Congruences between modular forms and related modules

Ciavarella, Miriam

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 507-514 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We fix $\ell$ a prime and let $M$ be an integer such that $\ell \operatorname{\not|} M$; let $f \in S_2(\Gamma_1(M\ell^2))$ be a newform supercuspidal of fixed type at $\ell$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global Galois deformation problem associated to $\bar\rho_f$. Moreover $M_f$ is free of rank 2 over $T$. If $f$ occurs at minimal level, as a consequence of our results and by the classical Ihara's lemma, we prove a theorem of raising the level and a result about congruence ideals. The extension of this results to the non minimal case is an open problem.

Résumé

-- Fissiamo $\ell$ un primo e $M$ un intero tale che $\ell \operatorname{\not|} M$; sia $f \in S_2(\Gamma_1(M\ell^2))$ una forma nuova supercuspidale di tipo fissato a $\ell$ e speciale in un insieme finito di primi. Per un'algebra di quaternioni indefinita su $Q$, di discriminante che divide il livello di $f$, associamo a $f$ un'algebra di Hecke locale quaternionica $T$. L'algebra $T$ agisce su un modulo $M_f$ proveniente dalla coomologia di una curva di Shimura. Applicando il criterio di Taylor-Wiles e il teorema di Savitt, rivediamo $T$ come l'anello di deformazione universale di un problema di deformazione globale di Galois associato a $\bar\rho_f$. In particolare $M_f$Mf è libero di rango 2 su $T$. Nel caso particolare in cui $f$ sia di livello e minimale, come conseguenza dei nostri risultati e grazie al lemma di Ihara classico, proviamo un teorema di alzamento di livello e un risultato sugli ideali di congruenza. L'estensione al caso non minimale è un problema aperto.

Détail

Publié le : 2006-06-01

MR 2233148 | Zbl 1178.11044

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