( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q

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1 2017 : msjmeeting-2017sep-00f006 p Langlands ( ) 1. Q, Q p Q Galois G Q p (p Galois ). p Galois ( p Galois ), L Selmer Tate-Shafarevich, Galois. Dirichlet ( Dedekind s = 0 ) Birch-Swinnerton-Dyer ( L s = 1 Tate-Shafarevich ),, p Galois Bloch- ([BK90]),. L Galois ( )., Dirichlet,, p Galois ([Ka04]). 90 ([Ka93a], [FK06]), p Galois, p Galois ( ). p Galois.,, Q(ζ n ) (n 1) Euler ), Galois., p Galois Langlands, p Galois L L, p Galois L. L, L., L Galois., l Q l Galois G Ql p, Galois ( :16K17556) 2010 Mathematics Subject Classification: 11F80 (primary), 11F85, 11S25 (secondary), p Langlands [email protected]

2 ( ),, ( [Ka93b],[FK06]).,. p Galois L, Langlands p p Galois (, ) p., Breuil, Colmez([Co10]), Q p Galois G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) (GL 2 (Q p ) p Langlands )., GL 2 (Q p ) p Langlands, G Qp 2 p ( ), ( ) ([Na17b]). p Langlands?. 2., 2.1., ? p Galois V,,. Q, L (V ), (Hodge p ). Q l, (L, ) (L, ), l = p de Rham )D dr (V ), Bloch-Kato exponential., V (V ) 1 z geom (V ) (V ) (Q ), ε geom l (V ) (V ) (Q l ) (, Q V, L Beilinson ). p Galois p Galois

3 ., Z p [[T 1,..., T d ]] G Q G Ql T., X := Spec(Z p [[T 1,..., T d ]][1/p]) x X, T x V x p Galois (, T x X p Galois V x Spf(Z p [[T 1,..., T d ]]) ). T, T Galois, T 1 Z p [[T 1,..., T d ]]. (T ) z(t ) (T ) (Q ), ε l (T ) (T ) (Q l ), x X V x, z(t ), ε l (T ) x z(v x ), ε l (V x ) z geom (V x ) = z(v x ) (V x ) (Q ), ε geom l (V x ) = ε l (V x ) (V x ) (Q l ). 1, z(t ), ε l (T ), Bloch- d = 0, Z p Galois T. X = Spec(Q p ) = {x} 1, V x = T [1/p]., V x, Q p z geom (V x ), ε geom l (V x ) (V x ),, T Z p (T ) (V x )., Q V x L p Bloch- ([BK90]), Q l (l = p Bloch- exponential ) p Perrin-Riou ([Pe95]).,,.., Q 1 p Galois Γ := Gal(Q(ζ p )/Q), Λ := Z p [[Γ]]. Z p [[[T ]] Zp Z p [(Z p /2p) ],., V p Galois, T Galois V Z p. Λ Galois Dfm(T ) := T Zp Λ Galois g x y T Zp Λ g(x y) := gx [ḡ] 1 y (, ḡ Γ g Q(ζ p ) [ḡ] Λ ḡ ). Dfm(T ) T. Shapiro, Q p Dfm(T ) Λ HIw(Q i p, T ) := H i (Q p, Dfm(T )) lim H i (Q p (ζ p n), T ) cor

4 (, Dfm( ) Iw. ε Iw (T ) := ε(dfm(t )), Iw (T ) := (Dfm(T ))).,., Dfm(T ) T, T., µ p n := {x Q x pn = 1}, G Q Z p Z p (1) := lim µ p n., Kummer 1 p n n ζ p n µ p n (1 ζ p n) Z[ζ p n, 1/p] c n H 1 (Z[1/p, ζ p n], Z p (1)) H 1 (Q(ζ p n), Z p (1)). 1 p n {ζ p n} n 1 ζ p p n+1, = ζ p n c := (c n ) n 1 H 1 Iw(Z[1/p], Z p (1)) H 1 Iw(Q, Z p (1)). ([Ka93a]), Dfm(Z p (1)) z Iw (Z p (1)) Iw (Z p (1)) (, Iw (Z p (1)) Λ Frac(Λ), Iw (Z p (1)) Mazur-Wiles, Rubin Q ). 2, ([Ka04]), ( ). f(τ) := q + n 2 a n (f)q n (τ C, Im(τ) > 0, q := exp(2πiτ)) N k 2 Hecke. Q C, Q Q p a n (f) Q p, Q p {a n (f)} n 2 Q p F f., Eichler-, Deligne, G Q F f 2 V f, L L(V f, s) f L L(f, s) := n 1 a n (f) n s (Re(s) >> 0). V f, p {c n,f H 1 (Z[1/Np, ζ p n], V f ) n 1} c f H 1 Iw(Z[1/Np], V f )., Dfm(Z p (1)) z Iw (V f ) Iw (V f ) Λ Frac(Λ) (Dfm(V f ) ) ([Ka04], [Na17b])., Frac(Λ), Q,

