2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)

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(Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s) R(s) > Riemann ζ(s) ζ( ) = + 2 + 3 + 4 +... *2 ζ(s) ian.sprung@gmail.com * p p p *2 ζ( ) Euler

2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) =.3.5 2 2.3 2.7 ζ( 8) = 2 4, ζ( 0) =.3.5 2 2.3., ζ( 2) = 69 2 3.3 2.5.7.3 69 Kummer ζ( 2n) 2.. (Kummer )ζ( 2), ζ( 4),..., ζ( (p 3)) p p Q(ζ p ) (ζ p x p = 0 *4 K K Cl(K) = K O K O K K K = Q(ζ p ) O Qζ p = Z[ζ p ] Cl(K) O K Cl(K) = O K Cl(K) 847 Lamé Fermat ( p x p + y p = z p Q(ζ p ) #Cl(Q(ζ p )) = Kummer Lamé p = 69 Kummer p Q(ζ p ) Lamé ζ( 2n), n 6 2.2. p = 3, 7,, 3 Fermat ζ( 2n) Fermat [Kummer] ζ( 2n) Kummer 2.3. Kummer n (p ) m n (mod (p )p a ) ( p m )ζ( m) ( p n )ζ( n) (mod p a+ ) p p Hensel 897 π k : Z/p k+ Z Z/p k Z Z p := lim Z/p k Z k *3 Euler *4 Y p p 2 π2 p 2 6

3 p 3, 3., 3.4, 3.4, 3.45, 3.459, 3.4592,... 3 0 0 + 0 + 4 0 2 + 0 3 + 5 0 4 +... Z p... + a 4 p 4 + a 3 p 3 + a 2 p 2 + a p + a 0 p 0... Z/p 4 Z Z/p 3 Z Z/p 2 Z Z/pZ 0... a 3 p 3 + a 2 p 2 + a p + a 0 p 0 a 2 p 2 + a p + a 0 p 0 a p + a 0 p 0 a 0 p 0 p a Q a = p e b b, c p bc c a p = p e 5 3 5 5 5 3 n, m n m p n m p Q R p Q p R R = C Q p Q p *5 p Kummer p n m p ( p m )ζ( m) ( p n )ζ( n) p Riemann ζ p 3 964 Leopoldt Kummer p ζ 3.. [ L] p r ζ p ( r) = ( p r )ζ( r) p ζ p : Z p Q p Leopoldt Riemann ζ ζ p p p ( is p-adically interpolated) Leopoldt ζ p (s) s Z p, s p s = Riemann ζ Cl(Q(ζ p )) 956 Seattle *5 C Q p p C p p

4 F Q F F Galois Γ = Gal(F /F ) Z p Galois Z p p n Z p Γ Γ pn F n = F Γpn F = F 0 F F 2... F n... *6 F n Γ Cl(F n ) 3.2. (#Cl(F n ) p p en λ, µ, ν n e n = λn + µp n + ν 959 [ 59] 960 F /F Γ (Γ-extensions) Z p 3.3. Z p p F := Q(ζ p ) ζ p n p n F n := F (ζ p n) F = F (ζ p n) = Q(ζ p n) =: Q(ζ p ) F F Z p n n 3.4. Z p p Q 0 := Q p Z p Gal(Q(ζ p n)/q) = (Z/p n Z) = Z/p n Z Gal(Q(ζ p )/Q) = lim Z/p n Z n = Z p Z p = Gal(Q(ζ p )/Q 0 ) = Z/(p )Z Q n Q(ζ p n+) Γ(n ) Γ Q(ζ p n) Q(ζ p ) Q := Q(ζ p ) Q(ζ p ) Q 0 := Q = Q(ζ p ) Γ(n ) Q n := Q(ζ p n) Γ Γ(n) : Gal(Q n /Q) = Z p /p n Z p = Z/p n Z Q /Q 0 Z p (Iwasawa s cyclotomic Z p -extension) 3.5. F := Q(ζ ) R. 3.3 R Z p 959 Jean-Pierre Serre Bourbaki [Serre] Γ Λ Λ = Z p [[X]] (Iwasawa algebra) Serre Gal(Q n /Q) = Γ(n) Q n Cl(Q n ) Cl(Q n ) p-sylow Cl(Q n )[p] p Γ(n) *6 Galois Γ pn F n

