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Quadratic twists of elliptic curves (or, more generally, abelian varieties) are familiar objects in arithmetic geometry. I would like to extend that definition to the category of 1-motives over global fields, but I have no idea where to begin. Unfortunately, a google search didn't help much. Can anyone suggest literature that might be related to this matter or pitch in with some ideas? Thanks a lot in advance and Happy New Year 2016!

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  • $\begingroup$ Maybe Bloch's "A note on height pairings, Tamagawa numbers, and the BSD Conjecture" would be helpful. I think he considers L-functions associated to 1-motives via generalized Picard varieties. $\endgroup$
    – Jesse Silliman
    Commented Dec 30, 2015 at 22:58
  • $\begingroup$ Did you try to read webusers.imj-prg.fr/~bruno.kahn/preprints/1009.1900v1.pdf (or a newer version of this text)? $\endgroup$
    – Mikhail Bondarko
    Commented Dec 31, 2015 at 5:57
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    $\begingroup$ I'm no expert on 1-motives, but looking at the definitions, it seems that there is no reason to stop at quadratic twists. If a 1-motive is supposed to be a mixed motive with Hodge--Tate weights in $\{0, 1\}$, then surely it should be possible to tensor a 1-motive with any Artin motive to get a new 1-motive? I don't seee how quadratic twists are special here. $\endgroup$
    – David Loeffler
    Commented Dec 31, 2015 at 9:01
  • $\begingroup$ Dear David, Jesse and Mikhail. Thank you very much for your helpful comments. I will follow your advice. There is certainly no reason to stop at quadratic twists (I was thinking of applications in this setting when I asked my question). Just last night I came accross a paper by Mazur, Rubin and Silverberg (arxiv.org/abs/math/0609066) where they provide valuable information on twists of arbitrary commutative algebraic groups (e.g. semiabelian varieties). Perhaps, in conjuntion with the references that you suggest, that paper is a good starting point for me... $\endgroup$
    – Cristian D. Gonzalez-Aviles
    Commented Dec 31, 2015 at 14:38

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