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Potentially semi-stable deformation rings for discrete series extended types. (Anneaux de déformations potentiellement semi-stables pour les types étendus de la série discrète.) (English. French summary) Zbl 1330.11039
Let \(K/\mathbb{Q}_p\) be a finite extension, and \(\overline{\rho}\) be a mod \(p\) Galois representation of the absolute Galois group \(G_K\). Potentially semi-stable deformation rings for \(\overline{\rho}\) with fixed determinant, Hodge-Tate weights and inertial type have been previously constructed by M. Kisin [J. Am. Math. Soc. 21, No. 2, 513–546 (2008; Zbl 1205.11060)]. An inertial type is a representation of the inertia group \(I_K\) with \(p\)-adic coefficients, which extends to a representation of the Weil group \(W_K\). In this article, the author analyzes potentially semi-stable deformations of \(\overline{\rho}\) with fixed determinant, Hodge-Tate weights and extended inertial type. The author defines the latter as a representation of \(W_K\), and in the article the results are concentrated in the case where this extended inertial type is of discrete series type. She shows that the corresponding deformation ring is a \(p\)-torsion free, reduced quotient of the Kisin deformation ring, it is supported on a set of irreducible components of it, and it is equidimensional after inverting \(p\).
The author then proceeds to express the Hilbert-Samuel multiplicity of the special fibre of these deformation rings, in terms of representations of a certain group \(\mathcal{G}\) which is a quotient of \(\mathcal{O}_D\), where \(D\) is the unique non-split quaternion algebra over \(K\). The formula is proved when \(K=\mathbb{Q}_p\), or for general K when \(\overline{\rho}\) satisfies some additional assumptions, including that the Hodge-Tate weights are \((0, 1)\). The result is obtained from a combination of a reformulation by T. Gee and D. Geraghty of the usual Breuil-Mézard conjecture and modularity lifting theorems on a quaternion algebra ramified at infinity and at primes dividing \(p\) [“The Breuil-Mezard conjecture for quaternion algebras”, Preprint, arXiv:1309.0019]. A consequence of the main formula is that when the choice of extended inertial type divides the deformation ring in two parts, then each of them has the same multiplicity. Finally, the author ends with an application of the results to the existence of congruences mod \(p\) of certain modular forms.

11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
Full Text: DOI arXiv
[1] Ahlgren, S.; Barcau, M., Congruences for modular forms of weights two and four, J. Number Theory, 126, 2, 193-199, (2007) · Zbl 1144.11036
[2] Allen, P., Deformations of polarized automorphic Galois representations and adjoint Selmer groups, (2014)
[3] Berger, L.; Colmez, P., Représentations \(p\)-adiques de groupes \(p\)-adiques. I. Représentations galoisiennes et \((ϕ ,Γ )\)-modules, 319, Familles de représentations de de Rham et monodromie \(p\)-adique, 303-337, (2008), Société Mathématique de France, Paris · Zbl 1168.11020
[4] Breuil, Ch.; Conrad, B.; Diamond, F.; Taylor, R., On the modularity of elliptic curves over \(\mathbb{Q}\): wild 3-adic exercises, J. Amer. Math. Soc., 14, 4, 843-939, (2001) · Zbl 0982.11033
[5] Breuil, Ch.; Diamond, F., Formes modulaires de Hilbert modulo \(p\) et valeurs d’extensions entre caractères galoisiens, Ann. Sci. École Norm. Sup. (4), 47, 5, 905-974, (2014) · Zbl 1309.11046
[6] Bushnell, C. J.; Henniart, G., The local Langlands conjecture for \(\rm GL(2), 335,\) xii+347 pp., (2006), Springer-Verlag, Berlin · Zbl 1100.11041
[7] Breuil, Ch.; Mézard, A., Multiplicités modulaires et représentations de \({\rm GL}_2({\mathbb{Z}}_p)\) et de \({\rm Gal}(\overline{\mathbb{Q}}_p/{\mathbb{Q}}_p)\) en \(ℓ =p,\) Duke Math. J., 115, 2, 205-310, (2002) · Zbl 1042.11030
