Un fil d’Ariane pour ce workshop

1(Main Tools)• Modularity Lifting Theorems– MLT for residually reducible representations [SW1],– MLT for potentially Barsotti-Tate deformations [K1],– (MLT for crystalline deformations of intermediate weights [K3]),– MLT for unitary groups [CHT], etc.
• Potential Modularity Theorems– PMT by using Hilbert-Blumenthal modular varieties[T2] (ordinary case), [T3] (crystalline of low weight case),– PMT by using a family of Calabi-Yau varieties [HSBT] (GSp case).
• existence of Strictly Compatible Systems[KW1], [Kh1], and [KW3]– crystalline liftings of low weights,– weight 2 liftings etc.

(Conjectures)• MLT for unitary groups & PMT by using Calabi-Yau familyÃSato-Tate conjecture (under mild condition),• MLT’s & the existence of several kinds of SCS’sÃSerre’s conjecture.

(Influence, or logical dependence)• Wiles’ (3, 5)-trickÃPMT,• Kisin’s modified TW argument in non-minimal casesÃMLT for unitary groups in non-minimal cases,• PMT & MLTÃthe existence of SCS’s,• (p-adic local LanglandsÃMLT for crystalline deformations of intermediate weights),• (p-adic local LanglandsÃBreuil-Mézard conjectureÃMLT for potentially semistabe deformations of arbitrary weights).
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written by Go Yamashita (gokun@kurims.kyoto-u.ac.jp)
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Taylor-Wiles system ([W1], [TW], [D1])• HL1 (Q, ad0 ρ) ∼= Homk (mRL /(λ, m2RL ), k),
its dim.= (the number of topological generators of the corresponding universal
deformation ring).
• dim HL1 (Q, ad0 ρ) = dim HL1 ∗ (Q, ad0 ρ(1))+ (sum of local terms)
by global Tate-Poitou duality, and– (local term at ∞)= −1,– (local term at p in the minimal case)≤ 1 by Fontaine-Laffaille theory,– (local terms (6= p) at “minimally ramified” deformations)= 0,– (local terms at TW-type deformations)= 1,– dim HL1 ∗ (Q, ad0 ρ(1)) − dim HL1 ∗Q (Q, ad0 ρ(1)) = #Qn ,∅n– HL1 ∗Q (Q, ad0 ρ(1)) = 0 by Cebotarev arguments,nÃ
(the number of topological generators of n-th level TW-type universal deformationring)= dim HL1 Qn (Q, ad0 ρ) ≤ dim HL1 ∗Q (Q, ad0 ρ(1)) + #Qn
n= #Qn = dim HL1 ∗ (Q, ad0 ρ(1)) (independent of n).
∅• TQn is free over O[∆Qn ] by de Shalit’s argument (need of mod p multiplicity one)
(resp. HQn is free over O[∆Qn ] by the argument in [D1] (no need of mod p multiplicity one)).
• TW system is not a compatible system with respect to n. We make a compatible
system from TW system by using the argument of “finite isomorphism classes”,
and take a projective limit. In the limit level, the situation is simple. So, we get∼∼R∞ → T∞ in the limit level. We deduce R∅ → T∅ & l.c.i. (resp. + freeness of H∅
over T∅ [D1]) in the finite level from the limit level.
• TΣ is reduced Ã#(O/ηΣ ) < ∞. (cf. we do not know a priori #(pΣ /p2Σ ) < ∞).
• Ihara’ lemma and its generalization +Gorenstein-ness of TΣ0 and TΣ (resp. no needof Gorenstein-ness [D1])Ãcalculation of #(ηΣ /ηΣ0 ) (resp. lengthO ΩΣ0 /ΩΣ , where ΩΣ := HΣ /(HΣ [pΣ ] +HΣ [IΣ ]))Ã#(pΣ0 /p2Σ0 )/(pΣ /p2Σ ) ≤ #(ηΣ /ηΣ0 )(resp. lengthO (pΣ0 /p2Σ0 )/(pΣ /p2Σ ) ≤ lengthO ΩΣ0 /ΩΣ )Ã∼RΣ → TΣ & l.c.i. (resp. + freeness of HΣ over TΣ ) implies∼
RΣ0 → TΣ0 & l.c.i. (resp. + freeness of HΣ0 over TΣ0 [D1]).

