# Note on Coleman's formula for the absolute Frobenius on Fermat curves

## TL;DR

This paper explores the p-adic properties of Coleman's formula for the Frobenius on Fermat curves, revealing that p-adic continuity underpins much of Coleman's explicit calculation through relations involving p-adic gamma functions and CM-periods.

## Contribution

It demonstrates that p-adic continuity explains a significant part of Coleman's explicit Frobenius formula on Fermat curves, linking functional equations of p-adic gamma functions to CM-period relations.

## Key findings

- p-adic continuity underlies Coleman's formula
- Functional equations of p-adic gamma functions relate to Frobenius
- Connections between p-adic gamma functions and CM-periods established

## Abstract

Coleman calculated the absolute Frobenius on Fermat curves explicitly. In this paper we show that a kind of $p$-adic continuity implies a large part of his formula. To do this, we study a relation between functional equations of the ($p$-adic) gamma function and monomial relations on ($p$-adic) CM-periods.