[8] Barcau, M.; Paşol, V., Mod \(p\) congruences for cusp forms of weight four for \(Γ _0(pN),\) Int. J. Number Theory, 7, 2, 341-350, (2011) · Zbl 1273.11076
[9] Conrad, B.; Diamond, F.; Taylor, R., Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc., 12, 2, 521-567, (1999) · Zbl 0923.11085
[10] Calegari, F.; Stein, W. A., Algorithmic number theory, 3076, Conjectures about discriminants of Hecke algebras of prime level, 140-152, (2004), Springer, Berlin · Zbl 1125.11320
[11] Darmon, H.; Diamond, F.; Taylor, R., Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993), Fermat’s last theorem, 2-140, (1997), Int. Press, Cambridge, MA · Zbl 0877.11035
[12] Emerton, M.; Gee, T., A geometric perspective on the breuil-Mézard conjecture, J. Inst. Math. Jussieu, 13, 1, 183-223, (2014) · Zbl 1318.11061
[13] Fontaine, J.-M., Périodes \(p\)-adiques (Bures-sur-Yvette, 1988), 223, Représentations \(ℓ \)-adiques potentiellement semi-stables, 321-347, (1994), Société Mathématique de France, Paris · Zbl 0873.14020
[14] Fontaine, J.-M., Périodes \(p\)-adiques (Bures-sur-Yvette, 1988), 223, Représentations \(p\)-adiques semi-stables, 113-184, (1994), Société Mathématique de France, Paris · Zbl 0865.14009
[15] Gee, T., Automorphic lifts of prescribed types, Math. Ann., 350, 1, 107-144, (2011) · Zbl 1276.11085
[16] Gérardin, P., Reductive groups and automorphic forms, I (Paris, 1976/1977), 1, Facteurs locaux des algèbres simples de rang \(4\). I, 37-77, (1978), Univ. Paris VII, Paris
[17] Gee, T.; Geraghty, G., The breuil-Mézard conjecture for quaternion algebras, Ann. Inst. Fourier (Grenoble), 65, 1, 1557-1575, (2015) · Zbl 1395.11085
[18] Gee, T.; Kisin, M., The breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math. Pi, 2, e1, 56 pp., (2014) · Zbl 1251.11044
[19] Ghate, E.; Mézard, A., Filtered modules with coefficients, Trans. Amer. Math. Soc., 361, 5, 2243-2261, (2009) · Zbl 1282.11042
[20] Gee, T.; Savitt, D., Serre weights for quaternion algebras, Compositio Math., 147, 4, 1059-1086, (2011) · Zbl 1282.11042
[21] Harris, M.; Taylor, R., The geometry and cohomology of some simple Shimura varieties, 151, viii+276 pp., (2001), Princeton University Press, Princeton, NJ · Zbl 1036.11027
[22] Hu, Y.; Tan, F., The Breuil-Mézard conjecture for non-scalar split residual representations, (2015)
[23] Imai, N., Filtered modules corresponding to potentially semi-stable representations, J. Number Theory, 131, 2, 239-259, (2011) · Zbl 1217.11112
[24] Khare, C., A local analysis of congruences in the \((p,p)\) case. II, Invent. Math., 143, 1, 129-155, (2001) · Zbl 0971.11028
[25] Kisin, M., Potentially semi-stable deformation rings, J. Amer. Math. Soc., 21, 2, 513-546, (2008) · Zbl 1205.11060
[26] Kisin, M., The fontaine-Mazur conjecture for \({\rm GL}_2,\) J. Amer. Math. Soc., 22, 3, 641-690, (2009) · Zbl 1251.11045
[27] Kisin, M., Moduli of finite flat group schemes, and modularity, Ann. of Math. (2), 170, 3, 1085-1180, (2009) · Zbl 1201.14034
[28] Matsumura, H., Commutative ring theory, 8, xiv+320 pp., (1989), Cambridge University Press, Cambridge · Zbl 0666.13002
[29] Paškūnas, V., On the breuil-Mézard conjecture, Duke Math. J., 164, 2, 297-359, (2015) · Zbl 1376.11049
[30] Saito, T., Hilbert modular forms and \(p\)-adic Hodge theory, Compositio Math., 145, 5, 1081-1113, (2009) · Zbl 1259.11060
[31] Taylor, R., On the meromorphic continuation of degree two \(L\)-functions, Doc. Math., 729-779, (2006) · Zbl 1138.11051
[32] Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math., 98, 2, 265-280, (1989) · Zbl 0705.11031
[33] Tokimoto, K., On the reduction modulo \(p\) of representations of a quaternion division algebra over a \(p\)-adic field, J. Number Theory, 150, 136-167, (2015) · Zbl 1321.11117
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