9Program:
17/Mar(Mon)15:30-17:00
Go Yamashita (RIMS)“Review of Taylor-Wiles system”
17:30-19:00
Go Yamashita (RIMS)“Galois representations associated to Hilbert modular forms
via congruence after Taylor”, and“Global-local compatibility after Carayol.”
18/Mar(Tue)9:00-10:30
Go Yamashita (RIMS)“Modularity lifting for potentially Barsotti-Tate deformations after Kisin I.”
11:00-12:30
Seidai Yasuda (RIMS)“Base change argument of Skinner-Wiles”, and“Integral p-adic Hodge theory after Breuil and Kisin.”
14:30-16:00
Seidai Yasuda (RIMS)“Modularity lifting for potentially Barsotti-Tate deformations after Kisin II.”
16:30-18:00
Go Yamashita (RIMS)“Modularity lifting for crystalline deformationsof intermediate weights after Kisin.”
19/Mar(Wed)9:00-10:30
Seidai Yasuda (RIMS)“p-adic local Langlands correspondence andmod p reduction of crystalline representations after Berger, Breuil, and Colmez.”
11:00-12:30
Go Yamashita (RIMS)“Modularity lifting of residually reducible case after Skinner-Wiles.”
14:30-16:00
Seidai Yasuda (RIMS)“Potential modularity after Taylor.”
16:30-18:00
Go Yamashita (RIMS)“Taylor-Wiles system for unitary groups after Clozel-Harris-Taylor I.”
18:30–
Party
20/Mar(Thu)
9:00-10:30
Seidai Yasuda (RIMS)“Taylor-Wiles system for unitary groups after Clozel-Harris-Taylor II.”
11:00-12:30
Seidai Yasuda (RIMS)“Proof of Sato-Tate conjecture after Taylor et al.”
14:30-16:00
Go Yamashita (RIMS)“First step of the induction of the proof of Serre’s conjecture
after Tate, Serre, and Schoof.”
16:30-18:00
Go Yamashita (RIMS)“Proof of Serre’s conjecture of level one case after Khare.”
21/Mar(Fri)
9:00-10:30
Seidai Yasuda (RIMS)“Proof of Serre’s conjecture after Khare-Wintenberger.”
11:00-12:30
Seidai Yasuda (RIMS)“Breuil-Mézard conjecture and modularity lifting
for potentially semistable deformations after Kisin.”
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Talks 1:
(1) “Review of Taylor-Wiles system.”
We will give a review of the method of Taylor-Wiles system in [TW], and [D1]. We
also explain how the method of Taylor-Wiles system developed until now.
(2) “Galois representations associated to Hilbert modular forms via congruence after
Taylor.”We explain the construction of Galois representations associated to Hilbert modular
forms in the case of 2 | [F : Q] via congruences after Taylor [T1].
(3) “Global-local compatibility after Carayol.”
We explain the global-local compatibility of Langlands correspondence for Hilbert
modular forms in ` 6= p after Carayol [Ca1].
(4) “Modularity lifting for potentially Barsotti-Tate deformations after Kisin I.”
We explain axiomatically Kisin’s technique of Rred = T in [K1]. We study global
deformation rings over local ones, and a moduli of finite flat group schemes to get
informations about local deformation rings in [K1]. We can use this technique inthe non-minimal cases too.
(5) “Base change argument of Skinner-Wiles.”We explain Skinner-Wiles level lowering technique allowing solvable field extensions
in Kisin’s paper [K1].
(6) “Integral p-adic Hodge theory after Breuil and Kisin.”
We prepare the tools of integral p-adic Hodge theory used in [K1]. We can consider
them as variants of Berger’s theory.
(7) “Modularity lifting for potentially Barsotti-Tate deformations after Kisin II.”
The sequel to the previous talk.
(8) “Modularity lifting for crystalline deformations of intermediate weights after Kisin.”
We show Kisin’s modularity lifting theorem for crystalline deformations of intermediate weights [K3]. We use results of Berger-Li-Zhu [BLZ] and Berger-Breuil [BB1]about mod p reduction of crystalline representations of intermediate weights.
(9) “p-adic local Langlands correspondence and mod p reduction of crystalline representations after Berger, Breuil, and Colmez.”
We explain results of Berger-Li-Zhu and Berger-Breuil about mod p reduction of
crystalline representations of intermediate weights [BLZ], [BB1]. We use p-adic local Langlands ([C1], [C2], [BB2]) in the latter case.
(10) “Modularity lifting of residually reducible case after Skinner-Wiles.”We explain Skinner-Wiles’ modularity lifting theorem for residually reducible representations [SW1].

