Related papers: The Geometry of Hida Families II: $\Lambda$-adic $…

Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, \eta)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for…

Cohomology of Specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the Borel subgroup B of the general linear group GL_d(k), but this has never been exploited to prove new symmetric group…

For an algebraically closed non-archimedean extension $C/\mathbb{Q}_p$, we define a Tannakian category of $p$-adic Hodge structures over $C$ that is a local, $p$-adic analog of the global, archimedean category of $\mathbb{Q}$-Hodge…

We use the stacky approach to $p$-adic cohomology theories recently developed by Drinfeld and Bhatt--Lurie to generalise known comparison theorems in $p$-adic Hodge theory so as to accommodate coefficients. More precisely, we establish a…

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a…

We study the $\Lambda$-module structure of the ordinary parts of the arithmetic cohomology groups of modular Jacobians made out of various towers of modular curves. We prove that the ordinary parts of $\Lambda$-adic Selmer groups coming…

We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $\Gamma$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection…

We discuss recent developments in $p$-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for "compact $p$-adic manifolds" over new period maps on moduli spaces of abelian…

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale cohomology with nontrivial coefficients, as well as in the relative setting, i.e.…

We give a simple proof of an isomorphism between the two $\mathbb{C}[t]$-modules: the module of relative cohomologies $\Lambda^2/dH\land \Lambda^1$ and the module of Abelian integrals corresponding to a regular at infinity polynomial $H$ in…

Given a proper, smooth (formal) scheme over the ring of integers of $\mathbb C_p$, we prove that if the crystalline cohomology of its special fibre is torsion-free then the $p$-adic \'etale cohomology of its generic fibre is also…

We introduce a notion of ellipticity of complexes of linear pseudodifferential operators acting on sections of $A$-Hilbert bundles over smooth manifolds, $A$ being a $C^*$-algebra. We prove that the cohomology groups of an $A$-elliptic…

Andreotti-Vesentini, Ohsawa, Gromov, Koll\'ar, among others, have observed that Hodge theory could be extended to (non compact) K\"ahler complete manifolds, within the L^2 framework. Also, many vanishing theorems on projective or K\"ahler…

Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category $\mathcal C$ as modules over the smash extension $\mathcal C\# H$. We construct Grothendieck spectral sequences for the cohomologies as well as the…

We construct explicitly some analytic families of etale (phi,Gamma)-modules, which give rise to analytic families of 2-dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on…

It is known that for a prime $p\ne 2$ there is the following natural description of the homology algebra of an abelian group $H_*(A,\mathbb F_p)\cong \Lambda(A/p)\otimes \Gamma({}_pA)$ and for finitely generated abelian groups there is the…

We construct the cohomology groups with compact support of stacks of shtukas with $\mathbb Z_{\ell}$-coefficients. We construct the cuspidal cohomology groups and prove that they are $\mathbb Z_{\ell}$-modules of finite type. We prove that…

In this article we use the combinatorial and geometric structure of manifolds with embedded cylinders in order to develop an adiabatic decomposition of the Hodge cohomology of these manifolds. We will on the one hand describe the adiabatic…

In local relative $p$-adic Hodge theory, we show that the Galois cohomology of a finite height crystalline representation (up to a twist) is essentially computed via the (Fontaine--Messing) syntomic complex with coefficients in the…

We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\Int/p^n\Int$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By…