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Let $k$ be a field (perfect, or characteristic zero if you want - I'm especially interested in when $k$ is a number field). Consider the categories $\mathsf{G}_k=\{\text{commutative affine group schemes over $k$}\}$ and $\mathsf{A}_k=\{\text{abelian varieties over $k$ modulo isogeny}\}$. The category $\mathsf{G}_k$ is abelian (see this answer), and so is $\mathsf{A}_k$, though I don't know of a nice reference for this. Two questions:

  1. Is $K_0(\mathsf{A}_k)$ generated by Jacobians of curves? (This question was asked by Grothendieck quite a while ago - I am wondering if any progress has been made).

  2. Is anything at all known about $\mathsf{G}_k$? (I acknowledge that this is a very vague question - more of a reference request)

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    $\begingroup$ If $G$ is a commutative affine group scheme, isn't the product of countably many copies of $G$ a group scheme, forcing the $K$-theory to vanish? If so, I think you might want to consider finite-type affine commutative group schemes or finite commutative group schemes. $\endgroup$ – Will Sawin Aug 4 '13 at 19:37
  • $\begingroup$ In general, $\mathsf{G}_k$ is kind of huge. Even if you restrict to tori, your category includes the Galois representations to $GL_n(\mathbb{Z})$ for all $n$. $\endgroup$ – S. Carnahan Aug 4 '13 at 22:17
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    $\begingroup$ Well the Grothendieck group of that category isn't too difficult - it's just a lattice in the group of locally constant class functions $Gal(k) \to \mathbb Z$ $\endgroup$ – Will Sawin Aug 4 '13 at 22:38

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