false elliptic curves and principal polarizations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T01:05:00Z http://mathoverflow.net/feeds/question/93591 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93591/false-elliptic-curves-and-principal-polarizations false elliptic curves and principal polarizations unknown 2012-04-09T19:34:55Z 2012-04-09T20:12:44Z <p>Hi,</p> <p>Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a <em>false elliptic curve</em> over a field $K$ is a pair $(A/K,i)$ consisting of an abelian surface $A/K$ and a ring homomorphism $i : \mathcal O_\Delta\to End_K(A)$. Suppose that $\Delta$ is indefinite, i.e., $\Delta\otimes\mathbf R \simeq M_2(\mathbf R)$. There is an involution $*:\Delta\to\Delta$ that coincides with taking transpose under the previous isomorphism. It is well-known (easy?) that if $K$ is of characteristic zero there is a polarization of $A/K$ such that the corresponding Rosatti involution in $End_K(A)\otimes\mathbf Q$ corresponds to $* : \Delta\to\Delta$ by $i$. Moreover, this polarization is unique up to a rational number.</p> <p>Question 1: is it possible to find a (necessarily unique) principal polarization with this property?</p> <p>Question 2: is it possible to find such a (principal) polarization over a field $K$ of non-zero characteristic?</p> <p>Thanks!</p>