Local Factorization of p-adic Gamma Sums

Abstract

We revisit the proposed equality between discrete Fourier transforms of $p$-adic $\hat{I}{“}_p$--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 $\ddot{I}‡(\mathrm{mod}p)$, \[ Φ_p (χ):=\sum_{a=1}^{p-1}χ(a)\,\log_p Γ_p \!\left(\frac{a}{p-1}\right) = U_{1,p}\,L'_p(0,χ)\;+\;U_{2,p}\,L(0,χ), \] with constants $U_{1,p}\in {\mathbb{Q}}_p(\hat{I}{¼}_{p-1}{)}^{\times }$ and $U_{2,p}\in {\mathbb{Q}}_p(\hat{I}{¼}_{p-1})$ depending only on $p$ and the fixed branch of ${\log }_p$, but independent of $\ddot{I}‡$. Subtracting the $L(0,\ddot{I}‡)$--block yields a \emph{renormalized} local input \[ Φ^{ren}_p(χ):=Φ_p(χ)-U_{2,p}L(0,χ)=U_{1,p}\,L'_p(0,χ), \] uniformly in odd, nontrivial $\ddot{I}‡$. 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 $\ddot{I}‡$--independence of $(U_{1,p},U_{2,p})$.

5 pages