### A basis for the space of modular forms

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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...

We state a conjecture concerning modular absolutely irreducible odd 2-dimensional representations of the absolute Galois group over finite fields which is purely combinatorial (without using modular forms) and proof that it is equivalent to Serre’s strong conjecture. The main idea is to replace modular forms with coefficients in a finite field of characteristic $p$, by their counterparts in the theory of modular symbols.

Let $f$ be a weight $k$ holomorphic automorphic form with respect to ${\mathrm{\Gamma}}_{0}\left(N\right)$. We prove a sufficient condition for the integrality of $f$ over primes dividing $N$. This condition is expressed in terms of the values at particular $CM$ curves of the forms obtained by iterated application of the weight $k$ Maaß operator to $f$ and extends previous results of the Author.

In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL 2 cusp forms modular with respect to the congruence subgroups Γ1(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q 2−δ for any δ > 0, where N is the length of linear forms in the large sieve.

We prove a large sieve type inequality for Maass forms and holomorphic cusp forms with level greater or equal to one and of integral or half-integral weight in short interval.

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...