Transcription

1 c ( ) : takuya@math.kyushu-u.ac.jp : takuya/edu.html

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3 i Witt Witt 5.1 Gram-Schmidt Witt Witt p Q p Hensel Hilbert Hasse Hasse

4 ii Weil Weil Weil Leray Weil SL(, F ) O(E) SL(, F ) Weil O(E) SL(, F ) θ SL(, A) SL(, A) O(E A )

5 Dedekind M K M Euler χ(m) Gauss-Bonnet K dσ = πχ(m). M M S 1 (U(1, R) ) π : P M ω dω = π Ω M ( ) De Rham H dr(m) [Ω] π : P M Euler e(p ) H (M, Z) π (de Rham ) Euler e(p ) S 1 π : P M (obstruction) Z ( ) Z Q p F p Q p Z p C p =, 3, 5 S n Galois Q Galois (Dedekind ) ( ) 4

6 1 Z O(n) Euler A, B, C SL() Hecke, Shalika-, Jacquet, Labesse-Langlands 10 takuya/arithlec06.html 1. F F F V (bilinear form) (, ) : V V F v V V v (v, v ) F, V v (v, v) V V {v 1,..., v n } v = n i=1 x i v i, v = n j=1 x jv j á ë á ë (v1, v 1 )... (v 1, v n ) x 1 n n (v, v ) = x i x j(v i, v j ) = (x 1,..., x n ) i=1 j= (v n, v 1 )... (v n, v n ). x n (, ) á ë (v1, v 1 )... (v 1, v n ) T :=..... (v n, v 1 )... (v n, v n ) (, ) {v 1,..., v n } V (, ) (v, v) = (v, v ), v, v V

7 (symmetric) V {v 1,..., v n } T : t T = T. V (quadratic form) Q : V F Q(xv) = x Q(v), x F, v V ; (, ) Q : V V (v, v ) 1 Ä Q(v + v ) Q(v) Q(v ) ä F {V } Q (, ) Q Q(v) := (v, v) (, ) { V } (1.1) 1.3 F V Q (V, Q) ( (V, (, ) Q )) F (quadratic space) 1.1. (i) V = F a F Q(x) = ax (V, Q) (F, a) (ii) V = F Q((x, y)) = xy (V, Q) (hyperbolic plane) (H, Q H ) ( (H, (, ) H )) (V 1, Q 1 ), (V, Q ) V 1 V Q 1 Q Q 1 Q (v 1, v ) := Q 1 (v 1 ) + Q (v ), (v 1, v ) V 1 V (V 1 V, Q 1 Q ) (V 1, Q 1 ), (V, Q ) (direct sum) (V, Q) (V, Q ) (morphism) f : V V Q (f(v)) = Q(v), v V f : V V (V, Q) (V, Q ) (isometry) (V, Q), (V, Q ) (V, Q) (V, Q ) (V, Q) O(V ) = O(Q) := {g GL F (V ) (g.v, g.v ) Q = (v, v ) Q, v, v V } (V, Q) (orthogonal group) 1.. f : (V, Q) (V, Q ) (v, w) Q = (f(v), f(w)) Q = 0, v ker f, w V V/ker f Q(v + ker f) := Q(v) f : (V/ker f, Q) (im f, Q im f )

8 (V, Q), (V, Q ) v = (v 1,..., v n ), v = (v 1,..., v n) T := ( t v, v) Q = Ä (v i, v j ) Q äi,j, T := ( t v, v ) Q f : V V v, v A = Ä a i,j äi,j : f(v) = v A V Q f v ( t f(v), f(v)) Q = ( t A t v, v A) Q = t AT A f (V, Q) (V, Q ) t AT A = T n S, S (F n, S), (F n, S ) A GL(n, F ) t AS A = S 1.4 (Gram-Schmidt ). (i) (V, Q) 1.1 (i) n i=1 (F, a i ) (ii) a i F/(F ) (V, Q). (i) V V = 0 dim V = n Q = 0 (V, Q) n i=1 (F, 0) Q 0 a 1 := Q(v 1 ) 0 v 1 V V 1 := span {v 1 }, V 1 := {v V (v, v 1 ) Q = 0} = ker (, v 1 ) Q (V, Q) = (V 1, Q V1 ) (V 1, Q V 1 ) (V 1, Q V1 ) (F, a 1 ) dim V 1 = n 1 (ii) 1.5. (H, Q H ) ( 1.1 (ii) ) {v 1 := (1, 1), v := (1, 1)} Q(v 1 ) = 1, Q(v ) = 1, (v 1, v ) Q = 0 (H, Q H ) (F, 1) (F, 1)

