Hodge-Tate Weight

A Hodge-Tate weight of a -adic representation of the absolute Galois group of a finite extension is an integer appearing in its Hodge-Tate decomposition after extension to the completed algebraic closure . For a finite-dimensional representation, the convention here writes that decomposition as

The weights are the integers with multiplicities . The representation is Hodge-Tate when such a finite decomposition exists, and has cyclotomic action of exponent . Thus the cyclotomic representation has weight in this convention. Some authors use the opposite sign.

The regularity condition in the two-dimensional Fontaine-Mazur modularity assertion requires distinct Hodge-Tate weights.