5 , Iw (V f ). z Iw (V f ) ([Na17b]). p Galois., p, p, F( ), Galois T F.,, Dfm(T F ) ( [Oc06] )., Mazur Galois, F p Galois T, p T p Galois Z p [[T 1,..., T d ]]/I (I ) R univ ) T univ (T ). p Galois, T univ z(t univ ) (T univ ) (Q ), ε l (T univ ) (T univ ) (Q l ). Galois ( Galois Galois ) R = T ( (Hecke T ) ), R univ T univ., R = T T univ, ?, p Galois V z geom (V ) ε geom l (V ) (L ), p Galois., (2.1.1 ) Z p [[T 1,..., T d ]] Galois T. X := Spec(Z p [[T 1,..., T d ]][1/p])., X geom := {x X V x } X geom X, x X geom z geom (V x ) ε geom l (V x ) X, Galois., Q l, l p., Grothendieck, G Ql

6 p V Weil-Deligne W l (V ). V W l (V ) Deligne, Langlands ([De73]), Weil-Deligne, L l (W l (V )), ε l (W l (V )), V ε geom l (V ) (V )., l p G Ql p p. G Ql (l p) p = p,, Q l, l p ( [Ya09])., p ( Q ) l = p (G Qp p ) 2. G Qp 2 p., l = p l = p ( 1) G Qp p, G Qp p,. G Qp p V, Fontaine p ([Fo82], [Fo94]) Q p D crys (V ), D dr (V )., D crys (V ) Frobenius φ : D crys (V ) D crys (V ), D dr (V ) (Hodge ) {D i dr (V )} i Z. V Q p X H í et (X Q p Q p, Q p ), D dr (V ) X de Rham H i dr (X/Q p), X Z p X, D crys (V ) X Fp := X Zp F p H i crys(x Fp /Z p ) (. p Hodge!!). ( ), Fontaine = {crys, dr} dim Qp D (V ) = dim Qp (V ) V,, de Rham., X H í et (X Q p Q p, Q p ) de Rham, X Z p,., G Qp p Galois de Rham p. G Qp p = de Rham

7 , l = p p (!). ε geom p {de Rham } { p } (V ) (V ), de Rham V ε geom p (V ) (V )., l p Grothendieck l = p p ([Be02]) ( l = p!)., de Rham V Weil-Deligne W p (V ), V W p (V ) L p (W p (V )), ε p (W p (V )) l p. V W p (V ), l p V W l (V ) V, l = p V W p (V ) V., l = p ε geom p (V ) L p (W p (V )), ε p (W p (V )). ε geom p (V ),, D dr (V ) Hodge. D i dr (V )/D i+1 dr (V ) {0} i V Hodge-Tate, εgeom p (V ) V Hodge-Tate., ε geom p (V ), Bloch- exponential ([BK90]) exp V : D dr (V )/DdR(V 0 ) H 1 (Q p, V ) exponential exp V : H 1 (Q p, V ) D 0 dr(v ) ([Ka93a]).,, V G Q p, H 1 (Q, V ) loc p H 1 (Q p, V ) H 1 (Q, V ) loc p H 1 (Q p, V ) exp V DdR(V 0 ),, H 1 (Q, V ) L., ε geom p (V ). ε geom p (V ) = {L p (W p (V )) + ε p (W p (V )) + Hodge Tate + exp V + exp V } (, +, ) ( ).