5 p p Γ(n) Cl(Q n )[p] Cl(Q n )[p] Γ(n) Z Cl(Q n )[p] Z[Γ(n)] Cl(Q n )[p] p e Z/p e Z Z p Z/p e Z Cl(Q n )[p] Z p Cl(Q n )[p] Z p [Γ(n)] lim n Cl(Q n )[p] lim Z p [Γ(n)] n Λ(Γ) := Z p [[Γ]] := lim Z p [Γ(n)] Z Z p Z p γ n Z Z p γ (topological generator) γ + X Z p [[Γ]] = Z p [[X]] lim Cl(Q n )[p] n 964 Λ [ 64] M Λ Char(M) Λ M Z #M Z/(a )... Z/(a k ) k = M Z (#M) = ( a i ) Z Λ M Λ/(a )... Λ/(a k ) M a i a i+ a i k ( a i ) Λ Char(M) i= X := Hom(lim n Cl(Q n )[p], Q p /Z p ) Λ ζ p Char(X ) = (ζ p ) 3.5 K Cl(K) K H Galois Gal(H/K) *7 Cl(F ) M M := (p pro-p ) G := Gal(F /Q) = Γ p # = p 2 Λ = Z p [ ][[X]] i= *7 Riemann [SGA])

6 3.6. Λ ξ (pseudo-measure) σ G (σ )ξ Λ s = ξ(s) 3.7. ψ : Gal(Q(ζ p )/Q) Z p Galois *8 Gal(Q(ζ p n)/q) = (Z/p n Z) 3.8. ξ G λ : G Q p = G λdξ := λ ((g )ξ) λ(g) ( λ(g) ) 3.9. ( G ξ p k 2, G ψ k dξ p := ( p k )ζ( k) Riemann ζ -Leopoldt ζ p [ 69] ([Washington, 5.4] ) 3.0. I := ker(λ Z p ) f f() 3.. (Char(Gal(M /F )) = Iξ p Λ Galois ξ p I 984 Mazur Wiles [MW] Kolyvagin Rubin Euler [Rubin] 4 y = x + 2 y = (x ) (line) y = x 2 + y = (x ) (parabola) y 2 = x 2 + y 2 = 2x 2 + 2 y 2 = (x ) (hyperbola) (ellipse) y 2 = x 3 x y 2 = (x ) *8 Gal(Q(ζ p n)/q) ζ p n ζ a p n a (Z/pn Z)

7 (elliptic curve) Q Q x Q E : y 2 = x 3 + ax + b K E K E(K) y 2 = x 3 + ax + b K * 9 K = F p x F p p x 3 + ax + b F p F p 2 y2 = x 3 + ax + b y F p ( y) x 3 + ax + b E(F p ) p 2 2 = p 4.. (Hasse) E : y 2 = x 3 + ax + b Q a p = p #E(F p ) a p 2 p 4.2. E Q p = 999 90 #E(F 999 ) 2088 Riemann ζ 4.3. Q E : y 2 = x 3 + ax + b L s C L(E, s) = p a p p s ( + p 2s F p x 3 + ax + b p ± p s Hasse (R(s) > 3 2 ) C L 4.4. Q E p a p { p a p p (ordinary) p a p p (supersingular) 987 4.5. [Elkies] * 0 E Q [Silverman] [ST] 5 970 Mazur E Z p E(F 0 ), E(F ),..., E(F ) * *9 E(K) (O) {E(K), (O)} [ST],[Silverman] *0 * Mazur

