Let be a complete algebraically closed extension, and let be an abelian variety, with p-adic Tate module and dual abelian variety . There are then at least three natural candidates for a “Hodge-Tate map”
1. (Scholze) For any smooth proper rigid space , we have the Hodge-Tate spectral sequence
Taking , we get canonical identifications and , so the edge map gives a map , and we may define as its -dual.
2. (Tate, Fargues) Supposing is a p-divisible group, there is a very direct definition of a Hodge-Tate map for : writing for the Cartier dual, we have
and we define
For with good reduction, we can apply this right away to – but this really only works over , since doesn’t have any differentials over ! To push this through in general, let be the formal semiabelian scheme over obtained by completing the Neron model of along the identity component of its special fiber. The rigid generic fiber of is naturally a rigid analytic subgroup of (the rigid space associated with) , and in particular we get a canonical inclusion . This dualizes and compiles into a surjection . Now is an honest p-divisible group over , so the discussion above gives a map , and finally is naturally an -lattice in , and we may define as the composite map
3. (Coleman, Hodge-Tate periods and p-adic abelian integrals) Given an element with , choose a divisor on corresponding to . Since , we may choose rational functions on such that . The limit
exists and defines an element of depending only on (the nonholomorphic bits of the individual ‘s go to zero p-adically).
In practice, definitions 1 and 2 are both pretty useful. Definition 1 varies well in families, and e.g. plays a crucial role in Scholze’s torsion paper. Definition 2 captures more delicate integral behavior and simultaneously allows one to import information from the p-divisible groups universe; for example, the proof of Theorem D in this post uses definition 2 crucially. Fortunately, Scholze proves the equivalence of these two definitions in his CDM article (and it requires a pretty ridiculous diagram chase!). Coleman’s definition is a lot crazier, and I don’t think it’s been proven to agree with the others in general. To quote Coleman’s paper, “It is also possible to make sense of the limit…when has bad reduction; however, we have not established its connection with Hodge-Tate.” Anyone up for it?