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

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