8 p L * 2 L p (E, X) Λ ζ p L(E, s) p L(E, s) = 0, s N Mazur L L(E, s) = a n s = Dirichlet ns n ( ) χ : (Z/p m Z) C L(E, s, χ) = a n χ(n) n s s = n L(E,, χ) Ω χ( ) Ω + Néron ) Ω ( Néron χ( ) = ± Ω ± χ p * 3 p p L L p (E, X) Λ [Mazur] Selmer p K E Selmer Sel(E/K) Tate-Šafarevič (E/K) Cl(K) 0 E(K) Q p /Z p Sel(E/K)[p] (E/K)[p] 0 Sel(E/K) E K (E/K) # (E/K) < X E (Q ) := Hom(Sel E (Q ), Q p /Z p ) X E (Q ) Λ Char(X E (Q )) 5.. )[Mazur2] Char(X E (Q )) = (L p (E, X)) Λ 2004 [ ] 5.2. ( Char(X E (Q )) (L p (E, X)) Λ 2006 Euler H z H Λ p L z (Euler Char(X E (Q )) (L p (E, X)) * 4 )z H Char(X E (Q )) (L p (E, X)) z L p (E, X) Λ *2 Λ = Z p[[x]] 3.4 *3 a p p Mazur *4

9 p L p (E, X) Λ. Skinner Urban 5.3. (Mazur) # (Q n ) p p en λ, µ, ν n e n = λn + µp n + ν 6 Q E p p 970 Višik, Manin, Amice-Vélu 986 Mazur, Tate, Teitelbaum T 2 a p T + p α, α p L L p (E, α, X) L p (E, α, X) [Višik],[AV],[MTT] L p (E, α, X), L p (E, α, X) / Λ Q p [[X]] p Λ Hom(Sel E (Q ), Q p /Z p ) Λ Perrin-Riou 990 Bloch- -logarithms Galois a p = 0 p 6.. p 5 p a p a p = 0 Hasse a p 2 p 2003 Mazur Pollack [Pollack] 6.2. (Pollack) a p = 0 L p (E, α, X) Mazur,Tate,Teitelbaum p L L p (E, α, X) = log + p ( + X)L + p (E, X) + log p ( + X)L p (E, X)α, p L 2 (E, α, X) = log 2 ( + X)L + 2 (E, X) + 2 log 2 ( + X)L 2 (E, X)α. L ± p (E, X) Λ log ± p ( + X) log + p ( + X) = p m Φ 2m ( + X) p, Φ n (X) = X pn i n p a p = 0 α = α L p (E, α, X)+L p (E, α, X) = log + p (+X)L + p (E, X) Φ n (+X) ζ p n L p (E, α, X)+L p (E, α, X) X = ζ p 2m, m = 0 Pollack L p (E, α, X) + L p (E, α, X) L + p (E, X) p = 2, p = 3 6.3. [S] p L p (E, α, X) = log ϑ α( + X)L ϑ p(e, X) + log υ α( + X)L υ p(e, X) i=0

0 L ϑ/υ p (E, X) Λ log ϑ/υ p ( + X) p L p (E, α, X), L p (E, α, X) Pollack log ± p ( + X) L ± p (E, X) L ± p (E, X) Λ 6.3 L ϑ/υ p (E, X) 2003 a p = 0 [ ] 6.4. (p a p = 0 Col ± H Col ± Λ z L ± p (E, X) Col ± z Pollack L L ± p (E, X) ker Col ± Sel ± X ± = Hom(Sel ±, Q p /Z p ) 6.5. (a p = 0 p (L ± p (E, X)) = Char(X ± ) Λ (L ± p (E, X)) Char(X ± ) 6.6. [S] p Col ϑ Col υ H Col ϑ/υ Λ z L ϑ/υ p (E, X) ker Col ϑ/υ p Sel ϑ/υ 6.7. (p (L ϑ/υ p (E, X)) = Char(X ϑ/υ ) Λ (L ϑ/υ p (E, X)) Char(X ϑ/υ ) 7 Bertolini Darmon[BD] Coates Sujatha Venjakob[C SV] weight = 2 (weight > 2 * 5 Lei) a p = 0 [Lei] a p 0 *5

