Tohoku Mathematical Journal
2020
March
SECOND SERIES VOL. 72, NO. 1

Tohoku Math. J.
72 (2020), 63-76

Title ON \'ETALE FUNDAMENTAL GROUPS OF FORMAL FIBRES OF $P$-ADIC CURVES

Author Mohamed Saïdi

(Received March 13, 2017, revised May 25, 2018)
Abstract. We investigate a certain class of (geometric) finite (Galois) coverings of formal fibres of $p$-adic curves and the corresponding quotient of the (geometric) étale fundamental group. A key result in our investigation is that these (Galois) coverings can be compactified to finite (Galois) coverings of proper $p$-adic curves. We also prove that the maximal prime-to-$p$ quotient of the geometric étale fundamental group of a (geometrically connected) formal fibre of a $p$-adic curve is (pro-)prime-to-$p$ free of finite computable rank.

Mathematics Subject Classification. Primary 14H30; Secondary 11G20.
Key words and phrases. Formal fibre, $p$-adic curves, étale fundamental groups.

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