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For each global field $K$, can we always find an Abelian variety $A$ with $A(K)$ rank $0$?

By Lang-Neron, Mordell Weil is also true for finite type fields (fields finitely generated over their prime field). Can we also find Abelian varieties over these fields with rank 0?

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    $\begingroup$ Bhargava-Shankar works, but surely there’s a better way. $\endgroup$
    – alpoge
    Commented Jun 6, 2019 at 17:09
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    $\begingroup$ For number fields, this follows from Theorem 1.1 from Mazur-Rubin ("Ranks of twists of elliptic curves and Hilbert's Tenth Problem"). Since this was described as a previously open question (well at least when asked for elliptic curves instead of abelian varieties more generally, but I don't suppose that made it much harder), I don't think there is a really elementary way of answering your question. For function fields, it is conceivable that an easier argument is possible since so many things related to elliptic curves are easier in the function field case, but I don't have one. $\endgroup$
    – R.P.
    Commented Jun 6, 2019 at 18:18
  • $\begingroup$ What about proving nonvanishing of central values of L functions of weight k, level N modular forms (let’s say N is large and k = 2 so the probability of me saying something wrong is just slightly smaller) via lower bounding e.g. a second moment? I believe a theorem of Kolyvagin (et al.?) then gives that the corresponding quotient of the Jacobian of X_0(N) has rank 0... $\endgroup$
    – alpoge
    Commented Jun 6, 2019 at 18:48
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    $\begingroup$ @alpoge You probably mean Kolyvagin--Logachev, but that is only for certain very specific fields (anticyclotomic extensions of imaginary quadratic fields). You can get to anticyclotomic extensions of CM fields using work of Zhang but that is no help for a general number field $K$. $\endgroup$
    – David Loeffler
    Commented Jun 6, 2019 at 20:27
  • $\begingroup$ Indeed I was afraid of the restriction on the number field but couldn’t remember the reference, so I asked it as a question... Alas! Thanks for the clarification! $\endgroup$
    – alpoge
    Commented Jun 6, 2019 at 20:28

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