Duke Mathematical Journal

Selmer groups and the Eisenstein-Klingen ideal

Eric Urban

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Article information

Source
Duke Math. J., Volume 106, Number 3 (2001), 485-525.

Dates
First available in Project Euclid: 13 August 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1092403940

Digital Object Identifier
doi:10.1215/S0012-7094-01-10633-9

Mathematical Reviews number (MathSciNet)
MR1813234

Zentralblatt MATH identifier
1061.11027

Subjects
Primary: 11F80: Galois representations
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F75: Cohomology of arithmetic groups 11F85: $p$-adic theory, local fields [See also 14G20, 22E50] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35] 11R23: Iwasawa theory 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11S25: Galois cohomology [See also 12Gxx, 16H05]

Citation

Urban, Eric. Selmer groups and the Eisenstein-Klingen ideal. Duke Math. J. 106 (2001), no. 3, 485--525. doi:10.1215/S0012-7094-01-10633-9. https://projecteuclid.org/euclid.dmj/1092403940


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  • ––––, On the self-dual representations of a $p$-adic group, Internat. Math. Res. Notices 1999, 443–452.
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  • ––––, The CAP representations of $\GSp(4,\bfA)$, J. Reine Angew. Math. 383 (1988), 87–108.
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  • ––––, On the $l$-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), 289–310.
  • \lccR. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572.
  • \lccJ. Tilouine and E. Urban, Familles $p$-adiques à trois variables de formes de Siegel et de représentations galoisiennes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 5–10.
  • ––––, Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. École Norm. Sup. (4) 32 (1999), 499–574.
  • \lccE. Urban, Formes automorphes cuspidales pour $\GL_2$ sur un corps quadratique imaginaire: Valeurs spéciales de fonctions $L$ et congruences, Compositio Math. 99 (1995), 283–324.
  • ––––, Module de congruences pour $\GL(2)$ d'un corps imaginaire quadratique et théorie d'Iwasawa d'un corps $CM$ biquadratique, Duke Math. J. 92 (1998), 179–220.
  • ––––, On residually reducible representations on local rings, J. Algebra 212 (1999), 738–742.
  • ––––, Groupes de Selmer et fonctions $L$ $p$-adic pour les représentations modulaires adjointes, in preparation.
  • ––––, Sur les représentations $p$-adiques associées aux représentations cuspidales de ${\GSp_4}_{/\bfQ}$, preprint, 1998.
  • \lccR. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula), preprint, 1993.
  • ––––, Four dimensional Galois representations, preprint, 2000.
  • \lccA. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493–540.
  • ––––, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443–551. 1996)}, Proc. Nat. Acad. Sci. U.S.A. 94, Nat. Acad. Sci., Washington, D.C., 1997, 11121–11124.
  • \lccR. Howe and I. Piatetski-Shapiro, Some examples of automorphic forms on $\Sp_4$, Duke Math. J. 50 (1983), 55–106.
  • \lccG. Laumon, Sur la cohomologie à supports compacts des variétés de Shimura pour $\GSp(4)_{\bfQ}$, Compositio Math. 105 (1997), 267–359.
  • \lccB. Mazur and J. Tilouine, Représentations galoisiennes, différentielles de Kähler et “conjecture principales,” Inst. Hautes Études Sci. Publ. Math. 71 (1990), 65–103.
  • \lccB. Mazur and A. Wiles, Class fields of abelian extensions of $\bfQ$, Invent. Math. 76 (1984), 179–330.
  • \lccD. Prasad, On the self-dual representations of finite groups of Lie type, J. Algebra 210 (1998), 298–310.
  • ––––, On the self-dual representations of a $p$-adic group, Internat. Math. Res. Notices 1999, 443–452.
  • \lccS. Sen, An infinite-dimensional Hodge-Tate theory, Bull. Soc. Math. France 121 (1993), 13–34.
  • \lccD. Soudry, A uniqueness theorem for representations of $\GSO(6)$ and the strong multiplicity one theorem for generic representations of $\GSp(4)$, Israel J. Math. 58 (1987), 257–287.
  • ––––, The CAP representations of $\GSp(4,\bfA)$, J. Reine Angew. Math. 383 (1988), 87–108.
  • \lccR. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), 281–332.
  • ––––, On the $l$-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), 289–310.