1written by Go Yamashita (gokun@kurims.kyoto-u.ac.jp)

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(11) “Potential modularity after Taylor.”
We explain Taylor’s potential modularity [T2], [T3]. This is a variant of Wiles’(3, 5)-trick replaced by Hilbert-Blumenthal abelian varieties.
(12) “Taylor-Wiles system for unitary groups after Clozel-Harris-Taylor I.”
We explain Clozel-Harris-Taylor’s Taylor-Wiles system for unitary groups [CHT],
and Taylor’s improvement for non-minimal case by using Kisin’s arguments [T4].
(13) “Taylor-Wiles system for unitary groups after Clozel-Harris-Taylor II.”
The sequel to the previous talk.
(14) “Proof of Sato-Tate conjecture after Taylor et al.”
We show Sato-Tate conjecture after Taylor et al. under mild conditions. We use avariant of (3, 5)-trick replaced by a family of Calabi-Yau varieties [HSBT].
(15) “First step of the induction of the proof of Serre’s conjecture after Tate, Serre, andSchoof.”
We show the first step of the proof of Serre’s conjecture, that is, p = 2 [Ta2],
p = 3 [Se2], and p = 5 [Sc]. We use Odlyzko’s discriminant bound, and Fontaine’s
discriminant bound.
(16) “Proof of Serre’s conjecture of level one case after Khare.”
We explain Khare-Wintenberger’s constuction of compatible systems by using Taylor’s potential modularity [T2], [T3] and Böckle’s technique of lower bound of thedimension of global deformation rings [Bo]. We show Serre’s conjecture of level one
case after Khare [Kh1].
(17) “Proof of Serre’s conjecture after Khare-Wintenberger.”
We prove Serre’s conjecture after Khare-Wintenberger [KW2], [KW3].
(18) “Breuil-Mézard conjecture and modularity lifting for potentially semistable deformations after Kisin.”
We explain Breuil-Mézard conjecture, and Kisin’s approach of modularity lifting
theorem for potentially semistable deformations via Breuil-Mézard conjecture [K6].
References
[Se1]
[Ta1]
[FM]

Serre, J.-P. Sur les représentations modulaires de degré 2 de Gal(Q/Q). Duke. Math. J. 54(1)(1987), 179–230.
Tate, J. Algebraic cycles and poles of zeta functions. Arithmetic Algebraic Geometry, Proc. of
Purdue Univ. Conf. 1963, New York, (1965) 93–110.
Fontaine, J.-M., Mazur, B. Geometric Galois representations. Elliptic Curves, Modular Forms,
and Fermat’s last Theorem (Hong Kong 1993), Internat. Press, Cambridge, MA, 1995, 190–227.

[Se1]: Serre’s conjecture. [Ta1]: Sato-Tate conjecture. [FM]: Fontaine-Mazur conjecture.
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[W1]

Wiles, A. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141(3) (1995),
443–551.
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[TW]

Taylor, R., Wiles, A. Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141(3)(1995), 553–572.
[DDT] Darmon, H., Diamond, F., Taylor, R. Fermat’s last theorem. Elliptic Curves, Modular Forms, andFermat’s last Theorem (Hong Kong 1993), Internat. Press, Cambridge, MA, 1995, 1–154.
[S1]
Saito, T. Fermat conjecture I. Iwanami publisher, 2000.
[S2]
Saito, T. Fermat conjecture II. Iwanami publisher, 2008.
[D1]
Diamond, F. The Taylor-Wiles construction and multiplicity one. Invent. Math. 128 (1997) no.
2, 379–391.
[D2]
Diamond, F. On deformation rings and Hecke rings. Ann. of Math. 144 (1996), 137–166.
[CDT] Conrad, B., Diamond, F., Taylor, R. Modularity of certain potentially Barsotti-Tate Galois representations. J. Amer. Math. Soc. 12(2) (1999), 521–567.
[BCDT] Breuil, C., Conrad, B., Diamond, F., Taylor, R. On the modularity of elliptic curves over Q: wild
3-adic exercises. J. Amer. Math. Soc. 14(4) (2001), 843–939.
[F]
Fujiwara, K. Deformation rings and Hecke algebras for totally real fields. preprint.
[W1]: Fermat’s last theorem. [TW]: Taylor-Wiles system. [DDT]: Survey of the proof of
Fermat’s last theorem. [S1], [S2]: Books about Fermat’s last theorem. [D1]: Axiomizationand improvement of Taylor-Wiles system. The freeness of Hecke modules became the output
from the input. [D2]: Shimura-Taniyama conjecture for elliptic curves, which are semistable
at 3 and 5. [CDT]: Shimura-Taniyama conjecture for elliptic curves, whose conductor is
not divisible by 27. [BCDT]: Shimura-Taniyama conjecture in full generality. [F]: R = T
in totally real case.
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[W2]
[T1]
[H]

Wiles, A. On ordinary λ-adic representations associated to modular forms. Invent. Math. 94(1988), 529–573.
Taylor, R. On Galois representations associated to Hilbert modular forms. Invent. Math. 98(2)(1989), 265–280.
Hida, H. On p-adic Hecke algebras for GL2 over totally real fields. Ann. of Math. (2) 128(2)(1988), 295–384.

[W2]: Construction of Galois representations associated to Hilbert modular forms in the
2 | [F : Q] and nearly ordinary case (including parallel weight 1) by using Hida theory.
[T1]: Construction of Galois representations associated to Hilbert modular forms in the
2 | [F : Q] case by the congruences. [H]: GL2 Hida theory for totally real case.
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[Ca1]
[S3]
[S4]

Carayol, H. Sur les représentations `-adiques associées aux formes modulaires de Hilbert. Ann.
Sci. École Norm. Sup. (4) 19(3) (1986), 409–468.
Saito, T. Modular forms and and p-adic Hodge theory. Invent. Math. 129(3) (1997), 607–620.
Saito, T. Hilbert modular forms and and p-adic Hodge theory. preprint.

