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Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface: $$ \mathcal{E} : y^2 = x^3 - 27ux - 54v \hspace{1cm}\text{ over }\hspace{1cm} B : v^2 = u^3 + T \hspace{1cm}\text{ over }\hspace{1cm} \mathbb{F}_p(T). $$ One can see that $\mathcal{E}$ admits an order 4 automorphism $(x,y,u,v) \mapsto (-x,iy,u,-v)$, which we might call "CM by $\mathbb{Q}(i)$". Further, after base changing to $\mathbb{F}_q(T)$ if necessary to ensure $q \equiv 1 \pmod{4}$, one can show that the (in this case $2$-dimensional) transcendental lattice $V$ inside $H^2(\mathcal{E},\mathbb{Q}_\ell)$ is a sum of characters over $\mathbb{F}_q(T)$. Again, one is tempted to say this subspace $V$ has CM in some sense.

My question is - what is known about CM motives over function fields? It seems there are some key differences, for instance there is no ability to take the composite between your base and the field by which you have CM. Hence why I am loathe to say "$V$ has CM by $\mathbb{Q}(i)$" above. However, is there still an appropriate Tannakian category of CM motives in this setting? If so, what can we say about it? Is there anything written on this subject?

(Note: I am aware of the theory of $A$-motives, but I don't believe these are quite the right analogue here since the cohomology groups under consideration are $\mathbb{Q}_\ell$-modules, as opposed to modules over some $A$-algebra.)

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    $\begingroup$ Rank one Galois representations over $\mathbb F_q(T)$ are classified - they are all products of finite order characters with characters factoring through the Galois group of $\mathbb F_q$. Galois representations factoring through the Galois group of $\mathbb F_q$ all are tensor products of Galois representations of abelian varieties over $\mathbb F_q$. So we might guess that such motives are tensor products of one-dimensional Artin motives with motives arising from abelian varieties over $\mathbb F_q$. But I don't know in what cases we can prove this. $\endgroup$
    – Will Sawin
    Commented Apr 11, 2021 at 20:17
  • $\begingroup$ Thanks for the insight! I wasn't aware of the fact that all Galois representations from $\mathbb{F}_q$ were tensor products of Galois representations of abelian varieties. How does one see this? $\endgroup$
    – Angus McAndrew
    Commented Apr 11, 2021 at 21:12
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    $\begingroup$ It's not all Galois representations, but the ones arising from algebraic varieties, which necessarily have eigenvalues Weil numbers. I learned a long time ago on MO that you can use the fact that Weil numbers lie in CM fields to write them as products of weight 1 Weil numbers that look like the Frobenius eigenvalues of abelian varieties, and use Honda's theorem to conclude that these actually come from abelian varieties. $\endgroup$
    – Will Sawin
    Commented Apr 11, 2021 at 21:23
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    $\begingroup$ See Proposition 2.6 of Motives over Finite Fields by J.S. Milne jmilne.org/math/articles/1994aP.pdf $\endgroup$
    – Will Sawin
    Commented Apr 11, 2021 at 21:30
  • $\begingroup$ Ah ok, that makes more sense. Thanks for the reference! $\endgroup$
    – Angus McAndrew
    Commented Apr 11, 2021 at 21:32

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