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The geometry of Hida families I: \(\Lambda\)-adic de Rham cohomology. (English) Zbl 1441.11098
Summary: We construct the \(\Lambda\)-adic de Rham analogue of Hida’s ordinary \(\Lambda\)-adic étale cohomology and of Ohta’s \(\Lambda\)-adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of \(\mathbb{Q}_p\), we give a purely geometric proof of the expected finiteness, control, and \(\Lambda\)-adic duality theorems. Following M. Ohta [J. Reine Angew. Math. 463, 49–98 (1995; Zbl 0827.11025)], we then prove that our \(\Lambda\)-adic module of differentials is canonically isomorphic to the space of ordinary \(\Lambda\)-adic cuspforms. In the sequel [the author, Compos. Math. 154, No. 4, 719–760 (2018; Zbl 1441.11097)] to this paper, we construct the crystalline counterpart to Hida’s ordinary \(\Lambda\)-adic étale cohomology, and employ integral \(p\)-adic Hodge theory to prove \(\Lambda\)-adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and [the author, loc. cit.], we will be able to provide a “cohomological” construction of the family of \((\varphi,\Gamma)\)-modules attached to Hida’s ordinary \(\Lambda\)-adic étale cohomology by J. Dee [J. Algebra 235, No. 2, 636–664 (2001; Zbl 0984.11062)], as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in B. Mazur and A. Wiles [Compos. Math. 59, 231–264 (1986; Zbl 0654.12008)] and Ohta [loc. cit.].

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G18 Arithmetic aspects of modular and Shimura varieties
11R23 Iwasawa theory
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References:
[1] Angeniol, B.; Zein, F., Appendice: “La classe fondamentale relative d’un cycle”, Bull. Soc. Math. France Mém., 58, 67-93, (1978) · Zbl 0388.14003
[2] Cais, B., Canonical integral structures on the de Rham cohomology of curves, Ann. Inst. Fourier (Grenoble), 59, 2255-2300, (2009) · Zbl 1220.14019
[3] Cais, B., Canonical extensions of Néron models of Jacobians, Algebra Numb. Theory, 4, 111-150, (2010) · Zbl 1193.14058
[4] Cais, B.: The geometry of Hida families II: \(\Lambda \)-adic \((\varphi ,\Gamma ) \)-modules and \(\Lambda \)-adic Hodge theory. Compos. Math. (to appear) · Zbl 1441.11097
[5] Cartier, P., Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris, 244, 426-428, (1957) · Zbl 0077.04502
[6] Carayol, H., Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4), 19, 409-468, (1986) · Zbl 0616.10025
[7] Conrad, B.: Grothendieck Duality and Base Change. Lecture Notes in Mathematics, vol. 1750. Springer, Berlin (2000) · Zbl 0992.14001
[8] Dee, J., \(\Phi \)-\(\Gamma \) modules for families of Galois representations, J. Algebra, 235, 636-664, (2001) · Zbl 0984.11062
[9] Deligne, P.: Formes modulaires et representations e-adiques, Séminaire Bourbaki vol. 1968/69 Exposés 347-363, 139-172 (1971) · Zbl 0206.49901
[10] Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. (40), 5-57 (1971)
[11] Dieudonné, J., Grothendieck, A.: Éléments de géométrie algébrique, Inst. Hautes Études Sci. Publ. Math. 4,8,11,17,20,24,28,37 (1960-7)
[12] Deligne, P.; Illusie, L., Relèvements modulo \(p^2\) et décomposition du complexe de de Rham, Invent. Math., 89, 247-270, (1987) · Zbl 0632.14017
[13] Deligne, P.; Serre, J-P, Formes modulaires de poids \(1\), Ann. Sci. École Norm. Sup. (4), 7, 507-530, (1975) · Zbl 0321.10026
[14] Edixhoven, B.: Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one. J. Inst. Math. Jussieu 5(1), 1-34 (2006) With appendix A (in French) by J. F. Mestre and appendix B by Gabor Wiese · Zbl 1095.14019
[15] Fukaya, T., Kato, K.: On conjectures of Sharifi, Preprint (2012)
[16] Fontaine, J.-M., Messing, W.: \(p\)-adic periods and \(p\)-adic étale cohomology, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, pp. 179-207. Am. Math. Soc. Providence, RI (1987)
[17] Gross, B., A tameness criterion for Galois representations associated to modular forms (mod \(p\)), Duke Math. J., 61, 445-517, (1990) · Zbl 0743.11030
[18] Hartshorne, R.: Residues and Duality, Lecture notes of a Seminar on the Work of A. Grothendieck, given at Harvard 1963/64. With an Appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer, Berlin (1966)
[19] Hida, H., Galois representations into \({\rm GL}_2({ Z}_p[[X]])\) attached to ordinary cusp forms, Invent. Math., 85, 545-613, (1986) · Zbl 0612.10021