[Ca1]: Global-local compatibility for ` 6= p for totally real case. [S3]: Global-local compatibility for ` = p for Q. [S4]: Global-local compatibility for ` = p for totally real case.
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[SW1]
[SW2]

Skinner, C., Wiles, A. Residually reducible representations and modular forms. Inst. Hautes Études
Sci. Publ. Math., 89 (2000), 5–126.
Skinner, C., Wiles, A. Nearly ordinary deformations of irreducible residual representations. Ann.
Fac. Sci. Toulouse Math. (6) 10(1) (2001), 185–215.
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[SW3]

Skinner, C., Wiles, A. Base change and a problem of Serre. Duke Math. J. 107(1) (2001), 15–25.

[SW1]: Modularity lifting in the residulally reducible case. Taylor-Wiles arguments in the
Hida theoretic situations. [SW2]: Modularity lifting for the nearly ordinary deformations in
the residually irreducible case by the method of [SW1]. Minor remark: we do not need toassume that ρ|Gal(F /F (ζp )) is irreducible. [SW3]: Level lowering technique allowing solvable
field extensions.
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[K1]
[PR]
[G]
[I]
[B1]
[K2]

[K3]
[BLZ]
[BB1]
[K4]
[K5]
[K6]
[K7]

Kisin, M., Moduli of finite flat group schemes and modularity. to appear in Ann. of Math.
Pappas, G., Rapoport, M. Local models in the ramified case. I. The EL-case. J. Algebraic Geom.
12 (2003), 107–145.
Gee, T., A modularity lifting theorem for weight two Hilbert modular forms. Math. Res. Lett. 13(2006), no. 5, 805–811.
Imai, N., On the connected components of moduli spaces of finite flat models. preprint.
Breuil, C., Integral p-adic Hodge theory. Algebraic Geometry 2000, Azumino, Adv. Studies inPure Math. 36 (2002), 51–80.
Kisin, M., Crystalline representations and F -crystals. Algebraic geometry and number theory,
Progr. Math. 253, Volume in honor of Drinfeld’s 50th birthday, Birkhäuser, Boston (2006), 459–
496.
Kisin, M., Modularity for some geometric Galois representations. preprint.
Berger, L., Li, H., Zhu, H. J. Construction of some families of 2-dimensional crystalline representations. Math. Ann. 329(2) (2004), 365–377.
Berger, L., Breuil, C., Sur la réduction des représentations cristallines de dimension 2 en poid
moyens. preprint.
Kisin, M., Potentially semi-stable deformation rings. preprint.
Kisin, M., Modularity of 2-adic Barsotti-Tate representations. preprint.
Kisin, M., The Fontaine-Mazur conjecture for GL2 . preprint.
Kisin, M., Modularity of potentially Barsotti-Tate Galois representations. preprint.

[K1]: Furthur improvement of R = T for potentially Barsotti-Tate representations studying
global deformation rings over local ones. We study a moduli of finite flat group schemes to
get informations of local deformation rings. We can also use this technique in non-minimalcase. [PR]: Used in [K1] to get informations of a moduli of finite flat group schemes. [G]:Connectedness of the moduli of finite flat models considered in [K1] in the case where theresidue field is not Fp and the residual representation is trivial. [I]: Connectedness of themoduli of finite flat models considered in [K1] in the case where the residue field is not
Fp and the residual representation is not trivial. [B1]: Used in [K1] to study a moduli offinite flat group schemes in terms of linear algebra. [K2]: Generalization of [B1], which isa variant of Berger’s theory too. [K3]: Modularity lifting for crystalline representations ofintermediate weights by the method of [K1]. [BLZ]: Explicite construction of a family of
Wach modules. The determination of the mod p reduction of crystalline representations ofintermediate weights is used in [K3], and [KW1]. [BB1]: By using p-adic local Langlands([C1], [C2], and [BB2]), we determine the mod p reduction of crystalline representationsof intermediate weights, which are not treated in [BLZ]. This is used in [K3]. [K4]: Construction of potentially semistable deformation rings. [K5]: p = 2 version of [K1]. Used in[KW2] and [KW3]. [K6]: Proof of many cases of Breuil-Mézard conjecture by using p-adic
local Langlands ([C1], [C2], and [BB2]), and deduce a modularity lifting theorem in a highgenerality from this. [K7]: Survey of [K1], [T2], [T3], and [KW1].
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[T2]

Taylor, R. Remarks on a conjecture of Fontaine and Mazur. J. Inst. Math. Jussieu, 1(1) (2002),
125–143.
[T3]
Taylor, R. On the meromorphic continuation of degree two L-functions. Documenta Math. Extra
Volume: John Coates’ Sixtieth Birthday (2006), 729–779.
[BR]
Blasius, D., Rogawski, J. Motives for Hilbert modular forms. Invent. Math. 114 (1993), 55–87.
[HT]
Harris, M., Taylor, R. The geometry and cohomology of some simple Shimura varieties. Annalsof Math. Studies 151, PUP 2001.
[CHT] Clozel, L., Harris, M., Taylor, R. Automorphy for some `-adic lifts of automorphic mod ` Galois
representations.
[T4]
Taylor, R. Automorphy for some `-adic lifts of automorphic mod ` Galois representations II.
preprint.
[HSBT] Harris, M., Shepherd-Barron, N., Taylor, R. A family of Calabi-Yau varieties and potential automorphy. preprint.
[T2]: Potential modularity in the ordinary case. Variant of (3, 5)-trick replaced by HilbertBlumenthal abelian variety. [T3]: Potential modularity in the crystalline of lower weights
case. [BR]: Motive of Hilbert modular forms. Used in [T2] and [T3]. [HT]: local Langlands
for GLn by the “vanishing cycle side” in the sense of Carayol’s program. [CHT]: TaylorWiles system for unitary groups. Proof of Sato-Tate conjecture assuming a generalization of
Ihara’s lemma. [T4]: By using Kisin’s modified Taylor-Wiles arguments [K1], improvements
are made so that we do not need level raising arguments and the generalization of Ihara’s
lemma. [HSBT]: Proof of Sato-Tate conjecture under mild conditions. Variant of (3, 5)-trick
replaced by a family of Calabi-Yau varieties.
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[KW1]
[Kh1]
[KW2]
[KW3]
[Kh2]

