$ 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.

53 pages. Revised title and abstract; substantial revisions to exposition and proof organization. Theorem statements now explicitly record nonvanishing and simplicity; critical-window and endpoint arguments are clarified