This thesis' main results are upper bounds to the Gelfand-Kirillov dimension of certain p-adic representations and some applications of such bounds.

The Gelfand-Kirillov dimension is a generalization of the Krull dimension of modules ovef commutative rings to modules over completed group rings of p-adic Lie groups -- also known as Iwasawa algebras. These Iwasawa algebras act on the duals of p-adic Banach representations or smooth mod p representations of such groups. 

The main result is, for p>2 a prime number, that an admissible p-adic Banach representation of GL_2 of a p-adic field K has Gelfand-Kirillov dimension at most the degree [K:Q_p] as soon as its locally analytic vectors admit an infinitesimal character.

The second main result is a generalization of this statement to families of Banach representations with infinitesimal characters in families in the sense of Dospinescu-Paškūnas-Schraen. This result yields a generalization of the GK-bound of Breuil-Herzig-Hu-Morra-Schraen beyond K unramified for the smooth mod p representation given by a Galois isotypic component in the mod p cohomology of Shimura varieties.

The method of proof for both results is much in the spirit of the filtrations of fixed ``radius of analyticity'' of Schneider-Teitelbaum on the duals of the representations and the study of their associated gradeds. The second main result required a generalization of these filtrations to double radii.

Finally, this thesis has two more results. The most important one being that (in the setting of K unramified, only needing the results of Breuil-Herzig-Hu-Morra-Schraen) the GK-bound for mod p cohomology of Shimura varieties allows one, via miracle flatness of the patched module over R_\infty, to compute the derived pro-p Iwahori invariants of the mod p representation as complex over the non-derived pro-p Iwahori Hecke algebra, up to quasi-isomorphism. This derived module turns out to depend only on the mod p Galois representation (and a multiplicity) and not on the global choices that go into the construction of the smooth mod p representation of GL_2.