[Kh3]
[Ca2]
[Di1]
[Di2]
[Bo]
[Sa]
[Sc]
[Ta2]
[Se2]

Khare, C., Wintenberger, J.-P. On Serre’s conjecture for 2-dimensional mod p representations ofGal(Q/Q). preprint.
Khare, C. Serre’s modularity conjecture: the level one case. Duke Math. J. 134 (2006), 534–567.
Khare, C., Wintenberger, J.-P. Serre’s modularity conjecture (I). preprint.
Khare, C., Wintenberger, J.-P. Serre’s modularity conjecture (II). preprint.
Khare, C. Serre’s modularity conjecture: a survey of the level one case. to appear in Proceedingsof the LMS Durham conference L-functions and Galois representations (2005), eds. D. Burns,
K.Buzzard, J. Nekovar.
Khare, C. Remarks on mod p forms of weight one. Internat. Math. Res. Notices (1997), 127–133.
Carayol, H. Sur les représentations galoisiennes modulo ` attachées aux formes modulaires. Duke
Math. J. 59(1989), no. 3, 785–801.
Dieulefait, L. Existence of families of Galois representations and new cases of the Fontaine-Mazur
conjecture. J. Reine Angew. Math. 577 (2004), 147–151.
Dieulefait, L. The level 1 case of Serre’s conjecture revisited. preprint.
Böckle, G. A local-global principle for deformations of Galois representations. J. Reine Angew.
Math., 509 (1999) 199–236.
Savitt, D. On a conjecture of Conrad, Diamond, and Taylor. Duke. Math. J. 128 (2005), 141–197.
Schoof, R. Abelian varieties over Q with bad reduction in one prime only. Compositio Math. 141(2005), 847–868.
Tate, J. The non-existence of certain Galois extensions of Q unramified outside 2. Arithmetic
Geometry (Tempe, AZ, 1993), Contemp. Math., 174, Amer. Math. Soc., (1994) 153–156.
Serre, J.-P. Œuvres Vol. III. p.710. (1972–1984) Springer-Verlag, Berlin, 1986.

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[KW1]: Constuction of compatible system of minimally ramified lifts by using Taylor’s
potential modularity ([T2] and [T3]) and Böckle’s technique. Starting point of [Kh1], [KW2],
and [KW3]. [Kh1]: Proof of Serre’s conjecture for level one case. Construct more generalcompatible systems than [KW1]. [KW2]: Proof of Serre’s conjecture Part 1. [KW3]: Proof of
Serre’s conjecture Part 2. [Kh2]: Survey of [Kh1]. [Kh3]: Serre’s conjecture implies Artin’s
conjecture for two dimensional odd representations. [Ca2]: Carayol’s lemma used in [KW1],
and [KW2]. [Di1]: Existence of compatible system. [Di2]: Another proof of Serre’s conjectureof level one case, not using the distribution of Fermat primes. [Bo]: The technique of thelower bound of the dimension of global deformation rings by using local deformation rings
used in [KW1], and [Kh1]. [Sa]: Non-vanishing of certain local deformation rings and some
calculations of strongly divisible modules are used in [Kh1], [KW2], and [KW3]. [Sc]: Nonexistence of certain abelian varieties by using Fontaine’s technique and Odlyzko’s bound.
Used in [KW1] to show Serre’s conjecture for p = 5. [Ta2]: Proof of Serre’s conjecture for
p = 2. Minkowski’s bound is used. [Se2]: Proof of Serre’s conjecture for p = 3. Odlyzko’s
bound is used.
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[BM]
[B2]
[C1]
[C2]
[BB2]

Breuil, C., Mézard, A. Multiplicités modulaires et représentations de GL2 (Zp ) et de Gal(Qp /Qp )en ` = p. Duke Math. J. 115(2) (2002), 205–310.
Breuil, C. Sur quelques représentations modulaires et p-adiques de GL2 (Qp ) II. J. Inst. Mat.
Jussieu 2 (2003), 1–36
Colmez, P. Série principale unitaire pour GL2 (Qp ) et représentations triangulines de dimension
2. preprint.
Colmez, P. Une correspondance de Langlands locale p-adique pour les représentations semi-stablede dimension 2. preprint.
Berger, L., Breuil, C. Sur quelques représentations potentiellement cristallines de GL2 (Qp ).
preprint.

