Functiones et Approximatio Commentarii Mathematici

Congruences between modular forms and related modules

Miriam Ciavarella

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Abstract

Fix a prime $l$ and let $M$ be an integer such that $l\not|M$. Let $f\in S_2(\Gamma_1(M l^2))$ be a newform which is supercuspidal at $l$ of a fixed type related to the nebentypus and special at a finite set of primes. Let $\mathbf{T}^\psi$ be the local quaternionic Hecke algebra associated to $f$. The algebra $\mathbf{T}^\psi$ acts on a module $\mathcal M^\psi_f$ coming from the cohomology of a Shimura curve. It follows from the Taylor-Wiles criterion and a recent Savitt's theorem, that $\mathbf{T}^\psi$ is the universal deformation ring of a global Galois deformation problem associated to $\orho_f$. Moreover $\mathcal M^\psi_f$ is free of rank 2 over $\mathbf{T}^\psi$. If $f$ occurs at minimal level, we prove a result about congruences of ideals and we obtain a raising the level result. The extension of these results to the non minimal case is still an open problem.

Article information

Source
Funct. Approx. Comment. Math., Volume 41, Number 1 (2009), 55-70.

Dates
First available in Project Euclid: 30 September 2009

Permanent link to this document
https://projecteuclid.org/euclid.facm/1254330159

Digital Object Identifier
doi:10.7169/facm/1254330159

Mathematical Reviews number (MathSciNet)
MR2568796

Zentralblatt MATH identifier
1189.11025

Subjects
Primary: 11F80: Galois representations

Keywords
modular form deformation ring Hecke algebra quaternion algebra congruences

Citation

Ciavarella, Miriam. Congruences between modular forms and related modules. Funct. Approx. Comment. Math. 41 (2009), no. 1, 55--70. doi:10.7169/facm/1254330159. https://projecteuclid.org/euclid.facm/1254330159


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References

  • C. Breuil, A. Mézard, Multiplicités modulaires et représentations de $GL_2(\ZZ_p)$ et de $\galois(\overline\QQ_p/\QQ_p)$ en $\ell=p$, with an appendix by Guy Henniart, Duke Math. J. 115, no. 2, (2002), 205--310.
  • M. Ciavarella, Congruences Between Quaternionic Modular Forms and Related Modules, Tesi di Dottorato di Ricerca in Matematica, Università degli Studi di Torino, Italy, A.A.2003/2004.
  • M. Ciavarella, Congruences Between Modular Forms and Related Modules, Nota Preventiva, Bollettino U.M.I. VIII-Vol.IX-B-2, 2006, 507--514.
  • M. Ciavarella, Eisenstein ideal and reducible $\lambda$-adic representations unramified outside a finite number of primes, Bollettino U.M.I., Serie VIII-Vol.IX-B-3, 2006, 711--721.
  • M. Ciavarella, L. Terracini, About an analogue of Ihara's lemma for Shimura curves, preprint 2009.
  • B. Conrad, F. Diamond, R. Taylor, Modularity of certain Potentially Barsotti-Tate Galois Representations, Journal of the American Mathematical Society 12 (2) (1999), 521--567.
  • H. Darmon, F. Diamond, R. Taylor, Fermat's Last Theorem, Current Developments in Mathematics, 1995, International Press, 1--154.
  • H. Darmon, F. Diamond, R. Taylor, Fermat's Last Theorem. Elliptic Curves, Modular Forms and Fermat's Last Theorem (Hong Kong, 1993), 2nd edn, International Press, Cambridge, MA, 1997, 2--140.
  • E. de Shalit, Hecke Rings and Universal Deformation Rings. Modular Forms and Fermat's Last Theorem, G. Cornell, H. Silverman, and G. Stevens, Eds. Springer, 1997, 421--445.
  • B. de Smit, H.W. Lenstra, Explicit construction of universal deformation rings. Modular Forms and Fermat's Last Theorem, G. Cornell, H. Silverman et G. Stevens, Eds. Springer, 1997, 313--326.
  • F. Diamond, R. Taylor, Lifting modular $\modulo\ \ell$ representations, Duke Math. J. 74 (1994), 253--269.
  • F. Diamond, R. Taylor, Non-optimal levels of mod $\ell$ modular representations, Invent. Math. 115 (1994), 435--462.
  • F. Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379--391.
  • J.-M. Fontaine, B. Mazur, Geometric Galois representation, Conference on Elliptic Curves and Modular Forms, Hong Kong, 1993, International Press, 41--78.
  • K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint, Nagoya University, 1996.
  • P. Gerardin. Facteur locaux des algebres simples de rang 4.I. Groupes Réductif et Formes Automorphes I, Publications Mathématiques Univ Paris VII, 1978, 37--77.
  • H. Hida, On $p$-adic Hecke algebras for $GL_2$ over totally reals fields, Ann. of Math. 128 (1988), 295--384.
  • H. Jacquet, R. LANGLANDS, Automorphic forms on $GL_2$, vol. 114, Lecture Notes Math. Springer, 1970.
  • Y. Matsushima, G. Shimura, On the cohomology groups attached to certain vector valued differential forms on the product of the upper half planes, Ann. of Math. 78 (1963), 417--449.
  • B. Mazur, An introduction to the deformation theory of Galois representations. Modular Forms and Fermat's Last Theorem, G. Cornell, H. Silverman, and G. Stevens, Eds. Springer, 1997, 243--311.
  • B. Mazur, Deforming Galois Representations. In Galois Groups over $\QQ$, Ed. Ihara Ribet Serre, Springer, 1989.
  • D. Savitt, On a conjecture of Conrad, Diamond, and Taylor, preprint, 2004.
  • L. Terracini, A Taylor-Wiles system for quaternionic Hecke algebras, Compositio Mathematica 137 (2003), 23--47.
  • A. Wiles, Modular elliptic curves and Fermat last Theorem, Ann. of Math. 141 (1995), 443--551.