Hodge–Tate module

In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. Serre (1967) introduced and named Hodge–Tate structures using the results of Tate (1967) on p-divisible groups.

Definition

Suppose that G is the absolute Galois group of a p-adic field K. Then G has a canonical cyclotomic character χ given by its action on the pth power roots of unity. Let C be the completion of the algebraic closure of K. Then a finite-dimensional vector space over C with a semi-linear action of the Galois group G is said to be of Hodge–Tate type if it is generated by the eigenvectors of integral powers of χ.

See also

• p-adic Hodge theory
• Mumford–Tate group

References

• Faltings, Gerd (1988), "p-adic Hodge theory", Journal of the American Mathematical Society 1 (1): 255–299, doi:10.2307/1990970, ISSN 0894-0347, MR 924705
• Serre, Jean-Pierre (1967), "Sur les groupes de Galois attachés aux groupes p-divisibles", in Springer, Tonny A., Proceedings of a Conference on Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, pp. 118–131, ISBN 978-3-540-03953-2, MR 0242839
• Tate, John T. (1967), "p-divisible groups.", in Springer, Tonny A., Proc. Conf. Local Fields (Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827