[BM]: Breuil-Mézard conjecture, which says Hilbert-Samuel multiplicity of universal deformation rings is explicitly described by the terms of automorphic side. [B2]: Conjecture
about mod p reduction of crystalline representations of intermediate weights, which is partially proved in [BLZ] and [BB1]. This conjecture comes from the insight of “mod p reduction” of p-adic local Langlands. Used in [BB1], and [K6]. [C1]: p-adic local Langlands.
Construction of a bijection between trianguline irreducible two dimensional representationsof Gal(Qp /Qp ) between “unitary principal series” of GL2 (Qp ). Used in [BB1], and [K6].
[C2]: p-adic local Langlands. By using (ϕ, Γ)-modules, we construct a correspondence between two dimensional irreducible semistable representations of Gal(Qp /Qp ) between unitary representations of GL2 (Qp ). Used in [BB1], and [K6]. [BB2]: p-adic local Langlands.
We associate Banach representations of GL2 (Qp ) to two dimensional potentially crystalline
representations of Gal(Qp /Qp ). Used in [BB1], and [K6].
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Un fil d’Ariane pour ce workshop

2(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 family
Sato-Tate conjecture (under mild condition),• MLT’s & the existence of several kinds of SCS’s

Serre’s conjecture.

(Influence, or logical dependence)• Wiles’ (3, 5)-trick
PMT,• Kisin’s modified TW argument in non-minimal cases
MLT for unitary groups in non-minimal cases,• PMT & MLT
the existence of SCS’s,• (p-adic local Langlands
MLT for crystalline deformations of intermediate weights),• (p-adic local Langlands
Breuil-Mézard conjecture
MLT for potentially semistabe deformations of arbitrary weights).
2

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,n(the number of topological generators of n-th level TW-type universal deformation
ring)= 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#(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])calculation of #(ηΣ /ηΣ0 ) (resp. lengthO ΩΣ0 /ΩΣ , where ΩΣ := HΣ /(HΣ [pΣ ] +HΣ [IΣ ]))#(pΣ0 /p2Σ0 )/(pΣ /p2Σ ) ≤ #(ηΣ /ηΣ0 )(resp. lengthO (pΣ0 /p2Σ0 )/(pΣ /p2Σ ) ≤ lengthO ΩΣ0 /ΩΣ )∼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 rings– 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
we can study local framed deformation rings even if ρ|GQv is not irreducible.
• dim. of Selmer group + local contributionsb v∈Σ Rv .
the 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)∼
DVflF ,(ξ) → DVcrys
ξcheck explicitly the formally smoothness by constructing a lifting,(3) Domain: Consider a moduli of finite flat models GRvVF ,(a) Tate’s theorem
GRvVF is isomorphic to SpecRv after inverting p,(b) comparing GRvVF with a complete local ring of a Hilbert modular variety
GRvVF ⊗ F is normal, in particular, reduced,(c) Kisin’s theory of S-modules in the integral p-adic Hodge theory
the 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

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Potential 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
it suﬃces to find local points (at λ, ℘, and ∞) to get such an abelian variety.
• Ordinary case ([T2]):– Honda-Tate theory
find an abelian variety over a finite field,– Serre-Tate theory
find 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 theory
find a Q rational point on the twisted variety
find local points on the original variety,– studying mod ` representations of GL2 (OFλ )change 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 ρ
we have to consider a covering of the Calabi-Yau family.
• The Calabi-Yau family has big monodromy
the covering is geometrically connected
we can apply Moret-Bailly’s theorem.
• trivial reason, or Fontaine-Laﬀaille theory, or Serre-Tate theory
find local points.

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existence of Strict Compatible Systems ([KW1], [Kh1], and [KW3])• Savitt’s study of local deformation rings
local 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 , andthe number of topological generators of R over ⊗1
2– calculation of dim cokerθ + dim kerθb v∈Σ Rv
the number of relations of R over ⊗dim R ≤ 1.
• PMT
global deformation ring RF of ρ|GF is flat over O by MLT
R/(p) is finite by de Jong’s argument
R is flat over O
we get a minimally ramified lifting ρ (with some conditions) to characteristic 0.
• PMT
ρ|GF arises from an automorphic representaion π of GL2 (AF )we can make ρ a part of SCS’s by Brauer’s thoerem:∑Gρλ := i ni IndGQF (χi ⊗ ρπFi ,λ ),i∑Gwhere 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).

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COMPREHENSION CHECK 3
Taylor-Wiles system ([W], [TW], and [D1]).
1 -1. Explain which part of the axioms of Taylor-Wiles system means “R is enough small”,
and which part means “T is enough large”.
1 -2. We need some numerical coincidence to make Taylor-Wiles system. Explain this.
1 -3. How do we kill dual Selmer groups?
1 -4. Explain how Auslander-Buchsbaum theorem was used in Diamond-Fujiwara’s improvement of Taylor-Wiles system.
1 -5. Explain O[∆Q ]-structure of universal deformation rings.
1 -6. Explain O[∆Q ]-structure and its properties of Hecke modules.
1 -7. Why is the Taylor-Wiles system applicable only for “minimal case”?
1 -8. The theory of Taylor-Wiles system was improved, by Faltings ([TW, appendix]), andDiamond-Fujiwara ([D1]). Explain how the inputs and the outputs were changed
about the following things.
(a) T is locally complete intersection,∼(b) R → T, and(c) freeness of Hecke modules.
1 -9. The Gorenstein-ness of Hecke algebras was used in three ways in the original arguments of [W] and [TW]. They are used in minimal case, non-minimal case, and aring theoretic proposition 4. Explain these.
1 -10. What did we deduce the Gorenstein-ness of Hecke algebras from?
1 -11. Now, we do not need to show the Gorenstein-ness of Hecke algebras to use TaylorWiles sytem. How was it improved?
1 -12. How do we use the assumption that “ρ|Q(√(−1)(p−1)/2 p) is absolutely irreducible”?
Explain at least two usage concretely.
1 -13. Explain Ihara’s lemma.
1 -14. Explain (3, 5)-trick.
1 -15. Which does not exist?
(a) elliptic curve,(b) modular curve,(c) Shimura curve,(d) Frey curve, or(e) Fermat curve.

