Tate's theorem about abelian variteies in case of abelian scheme - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T02:42:42Z http://mathoverflow.net/feeds/question/96448 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96448/tates-theorem-about-abelian-variteies-in-case-of-abelian-scheme Tate's theorem about abelian variteies in case of abelian scheme TOM 2012-05-09T13:14:53Z 2012-05-24T16:29:57Z <p>For $k$ a finite field , $A,A'$ an abelian varieties over $k$, $G$ the Galois group of $k$, $l$ a prime number different from the characteristic of $k$ . Tate has proved that:</p> <p>$Q_l\otimes Hom_k(A,A')\rightarrow Hom_G(V_l(A),V_l(A'))$<br> is bijective , where $V_l(A)=Q_l\otimes_{Z_l}T_l(A)$ , $T_l(A)$ is the Tate module of $A$.</p> <p>Now consider a Scheme $S$ over $F_p$, and Abelian schemes $A,A'$ over $S$ , is there any known result similar to Tate's theorem for this situation?</p> <p>Thank you ! </p>