Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it, also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ on $C$ such that $\mathcal O(D)=\mathcal L$ and such that the field of definition of all $p_i$ is "small" in some sense. Are there any algorithms for it, for various notions of "small"? For example, "small" could mean the smallest degree, or the simplest Galois group (say, solvable with shortest derived series).
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$\begingroup$ My guess would be that, for a general curve, you won't be able to get anything better than a field whose degree is the degree of the line bundle and having Galois group the symmetric group. $\endgroup$– Felipe VolochCommented Aug 7, 2014 at 22:37
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$\begingroup$ The curve is NOT generic, rather it would be a modular curve, although of fairly high genus. $\endgroup$– Lev BorisovCommented Aug 7, 2014 at 22:50
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$\begingroup$ Is the line bundle generic? $\endgroup$– Dan PetersenCommented Aug 8, 2014 at 5:27
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$\begingroup$ No it is not generic. The curve and the bundle would be written with explicit equations. $\endgroup$– Lev BorisovCommented Aug 8, 2014 at 10:53
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