3written by Go Yamashita (gokun@kurims.kyoto-u.ac.jp)
4The assumption of Gorenstein-ness was removed soon by Lenstra about this ring theoretic proposition.

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COMPREHENSION CHECK
Galois representations associated to Hilbert modular forms, and congruences ([T1]).
2 -1. How do we use Shimura curves and the Jacquet-Langlands correspondence to construct Galois representations associated to Hilbert modular forms in the case where[F : Q] is odd or π is special or supercuspidal at some finite place?
2 -2. How do we construct “congruences between old forms and new forms” to construct
Galois representations associated to Hilbert modular forms in the case where [F : Q]
is even?
2 -3. How is the “error term” of congruences controllable?
2 -4. Explain how we show there are enough congruences by using a Hilbert modular
variety (not Shimura curve).
2 -5. Explain in the point of view of applications to R = T why it is useful to construct
Galois representations associated to Hilbert modular form in the case of even degree.

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COMPREHENSION CHECK
Global-local compatibility ([Ca1], [S3], and [S4]).
` 6= p case:
3 -1. Express supersingular loci adelically.
3 -2. Explain how principal series representations appear in the cohomology of the normalization of the special fiber of Shimura curves.
3 -3. Explain how (a part of) special representations appear in the cohomology of supersingular loci of the special fiber of Shimura curves.
3 -4. Explain how supercuspidal representations appear in the cohomology of vanishingcycle sheaves of Shimura curves, and how we can show that σp (π) depends only on
πp .
3 -5. Explain the relation between the cohomology of vanishing cycle sheaves of Shimura
curves, and (GL2 -case of) “vanishing cycle side” of Carayol’s program.
3 -6. Explain how we used the “congruence relation” in 3 -2 and 3 -3.
3 -7. Explain how we can calculate the monodromy operator by using Picard-Lefschetz
formula.
3 -8. Explain how we can show the global-local compatibility for extraordinary representations.
` = p case:
3 -9 Explain how we use Lefschetz trace formulae and weight spectral sequences to compare`-adic side and p-adic side of Weil-Deligne representations.
3 -10 Explain how we use the weight-monodromy conjecture to compare `-adic side andp-adic side of monodromy operators.
3 -11 The only way to compare `-adic side and p-adic side is to use geometry. However,there is not moduli interpretation of Shimura curves, so we cannot consider “KugaSato variety” in a naive way. Explain how we overcome this diﬃculty.

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COMPREHENSION CHECK
Kisin’s modified Taylor-Wiles system ([K1]).
4 -1. What is the merit (for Taylor-Wiles system) of using automorphic forms on a quaternion, which ramifies at all archemedian places?
4 -2. When we consider deformations, which are Barsotti-Tate at p only after wildly ramified extensions, then the universal deformation ring has bad singularity, and wecannot expect that it is locally complete intersection. How did Kisin’s modified
Taylor-Wiles system overcome this diﬃculty?
4 -3. Explain how Kisin’s modified Talyor-Wiles system can treat “non-minimal case”.
4 -4. The properties of local deformation rings are important for Kisin’s modified TaylorWiles system. Explain how the following things are used in the “Rred = T” theorem.
(a) local deformation rings are domain,(b) local deformation rings are formally smooth after inverting p, and(c) calculations of the dimensions of local deformation rings.
4 -5. Calculate the dimension of(a) local deformation rings for v ∈ Σ,(b) local deformation rings for v | p, and(c) global deformation rings.
4 -6. Explain the technique of investigating stalks of points in generic fiber of local deformation rings.
4 -7. How did we get the needed information about local deformation rings from the moduli
of finite flat group schemes?
4 -8. Explain the following things, which we did to get the needed information about themoduli of finite flat group schemes:(a) To relate it with complete local rings of a Hilbert modular variety.
(b) To calculate linear algebraic data.
4 -9. (a) Where is a “geometric incarnation” of Tate’s theorem {p-divisible groups/OK } ,→RepZ (GK ) in [K1]?
p(b) Where did we use Breuil’s theorem Repfltor (GK ) ,→ Reptor (GK∞ ) in [K1]?
(c) Where did we use Breuil’s theorem “crystalline representations of Hodge-Tateweights ⊂ {0, 1} come from p-divisible groups” in [K1]?
4 -10. Explain Kisin’s generalization or another proof of the above 4 -9 (b) and (c), in
terms of modules with connection on open unit disk.
4 -11. Explain Skinner-Wiles’ base change arguments [SW3].

