$p$-adic properties of motivic fundamental lines

Fouquet, Olivier

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 37-86.

Nous prouvons la conjecture de compatibilité des droites fondamentales $p$-adiques avec les spécialisations aux points motiviques pour une large classe de familles $p$-adiques de représentations galoisiennes (par exemple, les familles provenant de familles $p$-adiques de représentations automorphes du groupe des unités d’une algèbre de quaternions ou d’un groupe unitaire totalement défini) et en déduisons la compatibilité de la Conjecture Équivariante sur les Nombres de Tamagawa pour ces spécialisations. Néanmoins, nous montrons également que les droites fondamentales ne sont en général pas compatibles avec les spécialisations arbitraires à valeurs dans un anneau intègre de caractéristique zéro. Ceci indique qu’il est nécessaire de modifier la conjecture de [73] en utilisant la cohomologie complétée.

We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic families of automorphic representations of the unit group of a quaternion algebra or of a totally definite unitary group) and deduce the compatibility of the Equivariant Tamagawa Number Conjectures for them. However, we also show that fundamental lines are not compatible with arbitrary characteristic zero specializations with values in a domain in general. This points to the need to modify the conjectures of [73] using completed cohomology.

Classification : 11G40, 11F67, 11F70, 11R23, 11F33
Keywords: Iwasawa theory, $p$-adic automorphic forms
Mot clés : Théorie d’Iwasawa, formes automorphes $p$-adiques

[1] Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA $4$). Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lect. Notes in Math., 269 (1972) (avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat) | Zbl

[2] Beĭlinson, A. A. Higher regulators and values of $L$-functions, Current problems in mathematics (Itogi Nauki i Tekhniki), Volume 24, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 181-238 | Zbl

[3] Bernstein, I. N.; Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. I, Ann. Sci. École Norm. Sup. (4), Volume 10 (1977) no. 4, pp. 441-472 | DOI | Numdam | MR | Zbl

[4] Bernstein, J. N. Le ‘centre’ de Bernstein, Representations of reductive groups over a local field (Deligne, P., ed.) (Travaux en Cours), Hermann, Paris, 1984, pp. 1-32

[5] Blasius, D. Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory (Aspects Math.), Volume E37, Vieweg, Wiesbaden, 2006, pp. 35-56 | DOI | Zbl

[6] Bloch, S. Algebraic cycles and values of $L$-functions, J. reine angew. Math., Volume 350 (1984), pp. 94-108 | MR | Zbl

[7] Bloch, S. Algebraic cycles and values of $L$-functions. II, Duke Math. J., Volume 52 (1985) no. 2, pp. 379-397 | DOI | MR | Zbl

[8] Bloch, S. Algebraic cycles and higher $K$-theory, Adv. in Math., Volume 61 (1986) no. 3, pp. 267-304 | DOI | MR | Zbl

[9] Bloch, S.; Kato, K. $L$-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I (Progress in Math.), Volume 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400 | MR | Zbl

[10] Breuil, Ch.; Schneider, P. First steps towards $p$-adic Langlands functoriality, J. reine angew. Math., Volume 610 (2007), pp. 149-180 | MR | Zbl

[11] Burns, D.; Flach, M. Motivic $L$-functions and Galois module structures, Math. Ann., Volume 305 (1996), pp. 65-102 | DOI | MR | Zbl

[12] Burns, D.; Flach, M. On Galois structure invariants associated to Tate motives, Amer. J. Math., Volume 120 (1998) no. 6, pp. 1343-1397 | DOI | MR | Zbl

[13] Burns, D.; Flach, M. Tamagawa numbers for motives with (non-commutative) coefficients, Doc. Math., Volume 6 (2001), pp. 501-570 | MR | Zbl

[14] Caraiani, A. Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., Volume 161 (2012) no. 12, pp. 2311-2413 | DOI | MR | Zbl

[15] Caraiani, A.; Emerton, M.; Gee, T.; Gerahgty, D.; Paskunas, V.; Shin, S. W. Patching and the $p$-adic local Langlands correspondence, Camb. J. Math., Volume 4 (2016) no. 2, pp. 197-287 | DOI | MR

[16] Carayol, H. Sur les représentations $\ell$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. École Norm. Sup. (4), Volume 19 (1986) no. 3, pp. 409-468 | DOI | MR | Zbl

