# $p$-adic ordinary Hecke algebras for $\mathrm{GL}\left(2\right)$

Annales de l'institut Fourier (1994)

- Volume: 44, Issue: 5, page 1289-1322
- ISSN: 0373-0956

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topHida, Haruzo. "$p$-adic ordinary Hecke algebras for ${\rm GL}(2)$." Annales de l'institut Fourier 44.5 (1994): 1289-1322. <http://eudml.org/doc/75098>.

@article{Hida1994,

abstract = {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 $p$.},

author = {Hida, Haruzo},

journal = {Annales de l'institut Fourier},

keywords = {-adic nearly ordinary Hecke algebra; weight; -power level; cohomological Hecke eigenforms; Krull dimension; Leopoldt conjecture},

language = {eng},

number = {5},

pages = {1289-1322},

publisher = {Association des Annales de l'Institut Fourier},

title = {$p$-adic ordinary Hecke algebras for $\{\rm GL\}(2)$},

url = {http://eudml.org/doc/75098},

volume = {44},

year = {1994},

}

TY - JOUR

AU - Hida, Haruzo

TI - $p$-adic ordinary Hecke algebras for ${\rm GL}(2)$

JO - Annales de l'institut Fourier

PY - 1994

PB - Association des Annales de l'Institut Fourier

VL - 44

IS - 5

SP - 1289

EP - 1322

AB - 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 $p$.

LA - eng

KW - -adic nearly ordinary Hecke algebra; weight; -power level; cohomological Hecke eigenforms; Krull dimension; Leopoldt conjecture

UR - http://eudml.org/doc/75098

ER -

## References

top- [H] H. HIDA, Elementary theory of L-functions and Eisenstein series, LMSST 26, Cambridge University Press, 1993. Zbl0942.11024MR94j:11044
- [H1] H. HIDA, p-Ordinary cohomology groups for SL(2) over number fields, Duke Math. J., 69 (1993), 259-314. Zbl0941.11024MR94g:11031
- [H2] H. HIDA, On nearly ordinary Hecke algebras for GL(2) over totally real fields, Adv. Studies in Pure Math., 17 (1989), 139-169. Zbl0742.11026MR92f:11064
- [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. Zbl0838.11036MR98f:11043
- [H4] H. HIDA, On p-adic Hecke algebras for GL2 over totally real fields, Ann. of Math., 128 (1988), 295-384. Zbl0658.10034MR89m:11046
- [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). Zbl0811.11040
- [H6] H. HIDA, On abelian varieties with complex multiplication as factors of the jacobians of Shimura curves, Amer. J. Math., 103 (1981), 727-776. Zbl0477.14024MR82k:10029
- [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. Zbl0739.11019MR93b:11052
- [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. Zbl0778.11061MR93m:11044

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