Related papers: Artin Conjecture for p-adic Galois Representations…

Deligne has formulated extremely influential conjectures about certain special values of the $L$-functions of (Grothendieck) motives over a number field $F$. Given the conjectural dictionary between motives and 'algebraic' automorphic…

In 1927, E. Artin conjectured that all non-square integers $a\neq -1$ are a primitive root of $\mathbb{F}_p$ for infinitely many primes $p$. In 1967, Hooley showed that this conjecture follows from the Generalized Riemann Hypothesis (GRH).…

We develop a variant of Coleman and Perrin Riou's methods giving, for a de Rham $p$-adic Galois representation, a construction of $p$-adic $L$ functions from a compatible system of global elements. As a result, we construct analytic…

It is proved that every two-dimensional residual Galois representation of the absolute Galois group of an arbitrary number field lifts to a characteristic zero $p$-adic representation, if local lifting problems at places above $p$ are…

We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases…

Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $\phi_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois…

For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a…

Let $K$ be a local function field of characteristic $l$, $\mathbb{F}$ be a finite field over $\mathbb{F}_p$ where $l \ne p$, and $\overline{\rho}: G_K \rightarrow \text{GL}_n (\mathbb{F})$ be a continuous representation. We apply the…

We establish a geometrisation of the Breuil-M\'ezard conjecture for potentially Barsotti-Tate representations, as well as of the weight part of Serre's conjecture, for moduli stacks of two-dimensional mod p representations of the absolute…

If $K/\mathbb Q$ is a finite Galois extension with an almost monomial Galois group and if $s_0\in\mathbb C\setminus\{1\}$ is not a common zero for any two Artin L-functions associated to distinct complex irreducible characters of the Galois…

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions.…

We define a functorial "Artin map" attached to any small $\bf{Z}$-linear stable $\infty$-category, which in the case of perfect complexes over a global field recovers the usual Artin map from the idele class group to the abelianized…

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

Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil-M\'ezard conjecture for 2-dimensional mod p representations of the absolute Galois group of Qp. We also state…

The main objective of this article is to establish the $p$-adic Artin formalism for the algebraic $p$-adic $L$-functions attached to the adjoint representations of Coleman families of modular forms. In particular, we prove a factorization…

The primary goal of this article is to study $p$-adic Beilinson conjectures in the presence of exceptional zeros for Artin motives over CM fields. In more precise terms, we address a question raised by Hida and Tilouine on the order of…

In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a…

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point,…

Let K/F be an arbitrary Galois extension of number fields and r be a representation of Gal(K/F) into GSp(4,C). Let E_16 be the elemetary abelian group of order 16 and C_5 the cyclic group of order 5. If the image of r in the projective…

Let $p$ be an odd prime number. Let $f$ be a normalized Hecke eigen-cuspform that is non-ordinary at $p$. Let $K$ be an imaginary quadratic field in which $p$ splits. We study the Artin formalism for the two-variable signed $p$-adic…