Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire $X_0\left(N\right)$ comme un quotient du groupe ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$. En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de $N$ d’un relèvement de l’action des opérateurs de Hecke de $H_1(X_0\left(N\right),ptes,\mathbf{Z})$ sur ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$.

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...

In this paper we construct $p$-adic measures related to the values of convolutions of Hilbert modular forms of integral and half-integral weight at the negative critical points under the assumption that the underlying totally real number field $F$ has class number $h_F=1$. This extends the result of Panchishkin [Lecture Notes in Math., 1471, Springer Verlag, 1991 ] who treated the case that both modular forms are of integral weight. In order to define the measures, we need to introduce the twist operator...

We construct $p$-adic $L$-functions (in general case unbounded) attached to “motivic" primitive Hilbert cusp forms as a non-archimedean Mellin transform of the corresponding admissible measure. In order to prove the growth conditions of the appropriate complex-valued distributions we represent them as Rankin type representation and use Atkin–Lehner theory and explicit form of Fourier coefficients of Eisenstein series.

We study the critical values of the complex standard-$L$-function attached to a holomorphic Siegel modular form and of the twists of the $L$-function by Dirichlet characters. Our main object is for a fixed rational prime number $p$ to interpolate $p$-adically the essentially algebraic critical $L$-values as the Dirichlet character varies thus providing a systematic control of denominators of critical values by generalized Kummer congruences. In order to organize this information we prove the existence of...

We study the $p$-adic nearly ordinary Hecke algebra for cohomological modular forms on $GL\left(2\right)$ 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...

We show that certain quadratic base change $L$-functions for $\mathrm{Gl}\left(2\right)$ are non-negative at their center of symmetry.

We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic$p$-adic $L$-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$-adic $L$-functions to more general Shimura curves.

For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined $F_k(D;x):={\sum }_{\begin{array}{c}a,b,c\in \mathbb{Z},a<0\\ b^2-4ac=D\end{array}}max(0,{(a{x}^2+bx+c)}^{k-1})$. Here we use the theory of periods to give identities and congruences which relate various values of $F_k(D;x)$.