[17] Carayol, H. Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) (Contemp. Math.), Volume 165, American Mathematical Society, Providence, RI, 1994, pp. 213-237 | DOI | MR | Zbl

[18] Chenevier, G. Une application des variétés de Hecke des groupes unitaires (2009) (available at http://gaetan.chenevier.perso.math.cnrs.fr/pub.html)

[19] Chenevier, G. The $p$-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings, Automorphic forms and Galois representations (Durham, July 2011) (London Math. Soc. Lecture Note Ser.), Volume 414, Cambridge Univ. Press, Cambridge, 2014, pp. 221-285 | DOI | MR | Zbl

[20] Chenevier, G.; Harris, M. Construction of automorphic Galois representations, II, Camb. J. Math., Volume 1 (2013) no. 1, pp. 53-73 | DOI | MR | Zbl

[21] Clozel, L. Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) (Perspect. Math.), Volume 10, Academic Press, Boston, MA, 1990, pp. 77-159 | Zbl

[22] Clozel, L. Représentations galoisiennes associées aux représentations automorphes autoduales de $\mathrm{GL}\left(n\right)$, Publ. Math. Inst. Hautes Études Sci. (1991) no. 73, pp. 97-145 | DOI | MR | Zbl

[23] Clozel, L.; Harris, M.; Taylor, R. Automorphy for some $\ell$-adic lifts of automorphic mod $\ell$ Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008) no. 108, pp. 1-181 | DOI | MR

[24] Deligne, P. Théorie de Hodge. I, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 425-430 | Zbl

[25] Deligne, P. Théorie de Hodge. II, Publ. Math. Inst. Hautes Études Sci. (1971) no. 40, pp. 5-57 | DOI | Numdam | Zbl

[26] Deligne, P. La conjecture de Weil. I, Publ. Math. Inst. Hautes Études Sci. (1974) no. 43, pp. 273-307 | DOI | Numdam | MR

[27] Séminaire de Géométrie Algébrique du Bois-Marie (SGA $4\frac{1}{2}$). Cohomologie étale, Lect. Notes in Math., 569 (1977) (avec la collaboration de J.-F. Boutot, A. Grothendieck, L. Illusie et J.-L. Verdier)

[28] Deligne, P. Valeurs de fonctions $L$ et périodes d’intégrales, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math.), Volume XXXIII, American Mathematical Society, Providence, R.I., 1979, pp. 313-346 | Zbl

[29] Diamond, Fred; Taylor, R. Nonoptimal levels of mod $\ell$ modular representations, Invent. Math., Volume 115 (1994) no. 3, pp. 435-462 | DOI | MR

[30] Emerton, M. A local-global compatibility conjecture in the $p$-adic Langlands programme for ${\mathrm{GL}}_{2/\mathbb{Q}}$, Pure Appl. Math. Q, Volume 2 (2006) no. 2, pp. 279-393 | DOI | MR

[31] Emerton, M. On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math., Volume 164 (2006) no. 1, pp. 1-84 | DOI | MR | Zbl

[32] Emerton, M. Locally analytic representation theory of $p$-adic reductive groups: a summary of some recent developments, $L$-functions and Galois representations (London Math. Soc. Lecture Note Ser.), Volume 320, Cambridge Univ. Press, Cambridge, 2007, pp. 407-437 | DOI | MR | Zbl

[33] Emerton, M. Completed cohomology and the $p$-adic Langlands program, Proceedings ICM, Volume II (2014), pp. 319-342 | Zbl

[34] Emerton, M. Locally analytic vectors in representations of locally $p$-adic analytic groups, Mem. Amer. Math. Soc., American Mathematical Society, Providence, RI, to appear

[35] Emerton, M.; Helm, D. The local Langlands correspondence for ${\mathrm{GL}}_n$ in families, Ann. Sci. École Norm. Sup. (4), Volume 47 (2014) no. 4, pp. 655-722 | DOI | MR | Zbl

[36] Emerton, M.; Pollack, R.; Weston, T. Variation of Iwasawa invariants in Hida families, Invent. Math., Volume 163 (2006) no. 3, pp. 523-580 | DOI | MR | Zbl

[37] Faltings, G. Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, 1988), Johns Hopkins University Press, Baltimore, MD, 1989, pp. 25-80 | Zbl

