Let $D(s,f,g)$ be the Rankin product $L$-function for two Hilbert cusp forms $f$ and $g$. This $L$-function is in fact the standard $L$-function of an automorphic representation of the algebraic group $GL\left(2\right)\times GL\left(2\right)$ defined over a totally real field. Under the ordinarity assumption at a given prime $p$ for $f$ and $g$, we shall construct a $p$-adic analytic function of several variables which interpolates the algebraic part of $D(m,f,g)$ for critical integers $m$, regarding all the ingredients $m$, $f$ and $g$ as variables.

Let $K$ be a local field, $a\in K$ and $\varphi :x\to x^p+a$ where $p$ denotes the characteristic of the residue field. We prove that the minimal subsets of the dynamical system $(K,\varphi )$ are cycles and describe the cycles of this system.

We explore the question of how big the image of a Galois representation attached to a $\Lambda$-adic modular form with no complex multiplication is and show that for a “generic” set of $\Lambda$-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...

We show that the slopes of the $U_5$ operator acting on 5-adic overconvergent modular forms of weight $k$ with primitive Dirichlet character $\chi$ of conductor 25 are given by either$$\left\{\frac{1}{4}·⌊\frac{8i}{5}⌋:i\in \mathbb{N}\right\}\text{or}\left\{\frac{1}{4}·⌊\frac{8i+4}{5}⌋:i\in \mathbb{N}\right\},$$depending on $k$ and $\chi$.We also prove that the space of classical cusp forms of weight $k$ and character $\chi$ has a basis of eigenforms for the Hecke operators $T_p$ and $U_5$ which is defined over ${\mathbf{Q}}_5(\sqrt[4]{5},\sqrt{3})$.

This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent $p$-adic modular symbols. Specifically, we give a constructive proof of acontrol theorem (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated $p$-adic $L$-functions in this case. In...