Title:

Hodge numbers of semistable representations

Part I of this thesis concerns the relation, in padic Hodge theory, between the monodromy and the Hodge numbers of a filtered (φ, N)module D. Studying the interaction of the Hodge and Newton polygons with N, we deduce that a monodromy operator of large rank forces the Hodge numbers of D to be large: this is the content of Theorem I.14.4. This result can then be applied to various Galois representations. For instance, starting with a Hilbert modular form f over a totally real field F , it is known that we can attach to it a global padic Galois representation pf: GF͢ GL2(E), for E/Qp some ënite extension. Choosing a prime p of F above p, we can then study the local padic Galois representation pf, p: GGFp GL2(E). Assuming that pf;p is semistable with matching Hodge–Tate weights, we can then use Fontaine–Dieudonné theory to obtain a filtered (φ, N)module Dst (ρf, p). Applying Theorem I.14.4, we deduce that if the weights of f are too small, then ρf, p is in fact crystalline. We also present in section I.16.1 an example of a nonsplit semistable noncrystalline extension of crystalline characters which is not “trivial by cyclotomic”, even up to twists. In part II, we explore parallel results on the automorphic side of the Langlands correspondence. Concentrating on the case of Hilbert modular forms, the approach is to study the padic integrality properties of Hecke operators. This naturally leads to the study of integral models of Hilbert modular varieties and of their associated automorphic vector bundles. A careful study of these leads to the introduction of certain renormalisation factors for the action of Hecke operators (Proposition II.5.2). We then prove the integrality of these renormalised Hecke operators using the qexpansion principle (Proposition II.5.4). Finally, we use these integrality properties to arrive at conditions, dependent on the weight of a Hilbert modular form f , that guarantee that local components of f cannot be special; is the content of Τheorem II.6.2. For instance, in the case that there is a unique prime p above p in F, corresponding to the local component πf, p is a unique filtered ( φ,N, GFp)module Df, p, and the results of Theorem I.14.4 and Theorem II.6.2 exactly match: if the weights ντ of f do not average at least 2, the former theorem shows that Df,p is potentially crystalline, while the latter shows that πf, p cannot be special. Other interesting behaviour can occur depending on the splitting behaviour of p in F , such as conditions on all pairs of weights of f as in the final example of section 6.2.1.
