1
$\begingroup$

For an abelian variety $A$ with formal group $F$, how is the formal group $F^\ast$ of the dual abelian variety $A^{\vee}$ related to $F$? In general, for a formal group $F$, is there a concept of dual formal group $F^\ast$? If so, how is it defined? I think this is related to the cartier dual, but could you give a simple description of $F^*$?

Any reference or insight would be very helpful.

$\endgroup$
2
  • 4
    $\begingroup$ The theory of $p$-divisible groups (when $p$ is the residue characteristic of the ground field) is more suited to duality than that of formal groups. The formal group more or less forgets about the étale part of the $p$-divisible group, and similarly, the dual of a toric $p$-divisible group ($\mu_{p^\infty}$, say, whose formal group would be the multiplicative formal group law) is étale. $\endgroup$
    – ACL
    Mar 4, 2013 at 22:56
  • 2
    $\begingroup$ ACL: You've already said everything necessary; I am just linking the references [Tate's classic][1] or [Serre's Seminaire Bourbaki][2] [1]: fhoermann.org/Tate%2520-%2520p-Divisible%2520Groups.pdf [2]: numdam.org/item?id=SB_1966-1968__10__73_0 $\endgroup$
    – SGP
    Mar 4, 2013 at 23:03

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.