New York Journal of Mathematics
Volume 26 (2020), 1422-1443


Moshe Jarden and Nantsoina Cynthia Ramiharimanana

Embedding problems with bounded ramification over function fields of positive characteristic

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Published: December 8, 2020.
Keywords: embedding problems, bounded ramification, function fields of positive characteristic.
Subject: 12F12, 12E30.

Let K0 be an algebraic function field of one variable over a Hilbertian field F of positive characteristic p. Let K be a finite Galois extension of K0. We prove that every finite embedding problem 1-->H-->G-->Gal(K/K0)-->1 whose kernel H is a p-group is properly solvable.

Moreover, the solution can be chosen to locally coincide with finitely many, given in advance, weak local solutions. Finally, and this is the main point of this work, the number of prime divisors of K0/F that ramify in the solution field is bounded by the number of prime divisors of K0 that ramify in K plus the length of the maximal G-invariant sequence of subgroups of H.


The authors are indebted to Aharon Razon for carefully reading drafts of this work and for his useful comments. Special thanks go to the anonymous referee who suggested an essential simplification of an earlier version of the proof of Proposition 6.8.

Author information

Moshe Jarden:
School of Mathematical Sciences
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University
Ramat Aviv, Tel Aviv 69978, Israel


Nantsoina Cynthia Ramiharimanana:
Department of Mathematics
Clemson University
Clemson, SC 29634, USA