Question about specifying complex 1-motives - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:49:08Z http://mathoverflow.net/feeds/question/87186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87186/question-about-specifying-complex-1-motives Question about specifying complex 1-motives Justin Shih 2012-02-01T01:02:21Z 2012-02-01T01:02:21Z <p>A 1-motive over a field \$k\$ is an algebraic torus \$T\$, an abelian variety \$A\$, a group scheme \$G\$ that's an extension of \$A\$ by \$T\$, a finitely generated free abelian group \$L\$, and a group homomorphism \$L \longrightarrow G(k)\$.</p> <p>I'm currently reading a <a href="http://archive.numdam.org/article/CM_1985__56_3_271_0.pdf" rel="nofollow">paper of Carlson</a>, and I want to use his construction to identify something that came up in a problem that I'm working on. However, on the first page of that paper he defines a complex group scheme, but appears to leave out the requirement that \$T\$ and \$G\$ be group schemes. Later on (in section 4), he constructs the trace motive, and consistent with his definition, appears to only define the \$\mathbb{C}\$-points. I'm missing something -- but I don't really know what.</p> <blockquote> <p>Does the fact that the groups in the trace motive come from group schemes somehow follow from some general nonsense about \$\mathbb{C}\$? Is it long and unenlightening to write down? Or am I just completely misunderstanding the paper?</p> </blockquote>