Talk 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.

Talk 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].

Talk 3: "Global-local compatibility after Carayol.''

We explain the global-local compatibility of Langlands correspondence for Hilbert modular forms in l\neq p after Carayol [Ca1].

Talk 4: Modularity lifting for potentially Barsotti-Tate deformations after Kisin I."

We explain axiomatically Kisin's technique of R^\red=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 in the non-minimal cases too.

Talk 5: "Base change argument of Skinner-Wiles."

We explain Skinner-Wiles level lowering technique allowing solvable field extensions in Kisin's paper [K1].

Talk 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.

Talk 7: "Modularity lifting for potentially Barsotti-Tate deformations after Kisin II.''

The sequel to the previous talk.

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.

Talk 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.

Talk 10: "Modularity lifting of residually reducible case after Skinner-Wiles."

We explain Skinner-Wiles' modularity lifting theorem for residually reducible representations [SW1].

Talk 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.

Talk 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].

Talk 13: "Taylor-Wiles system for unitary groups after Clozel-Harris-Taylor II."

The sequel to the previous talk.

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 a variant of $(3,5)$-trick replaced by a family of Calabi-Yau varieties [HSBT].

Talk 15: "First step of the induction of the proof of Serre's conjecture after Tate, Serre, and Schoof."

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.

Talk 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 Bockle's technique of lower bound of the dimension of global deformation rings [Bo].

We show Serre's conjecture of level one case after Khare [Kh1].

Talk 17: "Proof of Serre's conjecture after Khare-Wintenberger."

We prove Serre's conjecture after Khare-Wintenberger [KW2], [KW3].

Talk 18: "Breuil-Mezard conjecture and modularity lifting for potentially semistable deformations after Kisin."

We explain Breuil-Mezard conjecture, and Kisin's approach of modularity lifting theorem for potentially semistable deformations via Breuil-M\'ezard conjecture [K6].

[Se1]

Serre, J.-P.

"Sur les representations modulaires de degre 2 de Gal(\bar{Q}/Q)."

Duke. Math. J. 54(1) (1987), 179--230.

[Ta1]

Tate, J.

"Algebraic cycles and poles of zeta functions."

Arithmetic Algebraic Geometry, Proc. of Purdue Univ. Conf. 1963, New York, (1965) 93--110.

[FM]

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.

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[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.

[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, and Fermat'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.

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[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]: Axiomization and 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]

Wiles, A.

"On ordinary \lambda-adic representations associated to modular forms."

Invent. Math. 94 (1988), 529--573.

[T1]

Taylor, R.

"On Galois representations associated to Hilbert modular forms."

Invent. Math. 98(2) (1989), 265--280.

[H]

Hida, H. "On p-adic Hecke algebras for GL over totally real fields."

Ann. of Math. (2) 128(2) (1988), 295--384.

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[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] by the congruences.

[H]: GL Hida theory for totally real case.

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[Ca1]

Carayol, H.

"Sur les representations l-adiques associees aux formes modulaires de Hilbert."

Ann. Sci. Ecole Norm. Sup. (4) 19(3) (1986), 409--468.

[S3]

Saito, T.

"Modular forms and and p-adic Hodge theory."

Invent. Math. 129(3) (1997), 607--620.

[S4]

Saito, T.

"Hilbert modular forms and and p-adic Hodge theory."

preprint.

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[Ca1]: Global-local compatibility for l\neq p for totally real case.

[S3]: Global-local compatibility for l= p for Q.

[S4]: Global-local compatibility for l= p for totally real case.

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[SW1]

Skinner, C., Wiles, A.

"Residually reducible representations and modular forms."

Inst. Hautes Etudes Sci. Publ. Math., 89 (2000), 5--126.

[SW2]

Skinner, C., Wiles, A.

"Nearly ordinary deformations of irreducible residual representations."

Ann. Fac. Sci. AToulouse Math. (6) 10(1) (2001), 185--215.

[SW3]

Skinner, C., Wiles, A.

"Base change and a problem of Serre."

Duke Math. J. 107(1) (2001), 15--25.

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[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 to assume that \bar{\rho}|_{Gal(\bar{F}/F(\zeta_p))} is irreducible.

[SW3]: Level lowering technique allowing solvable field extensions.

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[K1]

Kisin, M.,

"Moduli of finite flat group schemes and modularity."

to appear in Ann. of Math.

[PR]

Pappas, G., Rapoport, M.

"Local models in the ramified case. I. The EL-case."

J. Algebraic Geom. 12 (2003), 107--145.

[G]

Gee, T.,

"A modularity lifting theorem for weight two Hilbert modular forms."

Math. Res. Lett. 13 (2006), no. 5, 805--811.

[I]

Imai, N.,

"On the connected components of moduli spaces of finite flat models."

preprint.

[B1]

Breuil, C.,

"Integral p-adic Hodge theory."

Algebraic Geometry 2000, Azumino, Adv. Studies in Pure Math. 36 (2002), 51--80.

[K2]

Kisin, M.,

"Crystalline representations and F-crystals."

Algebraic geometry and number theory, Progr. Math. 253, Volume in honor of Drinfeld's 50th birthday, Birkhauser, Boston (2006), 459--496.

[K3]

Kisin, M.,

"Modularity for some geometric Galois representations."

preprint.

[BLZ]

Berger, L., Li, H., Zhu, H. J.

"Construction of some families of 2-dimensional crystalline representations."

Math. Ann. 329(2) (2004), 365--377.

[BB1]

Berger, L., Breuil, C.,

"Sur la reduction des representations cristallines de dimension 2 en poid moyens."

preprint.

[K4]

Kisin, M.,

"Potentially semi-stable deformation rings."

preprint.