[20] Hida, H., Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4), 19, 231-273, (1986) · Zbl 0607.10022
[21] Kitagawa, K: On Standard \(p\)-adic \(L\)-Functions of Families of Elliptic Cusp Forms, \(p\)-Adic Monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), Contemp. Math., vol. 165, pp. 81-110. Am. Math. Soc. Providence, RI (1994)
[22] Katz, N.M., Mazur, B.: Arithmetic moduli of elliptic curves. In: Annals of Mathematics Studies, vol. 108. Princeton University Press, Princeton Nj, pp. xiv+514 (1985) · Zbl 0576.14026
[23] Lazard, M.: Commutative Formal Groups. Lecture Notes in Mathematics, vol. 443. Springer, Berlin (1975) · Zbl 0304.14027
[24] Liu, Q.: Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, Oxford (2002) Translated from the French by Reinie Erné, Oxford Science Publications · Zbl 0996.14005
[25] Matsumura, H.: Commutative Ring Theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge (1989) Translated from the Japanese by M. Reid · Zbl 0666.13002
[26] Mazur, B.; Wiles, A., Analogies between function fields and number fields, Am. J. Math., 105, 507-521, (1983) · Zbl 0531.12015
[27] Mazur, B.; Wiles, A., Class fields of abelian extensions of \({ Q}\), Invent. Math., 76, 179-330, (1984) · Zbl 0545.12005
[28] Mazur, B.; Wiles, A., On \(p\)-adic analytic families of Galois representations, Compos. Math., 59, 231-264, (1986) · Zbl 0654.12008
[29] Nakajima, S., Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants, Math. Z., 190, 559-566, (1985) · Zbl 0559.14022
[30] Oda, T., The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4), 2, 63-135, (1969) · Zbl 0175.47901
[31] Ohta, M., On the \(p\)-adic Eichler-Shimura isomorphism for \(\Lambda \)-adic cusp forms, J. Reine Angew. Math., 463, 49-98, (1995) · Zbl 0827.11025
[32] Ohta, M., Ordinary \(p\)-adic étale cohomology groups attached to towers of elliptic modular curves. II, Math. Ann., 318, 557-583, (2000) · Zbl 0967.11016
[33] Raynaud, M.: Géométrie analytique rigide d’après Tate, Kiehl,\(\cdots \), Table Ronde d’Analyse non archimédienne (Paris, 1972), Soc. Math. France, Paris, pp.  319-327. Bull. Soc. Math. France, Mém. No. 39-40 (1974)
[34] Rosenlicht, M., Extensions of vector groups by abelian varieties, Am. J. Math., 80, 685-714, (1958) · Zbl 0091.33303
[35] Saby, N., Théorie d’Iwasawa géométrique: un théorème de comparaison, J. Numb. Theory, 59, 225-247, (1996) · Zbl 0870.11069
[36] Sen, S.: Continuous cohomology and \(p\)-adic Galois representations. Invent. Math. 62(1), 89-116 (1980/1981) · Zbl 0463.12005
[37] Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique \(p\), pp. 24-53. Universidad Nacional Autónoma de México and UNESCO, Mexico City, Symposium internacional de topología algebraica International symposium on algebraic topology (1958)
[38] Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Mathematics, Vol. 225, Springer-Verlag, Berlin, 1971, Séminaire de Géométrie Algébrique du Bois-Marie 1966-1967 (SGA 6), Dirigé par P. Berthelot, A. Grothendieck et L. Illusie. Avec la collaboration de D. Ferrand, J. P. Jouanolou, O. Jussila, S. Kleiman, M. Raynaud et J. P. Serre
[39] Sharifi, R., Iwasawa theory and the eisenstein ideal, Duke Math. J., 137, 63-101, (2007) · Zbl 1131.11068
[40] Sharifi, R., A reciprocity map and the two-variable \(p\)-adic \(L\)-function, Ann. Math. (2), 173, 251-300, (2011) · Zbl 1248.11085
[41] Tate, J.: \(p\)-Divisible Groups., Proc. Conf. Local Fields (Driebergen, 1966), pp. 158-183. Springer, Berlin (1967) · Zbl 0157.27601
[42] Tate, J., Residues of differentials on curves, Ann. Sci. École Norm. Sup. (4), 1, 149-159, (1968) · Zbl 0159.22702
[43] Tilouine, J., Un sous-groupe \(p\)-divisible de la jacobienne de \(X_1(Np^r)\) comme module sur l’algèbre de Hecke, Bull. Soc. Math. France, 115, 329-360, (1987) · Zbl 0677.14006
[44] Ulmer, D., On universal elliptic curves over Igusa curves, Invent. Math., 99, 377-391, (1990) · Zbl 0705.14024
[45] Wiles, A., On ordinary \(\lambda \)-adic representations associated to modular forms, Invent. Math., 94, 529-573, (1988) · Zbl 0664.10013
[46] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. Math. (2), 131, 493-540, (1990) · Zbl 0719.11071
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