### 2-adic and 3-adic part of class numbers and properties of central values of L-functions

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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...

In this paper, we extend the results of Ribet and Taylor on level-raising for algebraic modular forms on the multiplicative group of a definite quaternion algebra over a totally real field $F$. We do this for automorphic representations of an arbitrary reductive group $G$ over $F$, which is compact at infinity. In the special case where $G$ is an inner form of $\mathrm{GSp}\left(4\right)$ over $\mathbb{Q}$, we use this to produce congruences between Saito-Kurokawa forms and forms with a generic local component.

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$${\mathcal{D}}_{N}(s,f,g)=(\sum _{n=1}^{\infty}\psi \xi \left(n\right){n}^{k+\ell -2s-2}\left)\right(\sum _{n\phantom{\rule{4pt}{0ex}}=1}^{\infty}a\left(n\right)b\left(n\right){n}^{-s}).$$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 $\frac{{\mathcal{D}}_{N}(\ell +m,f,g)}{{\pi}^{\ell +2m+1}\<f,f\>}$ for integers $m$ with $0\le m\<k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $\u27e8f,f\u27e9$ is the Petersson self-inner product of $f$.

I hope this article will be helpful to people who might want a quick overview of how modular representations fit into the theory of deformations of Galois representations. There is also a more specific aim: to sketch a construction of a point-set topological'' configuration (the image of an infinite fern'') which emerges from consideration of modular representations in the universal deformation space of all Galois representations. This is a configuration hinted previously, but now, thanks to some...