Related papers: Conjectures, consequences, and numerical experimen…

We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S5 and A5…

These expository notes introduce $p$-adic $L$-functions and the foundations of Iwasawa theory. We focus on Kubota--Leopoldt's $p$-adic analogue of the Riemann zeta function, which we describe in three different ways. We first present a…

In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the…

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).…

Let $k$ be an imaginary quadratic number field, and $F/k$ a finite abelian extension of Galois group $G$. We show that a Gross conjecture concerning the leading terms of Artin $L$-series holds for $F/k$ and all rational primes which are…

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions,…

In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$…

Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the "admissible" $p$-adic $L$-functions to…

We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate…

We present an analogue of Greenberg-Vatsal's and Emerton-Pollack-Weston's results on congruences of $p$-adic $L$-functions for $p$-non-ordinary cuspidal eigenforms $f$ and $g$ of equal weight that are $p$-congruent. In particular, we prove…

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…

In this article, we study the Iwasawa theory for cuspidal automorphic representations of $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ over CM fields along anticyclotomic directions, in the framework of the Gan--Gross--Prasad conjecture for…

We prove Mazur and Rubin's Iwasawa-theoretic Gross-Zagier conjecture (under some restrictive hypotheses), which relates Heegner points in towers of number fields to the 2-variable p-adic L-function. The result generalizes Perrin-Riou's…

In the classical theory of $L$-series, the exact order (of zero) at a trivial zero is easily computed via the functional equation. In the characteristic $p$ theory, it has long been known that a functional equation of classical $s\mapsto…

Let $f \in S_{2r}(\Gamma_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $\chi$ be an anticyclotomic Hecke…

In this paper, we prove the Iwasawa main conjecture of totally real fields for certain specific non-commutative $p$-adic Lie extensions, using the integral logarithms introduced by Oliver and Taylor. Our result gives certain generalization…

We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…

This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly…

In this paper, we prove that the derivative of the Iwasawa power series associated to p-adic L-functions of $\mathbb Q(\zeta\_p)$ are not divisible by p. This extends previous results obtained by Ferrero and Washington in 1979.