elliptic curves over function fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:28:52Z http://mathoverflow.net/feeds/question/116969 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116969/elliptic-curves-over-function-fields elliptic curves over function fields pedro 2012-12-21T11:54:41Z 2012-12-21T11:54:41Z <p>Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for every $v$. By Tate local duality, we know that the image is self-dual (i.e. it is its own orthogonal complement under the Tate pairing).</p> <p>Let $q \ne p$, $q > 3$ be a prime with $F_q$ the finite field of $q$ elements. Let $C$ be a smooth, geometrically connected curve $F_q$, $K = F_q^s(C)$ ($K$ is a geometric function field) , where $F_q^s$ is the separable closure of $F_q$. Following Ellenberg Prop 2.1 (http://www.math.wisc.edu/~ellenber/CMECTFF.pdf), we know that the Kummer injection factors through the tame fundamental group. We also have the local descent map</p> <p>$E(K_v)/pE(K_v) \hookrightarrow H^1 (\pi_v, E[p])$, where $\pi_v$ is the local tame fundamental group ($\cong Gal(K_v^{tame}/K_v)$).</p> <p><strong>Questions:</strong></p> <p>Is there a version of Tate local duality for geometric function fields? </p> <p>If there is a Local duality version as in the number field case, is the image self-dual? </p>