Local Factorization of p-adic Gamma Sums

Abstract

We revisit the proposed equality between discrete Fourier transforms of $p$-adic $\Gamma_p$--values and $p$-adic $L$--derivatives for odd characters modulo a prime $p$. The clean identity is false in general. Building on Coleman reciprocity and the Gross--Koblitz formula, we prove an exact two-term decomposition: for each odd, nontrivial Dirichlet character $\chi \pmod p$, \[ \Phi_p (\chi):=\sum_{a=1}^{p-1}\chi(a)\,\log_p \Gamma_p \!\left(\frac{a}{p-1}\right) = U_{1,p}\,L'_p(0,\chi)\;+\;U_{2,p}\,L(0,\chi), \] with constants $U_{1,p}\in\mathbb{Q}_p(\mu_{p-1})^\times$ and $U_{2,p}\in\mathbb{Q}_p(\mu_{p-1})$ depending only on $p$ and the fixed branch of $\log_p$, but independent of $\chi$. Subtracting the $L(0,\chi)$--block yields a \emph{renormalized} local input \[ \Phi^{ren}_p(\chi):=\Phi_p(\chi)-U_{2,p}L(0,\chi)=U_{1,p}\,L'_p(0,\chi), \] uniformly in odd, nontrivial $\chi$. Plumbing these renormalized locals at every finite place into the Weil explicit formula (with the standard Li kernel at $\infty$) reproduces exactly the classical Li coefficients. We also record a short, reproducible verification protocol; a tiny table for $p=5,7$ illustrates the $\chi$--independence of $(U_{1,p},U_{2,p})$.