Serre-Tate canonical lifts for finite fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:37:53Z http://mathoverflow.net/feeds/question/105586 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105586/serre-tate-canonical-lifts-for-finite-fields Serre-Tate canonical lifts for finite fields Marty B. 2012-08-26T23:50:16Z 2012-08-27T00:22:09Z <p>A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field \$k\$ of positive characteristic to an abelian scheme over the ring of Witt vectors of \$k\$ and that we can also lift an endomorphism of the variety uniquely to an endomorphism of the canonical lifting. </p> <p>In the case where \$k\$ is finite, is the canonical lift a variety? Do we know its dimension? I'd be glad to read your answers and the references to those answers as well.</p> http://mathoverflow.net/questions/105586/serre-tate-canonical-lifts-for-finite-fields/105589#105589 Answer by Felipe Voloch for Serre-Tate canonical lifts for finite fields Felipe Voloch 2012-08-27T00:22:09Z 2012-08-27T00:22:09Z <p>As you just said, the canonical lift is an abelian scheme over the ring of Witt vectors \$W(k)\$. Now, if \$k\$ is finite of characteristic \$p\$, \$W(k)\$ is the ring of integers of the unramified extension (say \$K\$) of \$\mathbb{Q}_p\$ whose residue field is \$k\$. The canonical lift has a generic fiber which is an abelian variety over \$K\$ and has the same dimension as the abelian variety over \$k\$ you started with. I guess there is a bit of ambiguity in the literature where some people refer to the abelian scheme over \$W(k)\$ as the canonical lift whereas others refer to its generic fiber over the field of fractions of \$W(k)\$ as the canonical lift. You can recover the former from the latter by taking Néron models.</p>