×

zbMATH — the first resource for mathematics

On \(p\)-adic analytic families of Galois representations. (Appendix by Nigel Boston). (English) Zbl 0654.12008
Hida has produced continuous Galois representations \(\rho_ p: G_{\mathbb Q}\to \text{GL}_ 2(\mathbb Z_ p[[t]])\) such that the specializations \(t\mapsto (1+p)^{k-1}- 1\) \((k=2,3,...)\) are \(p\)-adic representations \(\rho_ p^{(k)}\) attached by Deligne to cuspidal newforms of weight \(k\).
In this paper the geometry behind Hida’s construction is studied. It involves the tower of Jacobians \(J_ 1(p^ n)\) of the modular curves \(X_ 1(p^ n)\); the contravariant Tate-modules \[ W_ n=\text{Hom}(J_ 1(p^ n)({\overline {\mathbb Q}}),{\mathbb Q}_ p/{\mathbb Z}_ p) \] and their projective limit \(W\). This \(W\) is a Galois module and a module for the Hecke algebra \(T\). For a suitable maximal ideal \({\mathfrak m}\) of \(T\), the completion \(T_{{\mathfrak m}}\) is isomorphic to \({\mathbb Z}_ p[[ t]]\) and \(W_{{\mathfrak m}}=W\otimes T_{{\mathfrak m}}\) is a free \(T_{{\mathfrak m}}\)-module of rank 2. Hence \(W_{{\mathfrak m}}\) induces a Galois representation \(\rho_ p\) as above. The representation \(\rho_ p\) is unramified outside \(p\). The action of the inertia group at \(p\) is studied via \(p\)-adic Hodge theory. For the representation attached in this way to the cusp form \(\Delta\) of weight 12 and level 1 one finds that for many values of \(p\) the image of \(\rho_ p\) contains \(\text{SL}_ 2({\mathbb Z}_ p[[ t]])\). Further \(\rho_ p^{(1)}\) is in general not the Deligne-Serre representation attached to a newform of weight one; \(\rho_ p^{(1)}\) is not semisimple and its \(p\)-adic Hodge structure is not semisimple.

