TOMIYAMA, Yoshiyuki

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Title Lifting Galois representations over arbitrary number fields : Aresume (Algebraic Number Theory and Related Topics 2008)Author(s)TOMIYAMA, YoshiyukiCitation数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu(2010), B19: 23-24Issue Date2010-06URL http://hdl.handle.net/2433/176885RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityRIMS Ko^kyu^roku BessatsuB19 (2010), 23{24Lifting Galois representations over arbitrary numberelds: A resumeByYoshiyuki TomiyamaLet k be a nite eld of characteristic p > 5. Let K be a number eld of nitedegree over Q and GK its absolute Galois group Gal( K=K). Let  : GK ! GL2(k) bea continuous representation and let W (k) be the ring of Witt vectors of k. We considerthe following question:Question 0.1. Is there a continuous representation  : GK ! GL2(W (k)) sat-isfying  =  mod p?This question has been motivated by a conjecture of Serre ([S]) saying that all oddabsolutely irreducible continuous representations  : GQ ! GL2(k) are modular. Thisimplies the existence of a lift to characteristic zero. This conjecture was proved byKhare and Wintenberger in [KW1, KW2].In [K], Khare proved the existence of a lift toW (k) for any  : GK ! GL2(k) whichis reducible. Thus we may assume that  is irreducible.For a place v of K, let Kv be the completion of K at v, and let Gv be its absoluteGalois group Gal( Kv=Kv). Let Ad0 be the k-vector space of all trace zero two-by-twomatrices over k on which GK acts by conjugation. Our main result is the following:Theorem 0.2. Assume that H2(Gv;Ad0) = 0 for each place v j p. Then lifts to a continuous representation  : GK ! GL2(W (k)) which is unramied outside anite set of places of K.For K = Q, Ramakrishna proved under very general conditions on  that there existlifts to W (k) in [R1, R2]. Gee ([G]) and Manoharmayum ([M]) proved, independently,that there exist lifts to W (k) for K satisfying [K(p) : K] > 2, where p is the groupof p-th roots of unity.Our method used in the proof of the Theorem is essentially that of Ramakrishna[R1, R2], but we follow the more axiomatic treatment presented in [Ta]. We denote by SReceived March 23, 2009. Revised June 1, 2009.2000 Mathematics Subject Classication(s): 11F80Graduate School of Mathematics, Kyushu University, Fukuoka 812-8581, Japan.e-mail: tomiyama@math.kyushu-u.ac.jpc 2010 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.24 Yoshiyuki Tomiyamaa nite set of places of K containing the places above p, the innite places and the placesat which  is ramied. Let KS denote the maximal algebraic extension of K unramiedoutside S and put GK;S = Gal(KS=K). Thus  factors through GK;S . The existenceof a lifting of  follows from the triviality of a certain class of H2(GK;S ;Ad0). Since itis dicult to calculate it directly, we reduce the calculations of global obstructions tothose of local obstructions, which are much better understood. By Taylor [Ta], it boilsdown to showing the vanishing of a certain Selmer group. We can prove the trivialityof this group by extending S by a suitably chosen nite set Q. With the choice of Q ofRamakrishna and Gee, they could not prove the triviality of the Selmer group for anarbitrary number eld. Our choice of Q is dierent from Ramakrishna's and Gee's. Formore details, see our preprint [To].References[G] T. Gee, Companion forms over totally real elds, II, Duke Math. J. 136 (2007), 275{284.[K] C. Khare, Base Change, Lifting and Serre's Conjecture, J. Number Theory 63 (1997),387{395.[KW1] C. Khare and J.-P. Wintenberger, Serre's modularity conjecture (I), preprint[KW2] C. Khare and J.-P. Wintenberger, Serre's modularity conjecture (II), preprint[M] J. Manoharmayum, Lifting Galois representations of number elds, preprint[R1] R. Ramakrishna, Lifting Galois representations, Invent. Math. 138 (1999), 537{562.[R2] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre andFontaine-Mazur, Ann. of Math. 156 (2002), 115{154.[S] J.-P. Serre, Sur les representations modulaires de degre 2 de Gal(Q=Q), Duke Math. J. 54(1987), 179{230.[Ta] R. Taylor, On icosahedral Artin representations, II, Amer. J. Math. 125 (2003), 549{566.[To] Y. Tomiyama, Lifting Galois representations over arbitrary number elds, preprint avail-able online at http://arxiv.org/abs/0809.3076.

Lifting Galois representations over arbitrary number fields : A resume (Algebraic Number Theory and Related Topics 2008)

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Lifting Galois representations over arbitrary number fields : A resume (Algebraic Number Theory and Related Topics 2008)

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