EUDML

Let $f = \sum_{n = 1}^{\infty} a (n) q^{n}$ and $g = \sum_{n = 1}^{\infty} b (n) q^{n}$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup $Γ_{0} (N)$ of $S L_{2} (Z)$ with “Nebentypus” character $ψ$ and $ξ$ and of weight $k$ and $ℓ$ , respectively. Define the Rankin product of $f$ and $g$ by $𝒟_{N} (s, f, g) = (\sum_{n = 1}^{\infty} ψ ξ (n) n^{k + ℓ - 2 s - 2}) (\sum_{n = 1}^{\infty} a (n) b (n) n^{- s}) .$ Supposing $f$ and $g$ to be ordinary at a prime $p \geq 5$ , we shall construct a $p$ -adically analytic $L$ -function of three variables which interpolate the values $\frac{𝒟_{N} (ℓ + m, f, g)}{π^{ℓ + 2 m + 1} < f, f >}$ for integers $m$ with $0 \leq m < k - 1,$ by regarding all the ingredients $m$ , $f$ and $g$ as variables. Here $⟨ f, f ⟩$ is the Petersson self-inner...

On $p$ -adic $L$ -functions of $G L (2) \times G L (2)$ over totally real fields

Haruzo Hida — 1991

Annales de l'institut Fourier

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 $G L (2) \times G L (2)$ 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.

Iwasawa modules attached to congruences of cusp forms

Haruzo Hida — 1986

Annales scientifiques de l'École Normale Supérieure

$p$ -adic ordinary Hecke algebras for $GL (2)$

Haruzo Hida — 1994

Annales de l'institut Fourier

We study the $p$ -adic nearly ordinary Hecke algebra for cohomological modular forms on $G L (2)$ over an arbitrary number field $F$ . We prove the control theorem and the independence of the Hecke algebra from the weight. Thus the Hecke algebra is finite over the Iwasawa algebra of the maximal split torus and behaves well under specialization with respect to weight and $p$ -power level. This shows the existence and the uniqueness of the (nearly ordinary) $p$ -adic analytic family of cohomological Hecke eigenforms...

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Global quadratic units and Hecke algebras.

Automorphism groups of Shimura varieties.

Anticyclotomic main conjectures.

On the search of genuine $p$ -adic modular $L$ -functions for $G L (n)$ . With a correction to: On $p$ -adic $L$ -functions of $G L (2) \times G L (2)$ over totally real fields

Convolution de Rankin p-adique.

On Congruence Divisors of Cusp Forms as Factors of Special Values of Their Zeta Functions.

Congruences of Cusp Forms and Special Values of Their Zeta Functions.

Kummer's Criterion for the Special Values of Hecke L-Functions of Imaginary Quadratic Fields and Congruences Among Cusp Forms.

A p-adic measure attached to the zeta functions associated with two elliptic modular forms. I.

Control theorems of $p$ -nearly ordinary cohomology groups for $SL (n)$

A $p$ -adic measure attached to the zeta functions associated with two elliptic modular forms. II

On $p$ -adic $L$ -functions of $G L (2) \times G L (2)$ over totally real fields

Iwasawa modules attached to congruences of cusp forms

$p$ -adic ordinary Hecke algebras for $GL (2)$