Odd Parts of Derivative Period Polynomials and a Logarithmic Transition Scale

Abstract

Let $f$ be a normalized level-one Hecke eigenform of even weight $k$, and let $Q_{f,m}$ be the period polynomial formed from the critical values of the $m$-th derivative of its completed $L$-function. We study the odd part $Q^-_{f,m}(z)=(Q_{f,m}(z)-Q_{f,m}(-z))/2$, retaining the zero at the origin forced by oddness. A unit-circle theorem for the full polynomial does not settle this problem: taking an odd part can create off-circle zeros even when the original polynomial has all of its zeros on the unit circle. 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, form a single real reciprocal quartet $\{\pm b,\pm b^{-1}\}$ with $0<b<1$. For each fixed weight, all nonzero zeros are simple and lie on the unit circle once $m$ is sufficiently large. Hence any failure of the real-or-unit-circle containment is confined to finitely many weight--derivative pairs. We also determine the large-weight transition of the possible quartet. Its critical scale is $m_c(k)=(k-1)\log((k-1)/π)$. If $m/m_c(k)\toθ\in(0,1)$, exactly one quartet occurs and its inner positive zero tends to $(1+θ)/2$; if $θ>1$, every nonzero zero is simple and lies on the unit circle. At the critical ratio $θ=1$, the same real-or-unit-circle containment remains valid. More precisely, if $|m-m_c(k)|/\log k\to\infty$, the sign of $m-m_c(k)$ determines the phase. We also obtain first-order formulas for the quartet on the resolved pre-critical side and for positive derivative orders $m=O(\log k)$. The proof combines an exact odd self-inversive completion, a boundary-sensitive winding count, and uniform saddle estimates for a split Mellin integral.