On Special Arithmetic Gamma Values and Thakur's Gauss Sums: From $\infty$-adic to $v$-adic

Source abstract

In this paper, we establish an explicit algebra isomorphism between the normalized $\infty$-adic arithmetic gamma values and their $v$-adic counterparts. The construction relies on Thakur's arithmetic Gauss sums, which arise naturally from the Gross-Koblitz-Thakur relations. Our result demonstrates the phenomenon that Gauss sums, commonly regarded as "finite" analogs of gamma functions, serve as a bridge connecting the $\infty$-adic and $v$-adic gamma values.