9 5 Witt Witt.1 Gram-Schmidt (V, Q) n i=1 (F, a i ) a i 0 (V, Q) (rank) rk V = rk Q a i 0 F /(F ) (V, Q) det V = det Q F = R R /(R ) {±1} a i 0, ±1 a i 1 1 p, q (p, q) (V, Q) (signature) sgn V = sgn Q (V, Q) (non-degenerate) I (type I) rk V = dim V v 0, V (v, w) Q 0 w V Q = 0 (V, Q) II O(V ) GL F (V ) 1.4 II.1. (i) C n (ii) R n 1. F R Gram-Schmidt (i) n m A m n m Q m R A = QR (QR ) (ii) n GL n (R) B n, n O n (R) := {g GL(n, R) g t g = 1 n } GL n (R) = B n O n (R)

10 6 Witt Witt. F R (i) n S T : t T ST = D, (D ) (ii) A n (R +) n Cartan GL n (R) = O n (R) A n O n (R) (g GL n (R) t gg ) (iii) GL n (R) ( n M n (R) R n ). Witt (V, Q) F v V Q(v) = 0 (isotropic), (anisotropic) X V (totally isotropic) Q X 0 (X, Q X ) II (V, Q) 0.. (V, Q) II (v, w) Q 0 v, w V Q(v + w) Q(v w) = 4(v, w) Q v + w, v w.3. (i) (H, Q H ) e 1 := (1, 0), e := (0, 1) 1.5 v 1, v (ii) 1 (F, a) (a 0 ) C.4 ( ). X (V, Q) X {e 1,..., e r } {e 1,..., e r} V (e i, e j) Q = 0, (e i, e j) Q = δ i,j, (1 i, j r) (, ) Q X X := span{e 1,..., e r} X. X := {v V (v, x) Q = 0, x X} Z : V = X Z. ϕ : Z z Ä x (x, z) Q ä X

11 .. Witt 7 ϕ(z) = 0 z X Z = 0 ϕ Q X x (x, ) Q Z rk ϕ Z = dim Z = dim Z dim X. ϕ {e 1,..., e r} X {e 1,..., e r } f i := ϕ 1 (e i ), (1 i r) e i := f i Q(f i) i 1 e i (f i, e (f i, e i ) k) Q e k Q k=1 (e i, e j ) Q = (f i, e j ) Q = δ i,j i > j i = j (e i, e j) Q =(f i, e j) Q Q(f i) (e i, e i 1 (f i, e i ) j) Q (f i, e k) Q (e k, e j) Q Q k=1 =(f i, e j) Q (f i, e j) Q = 0, Å (e i, e i) Q = Q f i Q(f ã Å ã i) Q(f i ) e i = Q(f i ) f i, e i = 0 (e i, f i ) Q (e i, f i ) Q Q (H, Q H ) r Ä (x 1, y 1 ),..., (x r, y r ) ä r x i e i + y i e i (V, Q) i=1.5 (Witt ). (V, Q) W f : (W, Q W ) (V, Q) O(V ) (V V ). u V (reflection) r u : V v v (u, v) Q Q(u) u V O(V ) f m(w ) := dim W + dim(w W ) m(w ) = 0 f V W (W, Q W ) w (.) L := span {w}, W 1 := {v W (v, w) Q = 0} (i) W = L W 1. (L p w : W v Ä (v, w) Q /Q(w) ä w L L = im p W, W 1 = ker p W )