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Kisin’s modification of TW argument ([K1], [K3], [K7])• We study a global deformation ring over local deformation ringsÃ– we can show Rred = T even if the local deformation rings at the places dividing
p have complicated singularity, and– we can show Rred = T without level raising in non-minimal cases.
• We consider framed deformations
Ãwe can study local framed deformation rings even if ρ|GQv is not irreducible.
• dim. of Selmer group + local contributionsb v∈Σ Rv .
Ãthe number of topological generators of R over ⊗• We have to study the following things about local framed deformation rings to apply
Kisin’s modified TW argument:(1) calculation of the dimensions of the local deformation rings,(2) to show that the local deformation rings are formally smooth after inverting p,and(3) to show that the local deformation rings are domains.
• The above (1), (2), and (3) are easy in the case of v - p. In the case of v | p :(1) Calculation of the dimension is easy,(2) Formally smooth after inverting p:
Breuil’s theorem (crystalline representations of HT weights in {0,1} come fromp-divisible groups)∼ÃDVflF ,(ξ) → DVcrysξÃcheck explicitly the formally smoothness by constructing a lifting,(3) Domain: Consider a moduli of finite flat models GRvVF ,(a) Tate’s theoremÃGRvVF is isomorphic to SpecRv after inverting p,(b) comparing GRvVF with a complete local ring of a Hilbert modular varietyÃGRvVF ⊗ F is normal, in particular, reduced,(c) Kisin’s theory of S-modules in the integral p-adic Hodge theoryÃthe special fiber GRv,non-ord
is connected by explicit linear algebra calVF ,0culations (repeat connecting a point to another point by P1 ),v,non-ord(d) H0 (SpecRv,non-ord [ p1 ]) ∼⊗ Qp ) by (a)= H0 (GRVFv,non-ord\v,non-ord∼) by (b) ∼) by formal GAGA
= H0 (GRVF= H0 (GRVFv,non-ord ∼∼) = {∗} by (c).
= H0 (GRVF ,04Potential Modularity Theorems ([T2], [T3], [HSBT])
GL2 case ([T2], [T3]):• We want to find a Hilbert-Blumenthal abelian variety A such thatρ∼= A[λ] ←− Tλ A ! T℘ A −→ A[℘] ∼= Indψ.
• We consider Hilbert-Blumenthal modular varieties, and try to find such an abelianvariety as a rational point of this modular variety.
• (We allow “potentiality”)
Moret-Bailly’s theorem
Ãit suffices to find local points (at λ, ℘, and ∞) to get such an abelian variety.
• Ordinary case ([T2]):– Honda-Tate theoryÃfind an abelian variety over a finite field,– Serre-Tate theoryÃfind an abelian variety over a local field.
• Crystalline of low weight case ([T3]):– We consider a twist of the modular variety, which is isomorphic over Q` , Qp1 ,Qp2 , and R,– CM theoryÃfind a Q rational point on the twisted varietyÃfind local points on the original variety,– studying mod ` representations of GL2 (OFλ )Ãchange of weights.

GSpn case ([HSBT]):• We use Calabi-Yau varieties instead of abelian varieties, and a Calabi-Yau family
instead of Hilbert-Blumenthal modular variety.
• The condition of the relation with ρÃwe have to consider a covering of the Calabi-Yau family.
• The Calabi-Yau family has big monodromyÃthe covering is geometrically connectedÃwe can apply Moret-Bailly’s theorem.
• trivial reason, or Fontaine-Laffaille theory, or Serre-Tate theoryÃfind local points.

5existence of Strict Compatible Systems ([KW1], [Kh1], and [KW3])• Savitt’s study of local deformation ringsÃlocal deformation rings we are considering are not zero.
• (Böckle’s method) For θi : H i (GQ,S , ad0 ρ) → ⊕v∈Σ H i (Qv , ad0 ρ),– calculation of dim kerθ1b v∈Σ Rv , andÃthe number of topological generators of R over ⊗1
2– calculation of dim cokerθ + dim kerθb v∈Σ Rv
Ãthe number of relations of R over ⊗Ãdim R ≤ 1.
• PMTÃglobal deformation ring RF of ρ|GF is flat over O by MLT
ÃR/(p) is finite by de Jong’s argument
ÃR is flat over O
Ãwe get a minimally ramified lifting ρ (with some conditions) to characteristic 0.
• PMTÃρ|GF arises from an automorphic representaion π of GL2 (AF )Ãwe can make ρ a part of SCS’s by Brauer’s thoerem:PGρλ := i ni IndGQF (χi ⊗ ρπFi ,λ ),iP
G
where 1 = i ni IndGQF χi (F/Fi ’s are elementary, in particular, solvable), and πFi isian automorphic representation of GL2 (AFi ) such that ρπFi ,℘ ∼= ρ|GFi (we can checkthat ρλ is a true representation).