I am looking for an explicit example of a function field other than rational with an automorphism which fixes places of different degrees. Also, is there a counterexample, namely a function field with all the automorphisms don't fix any places of different degree.
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$\begingroup$ Consider $\mathbb F_{13}(X)$ and in it places corresponding to irreducible polynomials $X^2+2,X^4+2$. They are both fixed by the automorphism induced by $X\mapsto -X$ $\endgroup$– WojowuCommented Dec 19, 2020 at 21:31
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1$\begingroup$ Thank you but \mathbb F_{13}(X) is rational function field, I was looking for a different function field such as hyperelliptic $\endgroup$– Engin ŞenelCommented Dec 19, 2020 at 21:36
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1$\begingroup$ Ah apologies, I missed that part in the question. $\endgroup$– WojowuCommented Dec 19, 2020 at 21:40
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$\begingroup$ Also a recommedation of a book or a thesis for begginers greatly appreciated. I couldnt find much information in Stichtenoth's Algebraic Function Fields and Codes about that question. I searched in web, but no luck. Algebraic Geometry's language can be hard to understand $\endgroup$– Engin ŞenelCommented Dec 19, 2020 at 21:46
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Take an elliptic curve with no rational 2-torsion points ($y^2=f(x)$) where $f(x)$ is a cubic that does not split over the base curve. Then the automorphism $P\to[−1]P$ fixes all the 2 torsion which has one rational point (of degree 1) and 3 non rational points corresponding to one rational point of degree 3.
To be very explicit: the function field could be the field of fractions of $\mathbb Q[x,y]/(y^2−(x^3+2))$ and the automorphism sends $y \to −y$.
This example will work in general for a hyperelliptic curve $y^2 = f(x)$ where $f(x)$ could be any squarefree polynomial of any degree. The 2 torsion corresponds to roots of $f(x)$ and $y \to -y$ is the involution. It's clear that you can get any combination of degrees for the fixed points if you let the genus grow.
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$\begingroup$ Thank you very much Asvin, if you could suggest a book for begginners about these subjects or other materials, I would be grateful $\endgroup$ Commented Dec 19, 2020 at 22:16
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$\begingroup$ Are you looking for anything in particular? If you just want to learn about curves Fulton's "Algebraic Curves" has a good reputation. $\endgroup$– AsvinCommented Dec 19, 2020 at 22:19
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$\begingroup$ I would prefer a source presents the language of algebraic function fields, cause it is used generally in coding theory. Also, Stichtenoth's Algebraic Function Fields and Codes is a perfect source for it, but sometimes I just cant find what ı am looking for in it. Algebraic Curves is not my speciality, and its language can be a problem. But in my spare times I will look at it. I think I can fill the gaps with a good book. So, thank you $\endgroup$ Commented Dec 19, 2020 at 22:33
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$\begingroup$ Maybe try this? books.google.com/books/about/… $\endgroup$– AsvinCommented Dec 19, 2020 at 23:42
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$\begingroup$ Thanks for your suggestions Asvin, I will look at them $\endgroup$ Commented Dec 20, 2020 at 9:32