Sia $D$ un corpo di quaternioni indefinito su $\mathbf{Q}$ di discriminante $\mathrm{\Delta }$ e sia $\mathrm{\Gamma }$ il gruppo moltiplicativo degli elementi di norma 1 in un ordine di Eichler di $D$ di livello primo con $\mathrm{\Delta }$. Consideriamo lo spazio $S_k\left(\mathrm{\Gamma }\right)$ delle forme cuspidali di peso $k$ rispetto a $\mathrm{\Gamma }$ e la corrispondente algebra di Hecke ${\mathbf{H}}^D$. Utilizzando una versione della corrispondenza di Jacquet-Langlands tra rappresentazioni automorfe di $D^{\times }$ e di $G{L}_2$, realizziamo ${\mathbf{H}}^D$ come quoziente dell'algebra di Hecke classica di livello $N\mathrm{\Delta }$. Questo risultato permette di...

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation". This property can be viewed as a special kind of overconvergence property. One can also go in the opposite direction by using differential functions that arise in a ramified situation to construct "new" (unramified) differential functions.

We study the endomorphism algebra of the motive attached to a non-CM elliptic modular cusp form. We prove that this algebra has a sub-algebra isomorphic to a certain crossed product algebra $X$. The Tate conjecture predicts that $X$ is the full endomorphism algebra of the motive. We also investigate the Brauer class of $X$. For example we show that if the nebentypus is real and $p$ is a prime that does not divide the level, then the local behaviour of $X$ at a place lying above $p$ is essentially determined...

On montre, sous certaines hypothèses un résultat en direction de la conjecture de Serre pour $GS{p}_4$ formulée dans un autre article avec F. Herzig : si la représentation résiduelle associée à une forme de Siegel de genre $2$, de niveau premier à $p$, $p$-ordinaire de poids $p$-petit, laisse stables deux droites (au lieu d’une) dans un plan lagrangien, alors cette forme possède une forme compagnon de poids prescrit. Notre méthode consiste à traduire, grâce au théorème de comparaison mod. $p$ de Faltings, l’existence...

L’objectif de cet article est de mesurer la complexité arithmétique de la courbe modulaire $X_0\left(N\right)$ en fonction du niveau $N$. Pour ce faire, on utilise un morphisme fini (de degré 1 sur son image) de $X_0\left(N\right)$ vers une variété fixe $X\left(1\right)\times X\left(1\right)$ et on calcule la hauteur au sens d’Arakelov de l’image $T_N$ de ce morphisme. La hauteur employée est directement reliée à la hauteur de Faltings des courbes elliptiques. On a besoin pour cela de considérer une théorie d’Arakelov pour les faisceaux inversibles hermitiens $L_1^2$-singuliers (au...

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k - n/2 + 1/2 for Γ₀(4), let f be the corresponding primitive form of weight 2k-n for SL₂(ℤ) under the Shimura correspondence, and Iₙ(h) the Duke-Imamoḡlu-Ikeda lift of h to the space of cusp forms of weight k for Spₙ(ℤ). Moreover, let ${\varphi }_{Iₙ\left(h\right),1}$ be the first Fourier-Jacobi coefficient of Iₙ(h), and ${\sigma }_{n-1}\left({\varphi }_{Iₙ\left(h\right),1}\right)$ be the cusp form in the generalized Kohnen plus space of weight k - 1/2 corresponding to ${\varphi }_{Iₙ\left(h\right),1}$ under the...

It is proved that for almost all prime numbers $k\le N^{1/4-ϵ}$, any fixed integer b₂, (b₂,k) = 1, and almost all integers b₁, 1 ≤ b₁ ≤ k, (b₁,k) = 1, almost all integers n satisfying n ≡ b₁ + b₂ (mod k) can be written as the sum of two primes p₁ and p₂ satisfying $p_i\equiv b_i\left(modk\right)$, i = 1,2. For the proof of this result, new estimates for exponential sums over primes in arithmetic progressions are derived.