[K5]

Kisin, M.,

"Modularity of 2-adic Barsotti-Tate representations."

preprint.

[K6] Kisin, M.,

"The Fontaine-Mazur conjecture for GL."

preprint.

[K7]

Kisin, M.,

"Modularity of potentially Barsotti-Tate Galois representations."

preprint.

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[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-minimal case.

[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 the residue field is not $\mathbb{F}_p$ and the residual representation is trivial.

[I]: Connectedness of the moduli of finite flat models considered in [K1] in the case where the residue field is not $\mathbb{F}_p$ and the residual representation is not trivial.

[B1]: Used in [K1] to study a moduli of finite flat group schemes in terms of linear algebra.

[K2]: Generalization of [B1], which is a variant of Berger's theory too.

[K3]: Modularity lifting for crystalline representations of intermediate 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 of intermediate weights is used in [K3], and [KW1].

[BB1]: By using p-adic local Langlands ([C1], [C2], and [BB2]), we determineof the mod p reduction of crystalline representations of 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 many cases of Breuil-Mezard conjecture by using p-adic local Langlands ([C1], [C2], and [BB2]), and deduce a modular lifting theorem in a high generality 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."

Annals of Math. Studies 151, PUP 2001.

[CHT]

Clozel, L., Harris, M., Taylor, R.

"Automorphy for some l-adic lifts of automorphic mod l Galois representations."

[T4]

Taylor, R.

"Automorphy for some l-adic lifts of automorphic mod l Galois representations II."

preprint.

[HSBT]

Harris, M., Shepherd-Barron, N., Taylor, R.

"A family of Calabi-Yau varieties and potential automorphy."

preprint.

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[T2]: Potential modularity in the ordinary case.

Variant of (3,5)-trick replaced by Hilbert-Blumenthal 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 GL_n by the "vanishing cycle side" in the sense of Carayol's program.

[CHT]: Taylor-Wiles system for unitary groups.

[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]

Khare, C., Wintenberger, J.-P.

"On Serre's conjecture for 2-dimensional mod p representations of Gal(\bar{Q}/\Q)."

preprint.

[Kh1]

Khare, C.

"Serre's modularity conjecture: the level one case."

Duke Math. J. 134 (2006), 534--567.

[KW2]

Khare, C., Wintenberger, J.-P.

"Serre's modularity conjecture (I)."

preprint.

[KW3]

Khare, C., Wintenberger, J.-P.

"Serre's modularity conjecture (II)."

preprint.

[Kh2]

Khare, C.

"Serre's modularity conjecture: a survey of the level one case."

to appear in Proceedings of the LMS Durham conference L-functions and Galois representations (2005), eds. D. Burns, K.Buzzard, J. Nekovar.

[Kh3]

Khare, C.

"Remarks on mod p forms of weight one."

Internat. Math. Res. Notices (1997), 127--133.

[Ca2]

Carayol, H.

"Sur les representations galoisiennes modulo l attachees aux formes modulaires."

Duke Math. J. 59(1989), no. 3, 785--801.

[Di1]

Dieulefait, L.

"Existence of families of Galois representations and new cases of the Fontaine-Mazur conjecture."

J. Reine Angew. Math. 577 (2004), 147--151.

[Di2]

Dieulefait, L.

"The level 1 case of Serre's conjecture revisited."

preprint.

[Bo]

Bockle, G.

"A local-global principle for deformations of Galois representations."

J. Reine Angew. Math., 509 (1999) 199--236.

[Sa]

Savitt, D.

"On a conjecture of Conrad, Diamond, and Taylor."

Duke. Math. J. 128 (2005), 141--197.

[Sc]

Schoof, R.

"Abelian varieties over Q with bad reduction in one prime only."

Compositio Math. 141 (2005), 847--868.

[Ta2]

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.

[Se2]

Serre, J.-P.

"Oeuvres 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 Bockle's technique.

Starting point of [Kh1], [KW2], and [KW3].

[Kh1]: Proof of Serre's conjecture for level one case.

Construct more general compatible 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 conjecture of level one case, not using the distribution of Fermat primes.

[Bo]: The technique of the lower 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]: Non-existence 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]

Breuil, C., Mezard, A.

"Multiplicites modulaires et representations de GL(Zp) et de Gal(\bar{Qp}/Qp) en l=p."

Duke Math. J. 115(2) (2002), 205--310.

[B2]

Breuil, C.

"Sur quelques representations modulaires e p-adiques de GL(Qp) II."

J. Inst. Mat. Jussieu 2 (2003), 1--36

[C1]

Colmez, P.

"Serie principale unitaire pour GL(Qp) et representations triangulines de dimension 2."

preprint.

[C2]

Colmez, P.

"Une correspondance de Langlands locale p-adique pour les representations semi-stable de dimension 2."

preprint.

[BB2]

Berger, L., Breuil, C.

"Sur quelques representations potentiellement cristallines de GL(Qp)."

preprint.

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[BM]: Breuil-Mezard 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 representations of Gal(\bar{Qp}/Qp) between ``unitary principal series'' of GL(Qp).

Used in [BB1], and [K6].

[C2]: p-adic local Langlands.

By using (\phi,\Gamma)-modules, we construct a correspondence between two dimensional irreducible semistable representations of Gal(\bar{Qp}/Qp) between unitary representations of GL(Qp).

Used in [BB1], and [K6].

[BB2]: p-adic local Langlands.

We associate Banach representations of GL(Qp) to two dimensional potentially crystalline representations of Gal(\bar{Qp}/Qp).

Used in [BB1], and [K6].

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Go Yamashita(RIMS),

Sidai Yasuda(RIMS)