Related papers: $\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families an…

We prove a rank-two potential automorphy theorem for mod $l$ representations satisfying an ordinary condition. Combined with an ordinary automorphy lifting theorem, we prove a rank-two, $p \ne l$ case of local-global compatibility for…

This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p…

For every finite group $H$ and every finite $H$-module $A$, we determine the subgroup of negligible classes in $H^2(H,A)$, in the sense of Serre, over fields with enough roots of unity. As a consequence, we show that for every odd prime…

Let $L$ be a finite extension of $\mathbb{Q}_p$, and $\rho_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of $\mathrm{Gal}_L$ with full monodromy rank. Via a study of Breuil's (simple) $\mathcal{L}$-invariants,…

We show that under a suitable oddness condition, irreducible mod $p$ representations of the absolute Galois group of an arbitrary number field have characteristic zero lifts which are unramified outside a finite set of primes and…

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For…

We extend the results of Deligne and Illusie on liftings modulo $p^2$ and decompositions of the de Rham complex in several ways. We show that for a smooth scheme $X$ over a perfect field $k$ of characteristic $p>0$, the truncations of the…

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for…

Let $G$ be a $p$-divisible group over a complete discrete valuation ring $R$ of characteristic $p$. The generic fiber of $G$ determines a Galois representation $\rho$. The image of $\rho$ admits a ramification filtration and a Lie…

We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.

We consider mod $p$ Hilbert modular forms for a totally real field $F$, viewed as sections of automorphic line bundles on Hilbert modular varieties in prime characteristic $p$. For a Hecke eigenform of arbitrary weight, we prove the…

We prove several results concerning the existence of potentially crystalline lifts with prescribed Hodge-Tate weights and inertial types of a given n-dimensional mod p representation of the absolute Galois group of K, where K/Q_p is a…

Let $F$ be a totally real field of even degree in which $p$ splits completely. Let $\overline{r}:G_F \rightarrow \mathrm{GSp}_4(\overline{\mathbb{F}}_p)$ be a modular Galois representation unramified at all finite places away from $p$ and…

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field,…

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity.

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such…

Let $f$ be a genus two cuspidal Siegel modular eigenform. We prove an adelic open image theorem for the compatible system of Galois representations associated to $f$, generalising the results of Ribet and Momose for elliptic modular forms.…

We prove the compatibility of the local and global Langlands correspondences at places dividing l for the l-adic Galois representations associated to regular algebraic conjugate self-dual cuspidal automorphic representations of GL_n over an…

Let $p$ be a prime number, $F$ a totally real number field unramified at places above $p$ and $D$ a quaternion algebra of center $F$ split at places above $p$ and at no more than one infinite place. Let $v$ be a fixed place of $F$ above $p$…

Let F be a number field, p a prime number. We construct the (multi-variable) p-adic L-function of an automorphic representation of $GL_2(A_F)$ (with certain conditions at places above p and $\infty$), which interpolates the complex…