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COMPREHENSION CHECK
Modularity lifting theorem for crystalline representasions of intermediate weights ([K3],[BLZ], [BB1], [C1], and [C2]).
5 -1. What do we use in the case of crystalline deformations of intermediate weights insteadof the moduli of finit flat group schemes?
5 -2. How did we use the information of mod p reduction of crystalline representations ofintermediate weights to show the modularity lifting?
5 -3. Explain the relation between the category of Wach modules (lattices) and the category of crystalline representations (and their lattices).
5 -4. Explain the following two methods of determing mod p reduction of crystalline representations of intermediate weights:(a) the method of Berger-Li-Zhu ([BLZ]), and(b) the method of Berger-Breuil ([BB1]).
5 -5. Explain how the p-adic local Langlands correspondence was used in [BB1].
5 -6. Explain the compatibility of p-adic local Langlands correspondence and mod p local
Langlands correspondence.
5 -7. Explain how the technique of “flatening” was used in Gabber’s appendix in [K3].

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COMPREHENSION CHECK
Modularity lifting theorem for residually reducible representations ([SW1]).
6 -1. Explain pseudo representaions.
6 -2. We do not have deformations to Hecke algebras in the residually reducible case, andonly have pseudo deformations to Hecke algebras. This makes arguments complicated. Explain this, and how we solved this technical problem.
6 -3. In Skinner-Wiles’ technique, we have to make base changes to make codimension ofloci of reducible representaion larger in the spectrum of a universal deformation ring.
Explain this.
6 -4. We use Hida theoretic Hecke algebras in the Skinner-Wiles case. So, the quotient
algebras modulo prime ideals do not have finite cardinality in general. Thus, we haveto modify Taylor-Wiles’ patching arguments. Explain this.
6 -5. We do not have ρ|F (ζp ) is absolutely irreducible in the residually reducible case [SW1],
and we do not need to assume it even in the residually irreducible and nearly ordinary
deformation case [SW2]. How do we modify the Taylor-Wiles arguments, especiallykilling dual Selmer groups?
6 -6. How do we show the existence of Eisenstein ideals by using p-adic L-function in[SW1]?
6 -7. How do we construct “nice” primes in [SW1]?
6 -8. How do we use Washington’s theorem about p-rank of ideal class groups of cyclotomicZ` -extensions in [SW1]?
6 -9. How do we use Mme Raynaud’s theorem in [SW1]?
6 -10. Explain how we show pro-modularity of other irreducible components from promodularity of an irreducible component.
6 -11. How do we use the (technical) concept of “nice” prime?

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COMPREHENSION CHECK
Taylor’s potential modularity theorem ([T2], and [T3]).
7 -1. How do we use a Hilbert-Blumenthal modular variety to do “(`, `0 )-trick”?.
7 -2. How do we use Moret-Bailly’s theorem to do “(`, `0 )-trick”?.
7 -3. Explain the following things to construct “local points”:(a) construction of Hilbert-Blumenthal abelian varieties over a finite field, and HondaTate theory, and(b) construction of a lifting of it to a local field, and Serre-Tate theory.
7 -4. Explain the arguments of raising level, and congruences between diﬀerent weights.
7 -5. How do we use principal ideal theorem in [T3]?
7 -6. Explain the following applications:(a) Fontaine-Mazur conjecture of degree 2, and(b) meromorphic continuation and functional equation of L-functions for (strongly
compatible system of) `-adic representations of degree 2.
7 -7. Explain how we use these potential modularity theorems for Khare-Wintenberger’s
proof of Serre’s conjecture.

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COMPREHENSION CHECK
Modularity lifting theorem for unitary groups, and proof of Sato-Tate conjecture ([CHT],[T4], and [HSBT]).
8 -1. Explain why we use unitary groups to consider GLn .
8 -2. We need some numerical coincidence to use Taylor-Wiles system. Explain this coincidence for unitary groups and essentially self-dual representations.
8 -3. Explain Taylor-Wiles type deformations and O[∆Q ]-structure of universal deformation rings for unitary groups.
8 -4. Explain O[∆Q ]-structure and its properties of Hecke modules for unitary groups.
8 -5. Explain how we used “Ramakrishna deformations” and “one more deformations” in[CHT].
8 -6. How do we avoid using Ihara’s lemma, and make the modularity lifting theorem fornon-minimal case unconditional?
8 -7. We have to know some information about the action of local deformation rings on
Hecke modules to use Kisin’s modified Taylor-Wiles arguments. How do we get the
information about it?
8 -8. How do we use the fact that the Calabi-Yau family we are considering has big monodromy to show the potential modularity?
8 -9. Explain how the potential modularity of Symn H 1 (E) deduce from the modularity
lifting thoerem for unitary groups, and a Calabi-Yau family.
8 -10. Explain how we deduce Sato-Tate conjecture from the potential modularity of Symn H 1 (E)for odd n’s.

