Part I. First steps in p-adic Hodge theory 4 1. Motivation 4 1.1. Tate modules 4 1.2. Galois lattices and Galois deformations 6 1.3. Aims of p-adic Hodge theory 7 1.4. Exercises 9 2. Hodge–Tate representations 10 2.1. Basic properties of CK 11 2.2. Theorems of Tate–Sen and Faltings 12 2.3. Hodge–Tate decomposition 15 2.4. Formalism of Hodge–Tate representations 17 2.5. Exercises 24 3. Étale φ-m...

The purpose of this paper is to study in greater detail the arithmetic KodairaSpencer morphism of an elliptic curve introduced in [Mzk1], Chapter IX, in the general context of the Hodge-Arakelov theory of elliptic curves, developed in [Mzk1-3]. In particular, after correcting a minor error (cf. Corollary 1.6) in the construction of this arithmetic Kodaira-Spencer morphism in [Mzk1], Chapter IX,...

The modified Hodge method (MHT) has been recommended by the CLSI for confirmation of suspected class A carbapenemase production in species of Enterobacteriaceae. This test and the Masuda method (MAS) have advantages over traditional phenotypic methods in that they directly analyze carbapenemase activity. In order to identify the potential interferences of these tests, we designed a panel compos...

Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingrı́msson, and the homology of ∆(G) was shown to be nonzero only in dimension n − 3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(∆(G)) where the dimension of the j component in the decomposition, H n−3(∆(G)), equals the absolute value of th...

Let G be a simple graph with n vertices. The coloring complex ∆(G) was defined by Steingrı́msson, and the homology of ∆(G) was shown to be nonzero only in dimension n − 3 by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group Hn−3(∆(G)) where the dimension of the j component in the decomposition, H n−3(∆(G)), equals the absolute value of th...

In this paper, we consider an arithmetic Hodge index theorem for a family of semi-stable curves, generalizing Faltings-Hriljac’s arithmetic Hodge index theorem for an arithmetic surface.

We prove that the lowest nonzero piece in the Hodge filtration of a mixed Hodge module is always weakly positive in the sense of Viehweg.

We prove the weak positivity of the kernels of Kodaira-Spencertype maps for pure Hodge module extensions of generically defined variations of Hodge structure.