Note on Coleman's formula for the absolute Frobenius on Fermat curves (Q6567979)

In 1990, \textit{R. F. Coleman} calculated explicitly the absolute Frobenius on Fermat curves [Lect. Notes Math. 1454, 173--193 (1990; Zbl 0757.14013)]. In this paper, the author rewrites it in the form of Theorem 2.4. For this purpose, he writes Morita's \(p\)-adic gamma function \(\Gamma_p\) in terms of Shimura's period symbol \(p_K\), its \(p\)-adic analogue \(p_{K,p}\), and modified Euler's gamma function \(\Gamma_\infty\). Therefore, a large part of Coleman's formula follows automatically, without explicit computation, under assuming certain \(p\)-adic continuity properties.