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COMPREHENSION CHECK
Serre’s conjecture for p = 2, 3, and 5 ([Ta2], [Se2], and [Sc]).
9 -1. How do we use Minkowski’s (root) discriminant bound and global class field theory
in Tate’s proof of Serre’s conjecture for p = 2?
9 -2. How do we use Odlyzko’s (root) discriminant bound and global class field theory in
Serre’s proof of Serre’s conjecture for p = 3?
9 -3. How do we use Odlyzko’s (root) discriminant bound, Fontaine’s (root) discriminantbound, and global class field theory in Schoof’s proof of Serre’s conjecture for p = 51?
9 -4. How do we show Odlyzko’s (root) discriminant bound by using Dedekind zeta function?
9 -5. How was the PD-structure used for Fontaine’s (root) discriminant bound?
9 -6. Explain the following things in Schoof’s proof [Sc]:(a) How do we use Herbrand’s theorem and p-adic L-function when we consider
extensions of µp by Z/pZ over Spec Z[1/`]?
(b) How do we use Burnside’s basis theorem?
(c) How do we use Taussky’s theorem about 2-groups?
9 -7. Show that any group of order less than 60 is solvable as follows 2:(a) Show that any p-group is nilpotent.
(b) Show that any group G of order pn q (p, q: primes) is solvable as follows:(i) Assume that G does not have non-trivial normal subgroup. Show that the
number of p-Sylow subgroups is q.
(ii) Let D be one of the maximal subgroups, which can be expressed as theintersection of two p-Sylow subgroups. Put H be the normalizer of D in
G. Then, show that H has at least two p-Sylow subgroups. (Hint: any
p-Sylow subgroup of H can be expressed as the intersection of a p-Sylow
subgroup with H.)(iii) Show that H has exactly q p-Sylow subgroups, and all of them contain D.
(iv) Show that D = {1}.
(v) Show that any intersection of diﬀerent two p-Sylow groups is {1}.
(vi) Show that by counting elements, G has unique q-Sylow subgroup, so it isnormal.
(c) Show that any group of order pn q 2 (p > q primes) is solvable by similar arguments as above.
(d) Show that groups of order 30 = 2.3.5 and of order 42 = 2.3.7 are solvable.

1Precisely speaking, he proved non-existence of certain abelian varieties having restricted reduction

conditions.
2If we use Feit-Thompson’s theorem (any group of odd order is solvable) and Burnside’s theorem (anygroup of order pn q m (p, q: primes) is solvable), then it is easy. However, these theorems are very diﬃcult.
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COMPREHENSION CHECKExistence of strictly compatible system, and proof of Serre’s conjecture ([KW1], [Kh1],[KW2], and [KW3]).
10 -1. Prove Fermat’s Last Theorem without “Langlands-Tunnell’s theorem” and “Ribet’s
level lowering” by the method of strictly compatible systems.
10 -2. How do we use Taylor’s potential modularity (twice) for the proof of the existenceof strictly compatible systems?
10 -3. For the existence of strictly compatible systems, we show the following things about
global deformation rings. Explain the proof of them.
(a) Lower bound of the dimension, and(b) Flatness over O.
10 -4. Explain kinds of the strictly compatible systems (3 kinds in the level one case, 4kinds in general).
10 -5. Explain Dieulefait’s another proof of Serre’s conjecture of level one case, which do
not use a distribution of Fermat primes [Di2].
10 -6. Write down how the inductions are arranged in [Kh1], and [KW2].
10 -7. Explain the following induction steps:(a) “add a good dihedral prime”,(b) “killing ramification”, and(c) “weight reduction”.
10 -8. Explain applications of Serre’s conjecture about:(a) some Fermat-like diophantine equations,(b) non-existence of some finite flat group schemes,(c) the finiteness of isomorphism classes of mod p representations of Gal(Q/Q) of
bounded ramifications,(d) modularity of abelian varieties of GL2 -type,(e) Artin’s conjecture of degree 2, and(f) Fontaine-Mazur conjecture of degree 2.

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COMPREHENSION CHECK
Modularity lifting theorem and potential modularity for p = 2 ([Kh1], [KW3], and [K5]).
11 -1. We have (k · id) ⊂ ad0 ρ in the case of p = 2. So, we do not have (ad0 ρ)∗ ∼= ad0 ρ inthis case. How do we modify the usual Galois cohomology calculations in this case([KW3])?
11 -2. How do we modify the arguments of killing dual Selmer groups in the case of p = 2, 3
in [Kh1] (p = 3) and [KW3] (p = 2, 3) 1?
11 -3. How do we treat the “neatness problem” in the case of p = 2, 3 in [Kh1] (p = 3)
and [KW3] (p = 2, 3)?
11 -4. We cannot take a finite place v above an odd prime such that
(1 − N v)((1 + N v)2 det ρ(Frv ) − N v(trρ(Frv ))2 ) ∈ F×in the case of p = 2. How do we overcome this diﬃculty in [KW3] and [K5]?
11 -5. We cannot use Breuil’s theory in the case of p = 2. So, we use Zink’s theory of
displays and windows in [K5]. How do we use this to overcome the diﬃculty?

1In [T3], the case p = 2, 3 are excluded.

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COMPREHENSION CHECK
Modularity lifting theorem and Breuil-Mézard conjecture ([K6], [BM], [C1], and [C2]).
12 -1. Explain Breuil-Mézard conjecture.
12 -2. How do we use Breuil-Mézard conjecture to show the modularity lifting theorem
for potentially semistable deformations? In particular, how do we overcome thediﬃculty that we do not know that the local deformation rings are domains?
12 -3. How do we use the p-adic local Langlands correspondence in Kisin’s proof of manycases of Breuil-Mézard conjecture.
12 -4. Explain Colmez’ functor in the p-adic local Langlands correspondence.
12 -5. Explain the compatibility of the p-adic local Langlands correspondence and classicallocal Langlands correspondence.
12 -6. Explain how we use the compatibility of p-adic local Langlands correspondence andclassical local Langlands correspondence in Kisin’s proof of many cases of BreuilMézard conjecture.

32