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

In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.

We study Artin $L$-functions on a finite $2$-dimensional complex $X_\Gamma$ arising from PGL$_3$ attached to finite-dimensional representations $\rho$ of its fundamental group. Some key properties, such as rationality, functional equation,…

We establish an effective Bertini-type theorem for hypersurfaces $X_f \colon f = 0$ defined over a finite field $k$ for which $f$ has no linear factors over the algebraic closure $\overline{k}$. Given a line $L$ defined over $k$ and a…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to an ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified…

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 study the theory of finite-order p-adic functions and distributions on ray class groups of number fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphic forms on GL(2) which may be…

We formulate a conjecture about extra zeros of p-adic L-functions at near central points which generalises the conjecture formulated in our previous paper. We prove that this conjecture is compatible with Perrin-Riou's theory of p-adic…

We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

This paper concerns our earlier conjecture about the equivalence of a derived completion construction applied to the representation spectrum of the absolute Galois group of a geometric field is equivalent to the algebraic K-theory of the…

We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we…

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to…

We prove that (under the assumption of the generalized Riemann hypothesis) a totally real multiquadratic number field $K$ has a positive density of primes $p \in \mathbb{Z}$ for which the image of the unit group $(\mathcal{O}_K)^{\times})$…

We construct examples of p-adic L-functions over universal deformation spaces for GL(2). We formulate a conjecture predicting that the natural parameter spaces for p-adic L-functions are not the usual eigenvarieties (parametrising…

We prove a new automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call `potential diagonalizability'. This result allows for `change of weight' and seems to be substantially more flexible…

We determine all irreducible representations of primary quasi-cyclotomic fields in this paper. The methods can be applied to determine the irreducible representations of any quasi-cyclotomic field. We also compute the Artin L-functions for…

For \psi a nontrivial additive character on the finite field F_q, the map t \mapsto \sum_{x \in F_q} \psi(f(x)+tx) is the Fourier transform of the map t \mapsto \psi(f(t))$. As is well-known, this has a cohomological interpretation,…

We prove the classical $l = p$ local-global compatibility conjecture for certain regular algebraic cuspidal automorphic representations of weight 0 for GL$_2$ over CM fields. Using an automorphy lifting theorem, we show that if the…

We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local…

Let $K/F$ be a quadratic extension of p-adic fields. The Bernstein-Zelevinsky's classification asserts that generic representations are parabolically induced from quasi-square-integrable representations. We show, following a method…