Tate module

Tate module

In mathematics, a Tate module is a Galois module constructed from an abelian variety "A" over a field "K". It is denoted

:"T""l"("A")

where "l" is a given prime number (the letter "p" is traditionally reserved for the characteristic of "K"; the case where "K" has characteristic p is of importance). By definition, if

:"A" ["l""n"]

denotes the kernel of multiplication by "l""n" on "A", then "T""l"("A") is the inverse limit of these abelian groups, over all integers "n" ≥ 1. Assuming we have a fixed separable closure of "K" in which the points of the "A" ["l""n"] are all defined, the absolute Galois group "G" of "K" acts on "T""l"("A"), which is a profinite group. In fact it is more, being also a module over the ring of "l"-adic integers "Z""l".

Classical results on abelian varieties show that if "K" has characteristic zero, or characteristic "p" where the prime number "p" ≠ "l", then "T""l"("A") is a free module over "Z""l" of rank 2"d", where "d" is the dimension of "A". In the other case, it is still free, but the rank may take any value from 0 to "d" (see for example Hasse-Witt matrix).

The Galois action via "G" is not so well understood, in the general case. It is a case of the Tate conjecture, for example, to determine the subspace on which "G" acts by the trivial representation, after an appropriate Tate twist. For a Tate twist one takes the Galois module that is the same Tate module construction, but applied to the multiplicative group; in other words take the inverse limit of the "l"-power roots of unity. For reasons ultimately explained by étale cohomology and its version of Poincaré duality, tensor powers of the Tate twist module are carried around in the theory, as an analogue of orientations.

The Tate module is named for John Tate.


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