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

We compute the Artin $L$-function of a diagonal hypersurface D_{\lambda} over a finite field associated to a character of a finite group acting on D_{\lambda} , and under some condition, express it in terms of hypergeometric functions and…

Let $K/\mathbb Q$ be a finite Galois extension, $s_0\in \mathbb C\setminus \{1\}$, ${\it Hol}(s_0)$ the semigroup of Artin L-functions holomorphic at $s_0$. If the Galois group is almost monomial then Artin's L-functions are holomorphic at…

In this paper we prove a result which establishes an equivalence between the representational assembly conjecture proposed by the author and a rigidity question, in the case of Galois groups which are pro-l groups. In additional work with…

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic…

We prove the factorization conjecture for triple-product $p$-adic $L$-functions formulated in a companion article in the special case when two of the (three) factors have complex multiplication.

We take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary Banach space representations of $G(L)$ when…

We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations…

For a quadratic field K, we investigate continuous mod p representations of the absolute Galois groups of K that are unramified away from p and infinity. We prove that for certain pairs (K,p), there are no such irreducible representations.…

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (\S \S 3.1, 3.2) testing if K (defined by an irreducible…

We introduce a new class of finite groups, called weak almost monomial, which generalize two different notions of "almost monomial" groups, and we prove it is closed under taking factor groups and direct products. Let $K/\mathbb Q$ be a…

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four…

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin…

Let $K/{\mathbb Q}$ be a finite Galois extension, and let $s_0\neq 1$ be a complex number. We present two new criteria for the Artin L-functions to be holomorphic at $s_0$.

Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form.…

Let pi be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K such that the central character of pi is invariant under the non-trivial automorphism of K. We show that pi is associated…

In this summary paper, we present the key ideas behind the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups, which states that complements of locally finite affine hyperplane arrangements with real equations and stable…

We conjecture the existence of special elements in odd degree higher algebraic K-groups of number fields that are related in a precise way to the values at strictly negative integers of the derivatives of Artin L-functions of finite…

Let $\pi$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to…

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an \'etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical…

The principal aim of this article is to attach and study $p$-adic $L$-functions to cohomological cuspidal automorphic representations $\Pi$ of $\mathrm{GL}(2n)$ over a totally real field $F$ admitting a Shalika model. We use a modular…