8 1 p ( [Ka93b]) Dfm(V ) (V ) (Benois-Berger[BB08], Loeffler-Venjakob- Zerbes[LVZ15]) (trianguline ) ([Na17a]), V derham V W p (V ) (, 1 )., ε p (W p (V )) Gauss. Gauss p Galois,. W p (V ) ( ). 1([Na17b]),. 1 2 p ( ), F F p, G Qp F 2 T. R univ T, T univ V ( )., (T univ ) ε p (T univ ) (T univ ),, x Spec(R univ [1/p]) T univ x V x de Rham, ε p (T univ ) x ε p (V x ) ε p (V x ) = ε geom p (V x ). ε p (T univ )., ε p (V x ) = ε geom p (V x ) W p (V ),. ε p (V x ) = ε geom p (V x ), GL 2 (Q p ) p Langlands ? p 3. A Q := Ẑ Z Q Q, π GL n (A Q ). L, π L L(π, s) L Λ(π, s) := L (π, s)l(π, s), Λ(π, s) ( ) Λ(π, s) = ε(π, s)λ(π, 1 s) (, π π )., ε(π, s), Q l π ε(π l, s) ε(π, s) = l ε(π l, s)

9 (, ). z(t ) (T ). S p. Q S l S Q Galois, Galois G Q,S := Gal(Q S /Q). T ( )Z p [[T 1,..., T d ]] G Q,S, T, T (1) := T Zp Z p (1)(T Tate ). T S (T )(G Q,S Galois ), l S T GQl l (T )., Galois Galois Poitou-Tate, S (T (1)) l S l (T ) R S (T ).,. T, T (1), z S (T ) S (T ), ε l (T ) l (T )., z S (T (1)) = l S ε l (T ) z S (T ) T ([Ka93b],[FK06])., z S (T ) z S (T (1)), p V z geom (V ) L L(V (1), s) s = 0.,., z(t ) ε p (T )., L L (π, s) (Hodge ) ε p (V ) Hodge-Tate S = {p}, G Q,S Z p (1) Dfm(Z p (1))., z Iw S (Z p (1)) Iw S (Z p (1)), ([Ka93b]) ( 2) ε Iw p (Z p (1)) Iw p (Z p (1)) 2, f, S := {l N} {p}., z S Iw(V f) Iw S (V f), 1 Dfm(V f ) GQ p ε Iw p (V f ) Iw p (V f ) p ([Na17b]). 2 f z Iw S (V f )

10 3. GL 2 (Q p ) p Langlands GL 2 (Q p ) p Langlands, G Qp 2 p GL 2 (Q p ) p Banach., (Frobenius )Weil-Deligne 2 GL 2 (Q p ) GL 2 (Q p ) Langlands p., C, p Langlands G Qp GL 2 (Q p ) p., p Langlands. Breuil([Br03a],[Br03b], [Br04]), Colmez([Co10]) (φ, Γ) ([Fo91]) Galois GL 2 (Q p ), Berger-Breuil([BBr10]), Kisin([Ki10]), Pasukunas([Pa13]), Colmez- Dospinescu-Paskunas([CDP14])., Colmez G Qp 2 p ( ) GL 2 (Q p ) p Banach ( ) ( ). p, p. Emerton([Em11]) Kisin([Ki09]), p, G Q 2 p Galois V Fontaine-Mazur (Wiles -, ). GL 2 (Q p ) p Langlands 1. Emerton Kisin Fontaine-Mazur,,, (φ, Γ) (Colmez )., Colmez, GL 2 (Q p ) p Langlands! Colmez, 2 w := ( ) GL 2 (Q p ), ( Langlands ),. 1 2! w. (φ, Γ), G Qp 2 p T, B GL 2 (Q p ) Π(T )

11 (, T ). Π(T ) w GL 2 (Q p ) p Langlands.. V G Qp 2 p, Π(V ) GL 2 (Q p ) Banach. Π(V ) Π(V ) := Hom cont F (Π(V ), F ) (F Π(V ) Q p ), x Π(V ), y Π(V ) [x, y] := x(y) F. Π(V ) y, x Π(V ) GL 2 (Q p ) g [g x, y] F. Π(V ) Π(V ) alg. Colmez Π(V ) alg {0} V de Rham regular (, V Hodge-Tate (, 2 ) k 2 > k 1 regular ). Colmez Emerton([Em11]),, GL 2 (Q p ) ( ) Π(V ) alg π LL (W p (V )) F Sym k 2 k 1 1 (F 2 ) F det k 1 (, π LL (W p (V )) Langlands W p (V ) GL 2 (Q p ) ( F ) ) ( Langlands )., GL 2 (Q p ) p Langlands ( ), Emerton W p (V ) ( π LL (W p (V )) supercuspidal ), ( ) (G Qp 2 p T ) ε p (T ), ε geom p (V ), ε p (T ) (w ) Kirillov (GL 2 (Q p ) smooth Q p ), GL 2 (Q p ) smooth π ( ε p (π) ), w GL 2 (Q p ) π., G Qp 2 p T, Banach Π(T ) w T ε p (T ) (T ).

