Congruences between modular forms and related modules

Miriam Ciavarella

Bollettino dell'Unione Matematica Italiana (2006)

• Volume: 9-B, Issue: 2, page 507-514
• ISSN: 0392-4041

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Abstract

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We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }|̸M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type at $\mathrm{\ell }$ 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 ${\overline{\rho }}_f$. 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.

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Ciavarella, Miriam. "Congruences between modular forms and related modules." Bollettino dell'Unione Matematica Italiana 9-B.2 (2006): 507-514. <http://eudml.org/doc/289625>.

@article{Ciavarella2006,
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.},
author = {Ciavarella, Miriam},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {507-514},
publisher = {Unione Matematica Italiana},
title = {Congruences between modular forms and related modules},
url = {http://eudml.org/doc/289625},
volume = {9-B},
year = {2006},
}

TY - JOUR
AU - Ciavarella, Miriam
TI - Congruences between modular forms and related modules
JO - Bollettino dell'Unione Matematica Italiana
DA - 2006/6//
PB - Unione Matematica Italiana
VL - 9-B
IS - 2
SP - 507
EP - 514
AB - 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.
LA - eng
UR - http://eudml.org/doc/289625
ER -

References

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