MSC:
11R39 Langlands-Weil conjectures, nonabelian class field theory
11F33 Congruences for modular and \(p\)-adic modular forms
11F11 Holomorphic modular forms of integral weight
11R37 Class field theory
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14G25 Global ground fields in algebraic geometry
14G20 Local ground fields in algebraic geometry
11F80 Galois representations
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] O., Atkin : Congruences for modular forms . Proceedings of the IBM Conference on Computers in Mathematical Research, Blaricium, 1966. North-Holland (1968) 8-19. · Zbl 0186.36302
[2] C. Curtis and I. Reiner : Representation theory of finite groups and associative algebras . Interscience, New York (1962). · Zbl 0131.25601
[3] P. Deligne and J.-P. Serre : Formes modulaires de poids 1 . Ann. Sci. Ecole Norm. Sup. 7 (1974) 507-730. · Zbl 0321.10026 · doi:10.24033/asens.1277 · numdam:ASENS_1974_4_7_4_507_0 · eudml:81946
[4] G. Faltings : Report on Hodge-Tate-Structures . (preprint. Princeton U. 1985).
[5] A. Grothendieck : Groupes de Monodromie en Géométrie Algébrique . Lecture Notes in Mathematics, 288. Springer-Verlag, Berlin- Heidelberg-New York (1972). · Zbl 0237.00013
[6] R. Hartshorne : Residues and Duality . Lecture Notes in Mathematics, 20 Springer-Verlag, Berlin-Heidelberg-New York (1966). · Zbl 0212.26101 · doi:10.1007/BFb0080482 · eudml:203789
[7] H. Hida : Iwasawa modules attached to congruences of cusp forms . To appear in Ann. Sci. E.N.S. · Zbl 0607.10022 · doi:10.24033/asens.1507 · numdam:ASENS_1986_4_19_2_231_0 · eudml:82176
[8] H. Hida : Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms (preprint) . · Zbl 0612.10021 · doi:10.1007/BF01390329 · eudml:143381
[9] N. Katz : P-adic properties of modular schemes and modular forms. Modular Functions of One Variable III . Lecture Notes in Mathematics, 350. Springer Verlag, Berlin-Heidelberg-New York (1973) 69-141. · Zbl 0271.10033
[10] N. Katz and B. Mazur : Arithmetic moduli of elliptic curves . Annals of Math. Studies, 108. Princeton Univ. Press (1985). · Zbl 0576.14026 · doi:10.1515/9781400881710
[11] W. Li : Newforms and functional equations . Math. Ann. 212 (1975) 285-315. · Zbl 0278.10026 · doi:10.1007/BF01344466 · eudml:182749
[12] B. Mazur : Isogenies of prime degree . Inv. Math. 44 (1978) 129-162. · Zbl 0386.14009 · doi:10.1007/BF01390348 · eudml:142524
[13] B. Mazur : Modular curves and the Eisenstein ideal . Publ. Math. IHES 47 (1978) 33-186. · Zbl 0394.14008 · doi:10.1007/BF02684339 · numdam:PMIHES_1977__47__33_0 · eudml:103950
[14] B. Mazur and A. Wiles : Classfields of abelian extensions of Q . Inv. Math. 76 (1984) 179-330. · Zbl 0545.12005 · doi:10.1007/BF01388599 · eudml:143124
[15] B. Mazur and A. Wiles : Analogies between function fields and number fields . Amer. J. Math. 105 (1983) 507-521. · Zbl 0531.12015 · doi:10.2307/2374266
[16] A. Odlyzko : On conductors and discriminants . · Zbl 0362.12006
[17] A. Ogg : On the eigenvalues of Hecke operators . Math. Ann. 179 (1969) 101-108. · Zbl 0169.10102 · doi:10.1007/BF01350121 · eudml:161768
[18] I. Schur : Arithmetische Untersuchungen über endliche Gruppen linearer Substitutionen , Sitzungsberichte Preuss. Ak. der Wiss. (1906) pp. 164-184. In: Gesammelte Abhandlungen I . Springer-Verlag, Berlin-Heidelberg -New York (1973) 177-197. · JFM 37.0160.01
[19] S. Sen : Continuous cohomology and p-adic Galois representations . Inv. Math. 62 (1981) 89-116. · Zbl 0463.12005 · doi:10.1007/BF01391665 · eudml:142768
[20] J.-P. Serre : Représentations l-adiques . Kyoto Int. Symp. on Algebraic Number Theory (1977) 177-193. · Zbl 0406.14015
[21] J.-P. Serre : Congruences et formes modulaires [d’après H.P.F. Swinnerton-Dyer] Sém . Bourbaki exp. 416. Lecture Notes in Mathematics 317. Springer-Verlag, Berlin-Heidelberg -New York (1973) 319-338. · Zbl 0276.14013 · numdam:SB_1971-1972__14__319_0 · eudml:109819
[22] J.-P. Serre : Linear representations of finite groups . Springer-Verlag, New York-Heidelberg -Berlin (1977). · Zbl 0355.20006
[23] J.-P. Serre and J. Tate : Good reduction of abelian varieties , Ann. of Math. 88 (1968) 492-517. · Zbl 0172.46101 · doi:10.2307/1970722
[24] H.P.F. Swinnerton-Dyer : On l-adic representations and congruences for coefficients of modular forms. Modular Functions of One Variable III . Lecture Notes in Mathematics, 350. Springer-Verlag, Berlin-Heidelberg - New York (1973) 1-56. · Zbl 0267.10032
[25] J. Tate : p-divisible groups . Proceedings of a conference on local fields (Driebergen 1966). Berlin-Heidelberg-New York, Springer-Verlag (1967) 158-183. · Zbl 0157.27601
[26] A. Wiles : Modular curves and the class group of Q(\zeta p) . Inv. Math. 58 (1980) 1-35. · Zbl 0436.12004 · doi:10.1007/BF01402272 · eudml:142712
[27] J. Rose : A Course on Group Theory . Cambridge Univ. Press, London/New York (1978). · Zbl 0371.20001
[28] W. Klingenberg : Lineare Gruppen über lokalen Ringen , Amer. J. Math. 83 (1961) 137-153. · Zbl 0098.02303 · doi:10.2307/2372725
[29] B.R. Mcdonald : Geometric Algebra over Local Rings . Marcel Dekker Inc., New York and Basel (1976). · Zbl 0346.20027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.