[AV] Amice, Y., Vélu, J. Distributions p-adiques associées aux séries de Hecke, in Journées Arithmétiques de Bordeaux (Bordeaux, 974), Astérisque 24-25, Société Mathématique de France, Montrogue (975), 9-3. [BD] Bertolini, M., Darmon, H. The p-adic L-functions of modular elliptic curves, Mathematics unlimited - 200 and beyond (200), 09-70. [Coates] Coates, J. Lecture Notes on a Course in Iwasawa Theory given at Cambridge University, Lent 2004. [C SV] Coates, J., Fukaya, T., Kato, K. Sujatha, R., Venjakob, O. The GL 2 main conjecture for elliptic curves without complex multiplication, Publications Mathématiques de l Institut des Hautes Études Scientifiques, 0 (2005), 63-208. [Elkies] Elkies, N. The existence of infinitely many supersingular primes for every elliptic curve over Q, Inventiones Mathematicae 89 (987), no. 3, 56-567. [ ],. [ 59] Iwasawa, K. On Γ-extensions of algebraic number fields, Bulletin of the American Mathematical Society 65 (959), 83-226. [ 64] Iwasawa, K. On some modules in the theory of cyclotomic fields, Journal of the Mathematical Society of Japan 6 (964), 42-82. [ 69] Iwasawa, K. On p-adic L-functions, Annals of Mathematics 89 (969), 98-205. [ ] Kato, K. p-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), 7-290. [ L] Kubota, T. and Leopoldt, H. Eine p-adische Theorie der Zetawerte. I. Einführung der p-adischen Dirichletschen L-Funktionen, Journal für die reine und angewandte Mathematik 24/25 (964), 328-339. [ ] Kobayashi, S. Iwasawa theory for elliptic curves at supersingular primes, Inventiones Mathematicae 52 (2003), no., -36. [Kummer] Kummer, E. Beweis des Fermat schen Satzes der Unmöglichkeit von x λ + y λ = z λ für eine unendliche Anzahl Primzahlen λ, Monatsberichte der Akademie der Wissenschaften Berlin (847), 32-39. [Lei] Lei, A. Iwasawa Theory for Modular Forms at Supersingular Primes arxiv:0904.3938v[mathnt]. [Mazur] Mazur, B. Courbes elliptiques et symboles modulaires, Séminaire Bourbaki 44 (97/972). [Mazur2] Mazur, B. Rational points of abelian varieties with values in towers of number fields, Inventiones Mathematicae 8 (972), 83-266. [MTT] Mazur, B., Tate, J., and Teitelbaum, J. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Mathematicae 84 (986), -48. [MW] Mazur, B., Wiles, A. Class fields of abelian extensions of Q, Inventiones Mathematicae 69 (984), 79-330. [Pollack] Pollack, R. The p-adic L-function at a supersingular prime, Duke Mathematical Journal 8 (2003), no.3, 523-558. [Riemann] Riemann, B. Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (859). [Rubin] Rubin, K. The main conjecture. Appendix to: Cyclotomic Fields by S. Lang, Graduate Texts in Mathematics 2. [Serre] Serre, J.-P. Classes des corps cyclotomiques (d après K. Iwasawa), Séminaire Bourbaki no. 74 (959). [SGA] Grothendieck, A. et al. Revêtements étales et groupe fondamental, 960/6, Lecture Notes in Mathematics 224 (97). [Silverman] Silverman, J. The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 06. [ST] Silverman, J. and Tate, J. Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics. [S] Sprung, F. Iwasawa theory for elliptic curves at supersingular primes: beyond the case a p = 0, arxiv:0903.349v[mathnt]. [Višik] Višik, M. Nonarchimedean measures associated with Dirichlet series, Matematičeskii Sbornik 99(4), no. 2 (976), pp. 248-260, 296. [Washington] Washington, L. Introduction to Cyclotomic fields, Graduate Texts in Mathematics 83.

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