[38] Faltings, G. Almost étale extensions, Cohomologies $p$-adiques et applications arithmétiques. II (Astérisque), Volume 279, Société Mathématique de France, Paris, 2002, pp. 185-270 | Zbl

[39] Fontaine, J.-M. Sur certains types de représentations $p$-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2), Volume 115 (1982) no. 3, pp. 529-577 | DOI | Zbl

[40] Fontaine, J.-M. Valeurs spéciales des fonctions $L$ des motifs, Séminaire Bourbaki, Vol. 1991/92 (Astérisque), Volume 206, Société Mathématique de France, Paris, 1992, p. 205-249, Exp. No. 751 | Numdam | Zbl

[41] Fontaine, J.-M. Le corps des périodes $p$-adiques, Périodes $p$-adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, Paris, 1994, pp. 59-111

[42] Fontaine, J.-M. Représentations $p$-adiques semi-stables, Périodes $p$-adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, Paris, 1994, pp. 113-184 | Zbl

[43] Fontaine, J.-M.; Mazur, B. Geometric Galois representations, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) (Ser. Number Theory), Volume I, Int. Press, Cambridge, MA, 1995, pp. 41-78 | Zbl

[44] Fontaine, J.-M.; Perrin-Riou, B. Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions $L$, Motives (Seattle, WA, 1991) (Proc. Sympos. Pure Math.), Volume 55, American Mathematical Society, Providence, RI, 1994, pp. 599-706 | MR | Zbl

[45] Fouquet, O. The Equivariant Tamagawa Number Conjecture for Modular Motives with coefficients in the Hecke algebra (2015) (arXiv:1604.06411)

[46] Fouquet, O.; Ochiai, T. Control theorems for Selmer groups of nearly ordinary deformations, J. reine angew. Math., Volume 666 (2012), pp. 163-187 | MR | Zbl

[47] Fujiwara, K. Deformation rings and Hecke algebras in the totally real case (2006) (arXiv:0602606)

[48] Fukaya, T.; Kato, K. A formulation of conjectures on $p$-adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society, Vol. XII (Amer. Math. Soc. Transl. Ser. 2), Volume 219, American Mathematical Society, Providence, RI, 2006, pp. 1-85 | MR | Zbl

[49] Genestier, A.; Tilouine, J. Systèmes de Taylor-Wiles pour ${\mathrm{GSp}}_4$, Formes automorphes. II. Le cas du groupe $\mathrm{GSp}\left(4\right)$ (Astérisque), Volume 302, Société Mathématique de France, Paris, 2005, pp. 177-290 | Zbl

[50] Godement, R.; Jacquet, H. Zeta functions of simple algebras, Lect. Notes in Math., 260, Springer-Verlag, Berlin-New York, 1972 | MR | Zbl

[51] Greenberg, R. Iwasawa theory for $p$-adic representations, Algebraic number theory (Adv. Stud. Pure Math.), Volume 17, Academic Press, Boston, MA, 1989, pp. 97-137 | DOI | MR | Zbl

[52] Greenberg, R. Iwasawa theory for motives, $L$-functions and arithmetic (Durham, 1989) (London Math. Soc. Lecture Note Ser.), Volume 153, Cambridge Univ. Press, Cambridge, 1991, pp. 211-233 | MR | Zbl

[53] Greenberg, R. Galois theory for the Selmer group of an abelian variety, Compositio Math., Volume 136 (2003) no. 3, pp. 255-297 | DOI | MR | Zbl

[54] Greenberg, R.; Vatsal, V. On the Iwasawa invariants of elliptic curves, Invent. Math., Volume 142 (2000) no. 1, pp. 17-63 | DOI | MR | Zbl

[55] Grothendieck, A. On the de Rham cohomology of algebraic varieties, Publ. Math. Inst. Hautes Études Sci. (1966) no. 29, pp. 95-103 | DOI | Numdam | MR | Zbl

[56] Grothendieck, A. Dix exposés sur la cohomologie des schémas, Advanced Studies in Pure Mathematics, 3, North-Holland Publishing Co., Amsterdam, 1968

[57] Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA $7$). Groupes de monodromie en géométrie algébrique. I, Lect. Notes in Math., 288 (1972) (avec la collaboration de M. Raynaud et D. S. Rim) | Zbl

[58] Harris, M.; Taylor, R. The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies, 151, Princeton University Press, Princeton, NJ, 2001 | MR | Zbl