12 , Dfm(T ) ε Iw p (T ) Iw (T ) ( ε p (T ) (T ) ). Dfm(T ) Iw (T ) HIw 1 (Q p, T ), HIw 1 (Q p, T ) Π(T ).,, V = T 2 p ( ) p 0 ( T ). g p := 0 1 ( ) 1/p 0, Π(V ) δ. Π(V ) Π(V ) 0 1/p g p (Π(V ) ) gp=1, g p δ (Π(V ) ) gp=δ. w. γ Γ w : (Π(V ) ) g p=1 ( ) χ(γ) (Π(T ) ) g p=δ GL 2 (Q p ) (χ : Γ Z p p ), (Π(V ) ) g p=1 (Π(V ) ) g p=δ Λ[1/p] ( (Π(V ) ) g p=δ )., Colmez, Λ[1/p] (Π(V ) ) gp=1 H 1 Iw(Q p, V ), (Π(V ) ) gp=δ H 1 Iw(Q p, V (1)) (!)., w w : H 1 Iw(Q p, V ) H 1 Iw(Q p, V (1))., ε Iw p (V ) Iw (V ), Dfm(V ) Tate {, } Tate : H 1 Iw(Q p, V ) H 1 Iw(Q p, V (1)) Λ[1/p] 2 H 1 Iw (Q p, V ) (H 1 Iw (Q p, V ) 2 Λ[1/p] ) 2 H 1 Iw(Q p, V ) Λ[1/p] : x 1 x 2 {x 1, w x 2 } Tate ε geom p (V ) ( Langlands ) V regular 2 de Rham ε geom p (V ) (V ) 4.1 ε p (V ) (V ). V Weil-Deligne W p (V ) ( π LL (W p (V )) supercuspidal ),., ε geom p (V ), x H Iw (Q p, V ) α : H 1 iw(q p, V ) sp H 1 (Q p, V ) exp V D 0 dr(v )

13 α(x) ε p (W p (V )) 2, ε geom p (V ) ε p (W p (V )) + α(x) ε p (V ) x H Iw (Q p, V ) w : H 1 Iw (Q p, V ) H 1 Iw (Q p, V (1)) β : H 1 Iw(Q p, V (1)) sp H 1 (Q p, V (1)) exp V (1) D 0 dr(v (1)) β(w x) ( Hodge-Tate, ). ε p (V ) β(w x), ε p (V ) = ε geom p (V ), ε p (W p (V )) + α(x)? β(w x) x (Π(V ) ) g p=1, w x (Π(V ) ) g p=δ Π(V ). Colmez Kirillov (B Π(V ) alg Q p ), Q p y α, y β Π(V ) alg, x (Π(V ) ) g p=1, x (Π(V ) ) g p=δ ( ) α(x) [x, y α ], β(x ) [x, y β ]. y α, y β,, Langlands Π(V ) alg π LL (W p (V )) F Sym k 2 k 1 1 (F 2 ) det k 1 π LL (W p (V )) Kirillov w y β y α ε p (π LL (W p (V ))). w y β ε p (π LL (W p (V ))) + y α GL 2 (Q p ) Langlands ε p (π LL (W p (V ))) = ε p (W p (V )), w y β ε p (W p (V )) + y α. ε p (V ) β(w x) [w x, y β ] [x, w y β ] ε p (W p (V )) + [x, y α ] ε p (W p (V )) + α(x) ε geom p (V )

