p-adic ordinary Hecke algebras for GL (2)
Annales de l'Institut Fourier, Volume 44 (1994) no. 5, p. 1289-1322

We study the p-adic nearly ordinary Hecke algebra for cohomological modular forms on GL(2) over an arbitrary number field F. We prove the control theorem and the independence of the Hecke algebra from the weight. Thus the Hecke algebra is finite over the Iwasawa algebra of the maximal split torus and behaves well under specialization with respect to weight and p-power level. This shows the existence and the uniqueness of the (nearly ordinary) p-adic analytic family of cohomological Hecke eigenforms parametrized by the algebro-geometric spectrum of the Hecke algebra. As for a size of the algebra, we make a conjecture which predicts the Krull dimension of the Hecke algebra. This conjecture implies the Leopoldt conjecture for F and its quadratic extensions containing a CM field. We conclude the paper studying some special cases where the conjecture holds under the hypothesis of the Leopoldt conjecture for F and p.

On étudie l’algèbre de Hecke quasi-ordinaire pour les formes modulaires cohomologiques pour GL (2) sur un corps de nombres F quelconque. On démontre le théorème de contrôle et l’indépendance de l’algèbre relative au poids. Donc l’algèbre de Hecke est finie sur l’algèbre d’Iwasawa du tore déployé maximal et se comporte bien par spécialisation relative à un poids et un niveau fini. On en déduit l’existence et l’unicité de la famille p-adique (quasi-ordinaire) des formes propres de Hecke cohomologiques qui sont paramétrisées par le spectre algébro-géométrique de l’algèbre de Hecke. Pour la taille de l’algèbre de Hecke, on peut faire une conjecture qui prédit la dimension de Krull de l’algèbre de Hecke. Cette conjecture implique la conjecture de Leopoldt sur F et sur ses extensions quadratiques contenant un corps CM. On conclut l’article par l’étude du cas particulier où la conjecture est valable sous l’hypothèse de la conjecture de Leopoldt sur F et p.

@article{AIF_1994__44_5_1289_0,
     author = {Hida, Haruzo},
     title = {$p$-adic ordinary Hecke algebras for ${\rm GL}(2)$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {44},
     number = {5},
     year = {1994},
     pages = {1289-1322},
     doi = {10.5802/aif.1434},
     zbl = {0819.11017},
     mrnumber = {95k:11065},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1994__44_5_1289_0}
}
Hida, Haruzo. $p$-adic ordinary Hecke algebras for ${\rm GL}(2)$. Annales de l'Institut Fourier, Volume 44 (1994) no. 5, pp. 1289-1322. doi : 10.5802/aif.1434. http://www.numdam.org/item/AIF_1994__44_5_1289_0/

[H] H. Hida, Elementary theory of L-functions and Eisenstein series, LMSST 26, Cambridge University Press, 1993. | MR 94j:11044 | Zbl 0942.11024

[H1] H. Hida, p-Ordinary cohomology groups for SL(2) over number fields, Duke Math. J., 69 (1993), 259-314. | MR 94g:11031 | Zbl 0941.11024

[H2] H. Hida, On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv. Studies in Pure Math., 17 (1989), 139-169. | MR 92f:11064 | Zbl 0742.11026

[H3] H. Hida, On the critical values of L-functions of GL(2) and GL(2) ˟ GL(2), Duke Math. J., 74 (1994), 431-530. | MR 98f:11043 | Zbl 0838.11036

[H4] H. Hida, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math., 128 (1988), 295-384. | MR 89m:11046 | Zbl 0658.10034

[H5] H. Hida, Modular p-adic L-functions and p-adic Hecke algebras, in Japanese, Sugaku 44, n° 4 (1992), 1-17 (English translation to appear in Sugaku expositions). | Zbl 0811.11040

[H6] H. Hida, On abelian varieties with complex multiplication as factors of the jacobians of Shimura curves, Amer. J. Math., 103 (1981), 727-776. | MR 82k:10029 | Zbl 0477.14024

[H7] H. Hida, On p-adic L-functions of GL(2) ˟ GL(2) over totally real fields, Ann. Institut Fourier, 41-2 (1991), 311-391. | Numdam | MR 93b:11052 | Zbl 0739.11019

[HT] H. Hida and J. Tilouine, Anti-cyclotomic Katz p-adic L-functions and congruence modules, Ann. Scient. Éc. Norm. Sup., 4-th series, 26 (1993), 189-259. | Numdam | MR 93m:11044 | Zbl 0778.11061