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I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \in \mathbb Z[T]$ satisfying certain properties (Riemann hypothesis, functional equation, etc.) is the zeta function of such a function field.

I know very little about algebraic geometry, and when I look for Honda-Tate I only find texts about isogenies of abelian varieties, so it is of little use to me stated that way.

Would someone be kind enough to provide a statement of Honda-Tate's theorem in the form above, or give a reference to such a statement ? My goal would be to prescribe (inverse) roots of Artin $L$-functions of a certain shape, say $\sqrt q$ with a given multiplicity, $\sqrt q e^{2i\pi/3}$ with a given multiplicty and so on.

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    $\begingroup$ For abelian varieties you can get the desired characteristic polynomial of Frobenius on the nose (this is Honda's paper). Then use that every abelian variety is dominated by a Jacobian of a curve (e.g. take a smooth complete intersection curve inside $A$). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 15, 2020 at 17:24
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    $\begingroup$ This gives a curve where the characteristic polynomial of Frobenius contains the given one (as a factor), but it's a very difficult problem to determine which exact polynomials occur. For example, it is only known in characteristic $2$ that $(1-t\sqrt{q})^{2g}$ occurs on a genus $g$ curve for every $g$ (this is due to van der Geer and van der Vlugt). (And even then I think it's only an asymptotic result, i.e. for $q = 2^r \to \infty$; for smaller $q$ you'd get some roots of unity that might be hard to control.) $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 15, 2020 at 17:30
  • $\begingroup$ Thank you for these references. So we cannot prescribe anything more than lower bounds on multiplicities if I understood correctly ? $\endgroup$
    – A. Bailleul
    Commented Jun 16, 2020 at 8:55
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    $\begingroup$ In some sense, but you cannot really fix $g$. Given a $q$-Weil polynomial of degree $2g$, it shows up in some curve of genus $g' \gg g$. There might be an upper bound on $g'$ depending on some stuff, but I'm not entirely sure. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 16, 2020 at 21:11

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