14 ε p (V ) ε geom p (V ). Langlands ε p (V ) = ε geom p (V ), ε p (V ) = ε geom p (V )( V ),, Π(V ) alg w, Π(V ) alg smooth π LL (W p (V ))., ε p (V ) = ε geom p (V ) Langlands! 5. 1 ε p (V ) = ε geom p (V ) Langlands GL 2 (Q p ) p Langlands = G Qp 2. G Q 2, Emerton ([Em11]). ([CEGGPS16a], [CEGGPS16b], [GN17] ), p Langlands, = G Q 2. [BB08] D. Benois, L. Berger, Théorie d Iwasawa des représentations cristallines. II, Comment. Math. Helv. 83 (2008), no. 3, [BBr10] L. Berger, C. Breuil, Sur quelques représentations potentiellement cristallines de GL 2 (Q p ), Astérisque 330, 2010, [Be02] L. Berger, Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), [BK90] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives. The Grothendieck Festschrift, Vol. I, , Progr. Math. 86, Birkhäuser Boston, Boston, MA [Br03a] C. Breuil, Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ). I, Compositio Math. 138 (2003), [Br03b] C. Breuil, Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ). II, J. Inst. Math. Jussieu 2 (2003), [Br04] C. Breuil, Invariant L et série spéciale p-adique, Ann. Scient. de l E.N.S. 37 (2004), [CEGGPS16a] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paskunas, S. Shin, Paching and the p-adic local Langlands correspondence, Cambridge Journal of Mathematics 4 (2016), no. 2, [CEGGPS16b] A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paskunas, S. Shin, Patching and the p-adic Langlands program for GL 2 (Q p ), preprint. [Co10] P. Colmez, Représentations de GL 2 (Q p ) et (φ, Γ)-modules, Astérisque 330 (2010),

15 [CDP14] P. Colmez, G. Dospinescu, V. Paskunas, The p-adic local Langlands correspondence for GL 2 (Q p ), Cambridge Journal of Mathematics, Volume 2, Number 1 (2014), [De73] P. Deligne, Les constantes des équations fonctionelles des fonctions L, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp Lecture Notes in Math., Vol. 349, Springer, Berlin, [Em11] M. Emerton, Local-global compatibility in the p-adic Langlands programme for GL /Q, preprint, available at emerton/preprints.html. [Fo82] J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d un corps local; construction d un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), [Fo91] J.-M. Fontaine, Représentations p-adiques des corps locaux, The Grothendieck Festschrift, vol 2, Prog. in Math. 87, Birkhäuser (1991), [Fo94] J.-M. Fontaine, Le corps des périodes p-adiques, Astérisque 223 (1994), [FK06] T. Fukaya, K. Kato, A formulation of conjectures on p-adic zeta functions in non commutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII (Providence, RI), Amer. Math. Soc. Transl. Ser. 2, vol. 219, Amer. Math. Soc., 2006, pp [FK12] T. Fukaya, K. Kato, On conjectures of Sharifi, preprint, [GN17] T. Gee, J. Newton, Patching and the completed homology of locally symmetric spaces, preprint. [Ka93a] K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B dr. Arithmetic algebraic geometry, Lecture Notes in Mathematics 1553, Springer- Verlag, Berlin, 1993, [Ka93b] K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B dr. Part II. Local main conjecture, unpublished preprint. [Ka04] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004), no. 295, ix, , Cohomologies p-adiques et applications arithmétiques. III. [Ki09] M. Kisin, The Fontaine-Mazur conjecture for GL 2, J. Amer. Math. Soc. 22 (2009), no. 3, [Ki10] M. Kisin, Deformations of G Qp and GL 2 (Q p )-representations, Astérisque 330 (2010), [LVZ15] D. Loeffler, O. Venjakob, S. L. Zerbes, Local ε-isomorphism, Kyoto J. Math. 55 (2015), no.1, [Na17a] K. Nakamura, A generalization of Kato s local ε-conjecture for (φ, Γ)-modules over the Robba ring, Algebra and Number Theory 11-2 (2017), [Na17b] K. Nakamura, Local epsilon isomorphisms for rank two p-adic representations of Gal(Q p /Q p ) and a functional equation of Kato s Euler system, to appear in Cambridge Journal of Mathematics. [Oc06] T. Ochiai, On the two-variable Iwasawa Main conjecture for Hida deformations, Compositio Mathematica, vol 142, [Pa13] V. Paskunas, The image of Colmez s Montreal functor, Publications Mathématiques de l IHÉS 118, Issue 1 (2013), [Pe95] B. Perrin-Riou, Fonctions L p-adiques des représentations p-adiques. Astérisque No. 229 (1995), 198 pp. [Ya09] S. Yasuda, Local constants in torsion rings, J. Math. Sci. Univ. Tokyo 16 (2009), no. 2,