[59] Helm, D. The Bernstein center of the category of smooth $W\left(k\right)\left[{\mathrm{GL}}_n\left(F\right)\right]$-modules, Forum Math. Sigma, Volume 4 (2016) (e11, 98 p.) | DOI | MR

[60] Helm, D. Curtis homomorphisms and the integral Bernstein center for ${GL}_n$ (2016) (arXiv:1605.00487)

[61] Helm, D. Whittaker models and the integral Bernstein center for ${\mathrm{GL}}_{\mathrm{n}}$, Duke Math. J., Volume 165 (2016) no. 9, pp. 1597-1628 | MR

[62] Helm, D.; Moss, G. Converse theorems and the Local Langlands Correspondance in families (2016) (arXiv:1610.03277)

[63] Henniart, G. Sur la conjecture de Langlands locale pour ${\mathrm{GL}}_n$, J. Théor. Nombres Bordeaux, Volume 13 (2001) no. 1, pp. 167-187 | DOI | MR

[64] Hida, H. Galois representations into ${\mathrm{GL}}_2\left({\mathbf{Z}}_p\left[\left[X\right]\right]\right)$ attached to ordinary cusp forms, Invent. Math., Volume 85 (1986) no. 3, pp. 545-613 | DOI | MR

[65] Hida, H. Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4), Volume 19 (1986) no. 2, pp. 231-273 | DOI | Numdam | MR | Zbl

[66] Hida, H. Modules of congruence of Hecke algebras and $L$-functions associated with cusp forms, Amer. J. Math., Volume 110 (1988) no. 2, pp. 323-382 | DOI | MR | Zbl

[67] Hida, H. Hecke fields of analytic families of modular forms, J. Amer. Math. Soc., Volume 24 (2011) no. 1, pp. 51-80 | DOI | MR | Zbl

[68] Hodge, W. V. D. The theory and applications of harmonic integrals, Cambridge University Press, Cambridge, England; Macmillan Company, New York, 1941 | Zbl

[69] Ihara, Y. On modular curves over finite fields, Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), Oxford Univ. Press, Bombay, 1975, pp. 161-202 | Zbl

[70] Illusie, L. Autour du théorème de monodromie locale, Périodes $p$-adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, Paris, 1994, pp. 9-57

[71] Iwasawa, K. On $p$-adic $L$-functions, Ann. of Math. (2), Volume 89 (1969), pp. 198-205 | DOI | Zbl

[72] Kato, K. Iwasawa theory and $p$-adic Hodge theory, Kodai Math. J., Volume 16 (1993) no. 1, pp. 1-31 | DOI | MR | Zbl

[73] Kato, K. Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B_{\mathrm{dR}}$. I, Arithmetic algebraic geometry (Trento, 1991) (Lect. Notes in Math.), Volume 1553, Springer, Berlin, 1993, pp. 50-163 | DOI | MR

[74] Kato, K. $p$-adic Hodge theory and values of zeta functions of modular forms, Cohomologies $p$-adiques et applications arithmétiques. III (Astérisque), Volume 295, Société Mathématique de France, Paris, 2004, pp. 117-290 | Zbl

[75] Langlands, R. P. Modular forms and $\ell$-adic representations, Modular functions of one variable, II (Antwerp, 1972) (Lect. Notes in Math.), Volume 349, Springer, Berlin, 1973, pp. 361-500 | DOI | MR | Zbl

[76] Manin, Yu. I. Periods of cusp forms, and $p$-adic Hecke series, Mat. Sb. (N.S.), Volume 92(134) (1973), p. 378-401, 503 | MR | Zbl

[77] Mazur, B. Rational points of abelian varieties with values in towers of number fields, Invent. Math., Volume 18 (1972), pp. 183-266 | DOI | MR | Zbl

[78] Mazur, B. Notes on étale cohomology of number fields, Ann. Sci. École Norm. Sup. (4), Volume 6 (1973), pp. 521-552 | DOI | Numdam | Zbl

[79] Mazur, B. On the arithmetic of special values of $L$ functions, Invent. Math., Volume 55 (1979) no. 3, pp. 207-240 | DOI | MR | Zbl

[80] Mazur, B. The theme of $p$-adic variation, Mathematics: frontiers and perspectives, American Mathematical Society, Providence, RI, 2000, pp. 433-459 | MR | Zbl

