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].

1

written by Go Yamashita (gokun@kurims.kyoto-u.ac.jp)
12(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.

¶³[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.

3[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.
¶³[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.

¶³[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.

¶³[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.

¶³[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.

¶³[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.
¶³[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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³

[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.

¶³[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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