  • \lccR. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572.
  • \lccJ. Tilouine and E. Urban, Familles $p$-adiques à trois variables de formes de Siegel et de représentations galoisiennes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 5–10.
  • ––––, Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. École Norm. Sup. (4) 32 (1999), 499–574.
  • \lccE. Urban, Formes automorphes cuspidales pour $\GL_2$ sur un corps quadratique imaginaire: Valeurs spéciales de fonctions $L$ et congruences, Compositio Math. 99 (1995), 283–324.
  • ––––, Module de congruences pour $\GL(2)$ d'un corps imaginaire quadratique et théorie d'Iwasawa d'un corps $CM$ biquadratique, Duke Math. J. 92 (1998), 179–220.
  • ––––, On residually reducible representations on local rings, J. Algebra 212 (1999), 738–742.
  • ––––, Groupes de Selmer et fonctions $L$ $p$-adic pour les représentations modulaires adjointes, in preparation.
  • ––––, Sur les représentations $p$-adiques associées aux représentations cuspidales de ${\GSp_4}_{/\bfQ}$, preprint, 1998.
  • \lccR. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula), preprint, 1993.
  • ––––, Four dimensional Galois representations, preprint, 2000.
  • \lccA. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493–540.
  • ––––, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443–551. 1996)}, Proc. Nat. Acad. Sci. U.S.A. 94, Nat. Acad. Sci., Washington, D.C., 1997, 11121–11124.
  • \lccR. Howe \and I. Piatetski-Shapiro, Some examples of automorphic forms on $\Sp_4$, Duke Math. J. 50 (1983), 55–106.
  • \lccG. Laumon, Sur la cohomologie à supports compacts des variétés de Shimura pour $\GSp(4)_{\bfQ}$, Compositio Math. 105 (1997), 267–359.
  • \lccB. Mazur \and J. Tilouine, Représentations galoisiennes, différentielles de Kähler et “conjecture principales,” Inst. Hautes Études Sci. Publ. Math. 71 (1990), 65–103.
  • \lccB. Mazur \and A. Wiles, Class fields of abelian extensions of $\bfQ$, Invent. Math. 76 (1984), 179–330.
  • \lccD. Prasad, On the self-dual representations of finite groups of Lie type, J. Algebra 210 (1998), 298–310.
  • ––––, On the self-dual representations of a $p$-adic group, Internat. Math. Res. Notices 1999, 443–452.
  • \lccS. Sen, An infinite-dimensional Hodge-Tate theory, Bull. Soc. Math. France 121 (1993), 13–34.
  • \lccD. Soudry, A uniqueness theorem for representations of $\GSO(6)$ and the strong multiplicity one theorem for generic representations of $\GSp(4)$, Israel J. Math. 58 (1987), 257–287.
  • ––––, The CAP representations of $\GSp(4,\bfA)$, J. Reine Angew. Math. 383 (1988), 87–108.
  • \lccR. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), 281–332.
  • ––––, On the $l$-adic cohomology of Siegel threefolds, Invent. Math. 114 (1993), 289–310.
  • \lccR. Taylor \and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572.
  • \lccJ. Tilouine \and E. Urban, Familles $p$-adiques à trois variables de formes de Siegel et de représentations galoisiennes, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 5–10.
  • ––––, Several-variable $p$-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. École Norm. Sup. (4) 32 (1999), 499–574.
  • \lccE. Urban, Formes automorphes cuspidales pour $\GL_2$ sur un corps quadratique imaginaire: Valeurs spéciales de fonctions $L$ et congruences, Compositio Math. 99 (1995), 283–324.
  • ––––, Module de congruences pour $\GL(2)$ d'un corps imaginaire quadratique et théorie d'Iwasawa d'un corps $CM$ biquadratique, Duke Math. J. 92 (1998), 179–220.
  • ––––, On residually reducible representations on local rings, J. Algebra 212 (1999), 738–742.
  • ––––, Groupes de Selmer et fonctions $L$ $p$-adic pour les représentations modulaires adjointes, in preparation.
  • ––––, Sur les représentations $p$-adiques associées aux représentations cuspidales de ${\GSp_4}_{/\bfQ}$, preprint, 1998.
  • \lccR. Weissauer, The Ramanujan conjecture for genus two Siegel modular forms (an application of the trace formula), preprint, 1993.
  • ––––, Four dimensional Galois representations, preprint, 2000.
  • \lccA. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493–540.
  • ––––, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), 443–551.