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 $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n$ and $g={\sum }_{n=1}^{\infty }b\left(n\right)q^n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup ${\Gamma }_0\left(N\right)$ of $S{L}_2\left(\mathbf{Z}\right)$ with “Nebentypus” character $\psi$ and $\xi$ and of weight $k$ and $\ell$, respectively. Define the Rankin product of $f$ and $g$ by $${𝒟}_N(s,f,g)=(\sum _{n=1}^{\infty }\psi \xi \left(n\right)n^{k+\ell -2s-2}\left)\right(\sum _{n=1}^{\infty }a\left(n\right)b\left(n\right)n^{-s}).$$ Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values $\frac{{𝒟}_N(\ell +m,f,g)}{{\pi }^{\ell +2m+1}\<f,f\>}$ for integers $m$ with $0\le m\<k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $⟨f,f⟩$ is the Petersson...

Let $f$ be a primitive cusp form of weight at least 2, and let ${\rho }_f$ be the $p$-adic Galois representation attached to $f$. If $f$ is $p$-ordinary, then it is known that the restriction of ${\rho }_f$ to a decomposition group at $p$ is “upper triangular”. If in addition $f$ has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members...

Special values of certain $L$ functions of the type $L(M,s)$ are studied where $M$ is a motive over a totally real field $F$ with coefficients in another field $T$, and $$L(M,s)=\prod _{𝔭}{L}_{𝔭}(M,𝒩{𝔭}^{-s})$$ is an Euler product $𝔭$ running through maximal ideals of the maximal order ${𝒪}_F$ of $F$ and $$\begin{array}{cc}\hfill L_{𝔭}(M,X{)}^{-1}& =(1-{\alpha }^{\left(1\right)}(𝔭\left)X\right)·(1-{\alpha }^{\left(2\right)}(𝔭\left)X\right)·...·(1-\alpha (d\left)\right(𝔭\left)X\right)\hfill \\ & =1+A_1\left(𝔭\right)X+...+A_d\left(𝔭\right)X^d\hfill \end{array}$$ being a polynomial with coefficients in $T$. Using the Newton and the Hodge polygons of $M$ one formulate a conjectural criterium for the existence of a $p$-adic analytic continuation of the special values....