[81] Nekovář, J. Selmer Complexes, Astérisque, 310, Société Mathématique de France, Paris, 2006 | Zbl

[82] Newton, J. Level raising for $p$-adic Hilbert modular forms, J. Théor. Nombres Bordeaux (to appear) (arXiv:1409.6533) | MR | Zbl

[83] Nizioł, W. On uniqueness of $p$-adic period morphisms, Pure Appl. Math. Q, Volume 5 (2009) no. 1, pp. 163-212 | DOI | MR | Zbl

[84] Ochiai, T. Control theorem for Greenberg’s Selmer groups of Galois deformations, J. Number Theory, Volume 88 (2001) no. 1, pp. 59-85 | DOI | MR | Zbl

[85] Ohta, M. On $\ell$-adic representations attached to automorphic forms, Japan. J. Math. (N.S.), Volume 8 (1982) no. 1, pp. 1-47 | MR | Zbl

[86] Perrin-Riou, B. Fonctions $L$ $p$-adiques des représentations $p$-adiques, Astérisque, 229, Société Mathématique de France, Paris, 1995 | Numdam | Zbl

[87] Rapoport, M.; Zink, Th. Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., Volume 68 (1982) no. 1, pp. 21-101 | DOI | Zbl

[88] Raynaud, M. 1-motifs et monodromie géométrique, Périodes $p$-adiques (Bures-sur-Yvette, 1988) (Astérisque), Volume 223, Société Mathématique de France, Paris, 1994, pp. 295-319 | Zbl

[89] Ribet, K. A. On modular representations of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ arising from modular forms, Invent. Math., Volume 100 (1990) no. 2, pp. 431-476 | DOI | MR | Zbl

[90] Ribet, K. A. Raising the levels of modular representations, Séminaire de Théorie des Nombres, Paris 1987–88 (Progress in Math.), Volume 81, Birkhäuser Boston, Boston, MA, 1990, pp. 259-271 | MR | Zbl

[91] Saha, J. P. Purity for families of Galois representations, Ann. Inst. Fourier (Grenoble) (to appear) | MR

[92] Saito, T. Weight spectral sequences and independence of $\ell$, J. Inst. Math. Jussieu, Volume 2 (2003) no. 4, pp. 583-634 | DOI | MR | Zbl

[93] Serre, J.-P. Facteurs locaux des fonctions zêta des variétés algébriques (Définitions et conjectures), Seminaire Delange-Pisot-Poitou (1969/70). Théorie des nombres, Volume 11, Secrétariat mathématique, 1970, pp. 1-15 | Numdam | Zbl

[94] Serre, J.-P.; Tate, J. Good reduction of abelian varieties, Ann. of Math. (2), Volume 88 (1968), pp. 492-517 | DOI | MR | Zbl

[95] Shin, S. W. Galois representations arising from some compact Shimura varieties, Ann. of Math. (2), Volume 173 (2011) no. 3, pp. 1645-1741 | DOI | MR | Zbl

[96] Tate, J. Algebraic cycles and poles of zeta functions, Arithmetical Algebraic Geometry (Purdue Univ., 1963), Harper & Row, New York, 1965, pp. 93-110 | Zbl

[97] Tate, J. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, Séminaire Bourbaki, Vol. 9, Société Mathématique de France, Paris, 1966, p. 415-440, Exp. No. 306 | Numdam | Zbl

[98] Taylor, R. Galois representations, Ann. Fac. Sci. Toulouse Math. (6), Volume 13 (2004) no. 1, pp. 73-119 | DOI | Numdam | Zbl

[99] Taylor, R.; Yoshida, T. Compatibility of local and global Langlands correspondences, J. Amer. Math. Soc., Volume 20 (2007) no. 2, pp. 467-493 | DOI | MR | Zbl

[100] Tsuji, T. $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math., Volume 137 (1999) no. 2, pp. 233-411 | DOI | Zbl

[101] Wiles, A. Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2), Volume 141 (1995) no. 3, pp. 443-551 | DOI | MR | Zbl

[102] Zelevinsky, A. V. Induced representations of reductive $𝔭$-adic groups. II. On irreducible representations of $\mathrm{GL}\left(n\right)$, Ann. Sci. École Norm. Sup. (4), Volume 13 (1980) no. 2, pp. 165-210 | DOI | Numdam | MR | Zbl