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There are many references (good books, papers, ...) available that treat global function fields (over finite fields) of one independent variable. To name a few, Stichtenoth's book "Algebraic Function Fields And Codes", Rosen's "Number Theory in Function Fields".

I was wondering if there are out there any references for function fields (in positive characteristic) with imperfect base field? For instance, function fields $F/k$ where $k = \mathbb{F}_q(t_1, \cdots, t_n)$. I am particularly interested in places, valuations, derivations, differentials and residues of differentials associated to such kind of function fields.

As an aside question, 'how much' of the theory from global function fields of one independent variable, for instance as treated in Stichtenoth, can be done with imperfect base field? I realize it might be difficult to answer such a question precisely but if someone could just give a rough idea of what can be done, it would be great.

Thanks.

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    $\begingroup$ If I recall correctly, some older books make no assumption on the constant field, or at least discuss what changes: Artin "Algebraic Numbers and Algebraic Functions", Deuring "Lectures on the Theory of Algebraic Functions of One Variable" and Chevalley "Introduction to the Theory of Algebraic Functions of One Variable" $\endgroup$ Commented Jan 17, 2022 at 4:13

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Not sure if this is what you have in mind, but the paper

Michael Szydlo, Elliptic fibers over non-perfect residue fields, J. Number Theory 104 (2004), no. 1, 75-99 (MR2021627)

is a detailed study of the possible reduction types of elliptic curves over non-perfect fields. So those are 1-dimensional (genus 1) function fields over imperfect fields.

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