12 8 Witt Witt (ii) m(w 1 ) = m(w ) 1. (W W = {v W 1 W (v, w) Q = 0} = W 1 W 1. ) (ii) g 1 O(V ) g 1 W1 w := g1 1 f(w) = f W1 Q(w + w ) + Q(w w ) = (Q(w) + Q(g 1 1 f(w)) = 4Q(w) 0 w ± w w + w w w r w+w r w (w) = w (w + w, w) Q (w + w ) Q(w + w ) = w + Q(w) + (w, w) Q Q(w) + (w, w) Q (w + w ) = w, r w w (w) = w Q(w) (w, w) Q Q(w) (w, w) Q (w w ) = w g O(V ) g (w) = w v W 1 (v, w ) Q = (g 1 (v), f(w)) Q = (f(v), f(w)) Q = (v, w) Q = 0 g W1 = id W1 g := g 1 g O(V ) (i) a F, w 1 W 1 g(aw + w 1 ) = ag 1 (w ) + g 1 (w 1 ) = af(w) + f(w 1 ) = f(aw + w 1 ) g W = f W.4 W {e 1,..., e m } {e 1,..., e m ; e 1,..., e m} (H, Q H ) m W W V, W := span{e 1,..., e m} f(w ) {f(e 1 ),..., f(e m )}.4 {f(e 1 ),..., f(e m ); f(e 1 ),..., f(e m ) } f m m W W x i e i + y i e i x i f(e i ) + y i f(e i ) V i=1 i=1 W W m(w W ) = dim(w W ) = dim W < 3 dim W = dim W + dim(w W ) = m(w ) W W

13 .. Witt 9 (V, Q) X V.4 X V (, ) Q X X V := (X X ) (V, Q) = (X X, Q X X ) (V, Q ), Q := Q V (.1) (V, Q ) v 0, V X F v X (V, Q) Witt (V, Q ) (V, Q) (anisotropic kernel) X, Y V dim X dim Y.5 f : X Y g O(V ) g 1 (Y ) X X dim X = dim Y Witt (V, Q) r = dim X (V, Q) Witt (Witt index) r(v ) = r(q)

14 Witt F Witt-Grothendieck W (F ) Witt-Grothendieck (V, Q) (V, Q ) (V, Q) (V, Q ) := (V V, Q Q : V V v v Q(v)Q (v ) F ) W (F ) (F, 1) F Witt-Grothendieck W (F ) (H, Q H ) F Witt Witt(F ) (V, Q) Witt(F ) (V, Q) W (F ) W (F ) F Witt 3. F = R F (V, Q) O(V ) (V, Q) O(V ) End R (V ) 4. Witt(C) Witt(R) 3. p Q p p x Q p m a/b, (m Z, a, b Z pz) x p := p m 0 p = 0 Q x y p x p 0, x Q; x p = 0 x = 0; ( ) x + y p x p + y p, x, y Q Q Q p Q p Q (a n ) n N p Cauchy ε > 0 N N a n a m p < ε, n, m > N

15 Q p Cauchy F (N, Q) p (1) n N ß m(n, Q) p := (a n ) n F (N, Q) p lim a n = 0 n F (N, Q) p d p ((a n ) n, (b n ) n ) := lim n a n b n p, (a n ) n, (b n ) n F (N, Q) p Q p := F (N, Q) p /m(n, Q) p (a n ) n F (N, Q) p m(n, Q) p (a 1 n ) n F (N, Q) p Q p 5. x + y p max( x p, y p ), x, y Q p Q p Z p := {x Q p x p 1} p n Z p = {x Q p x p p n } = {x Q p x p < p 1 n }, n N 0 x Q p {x + p n Z p } n N R Q R Q p R x, y R nx > y n N 1 Q p n N np p < 1 = 1 p 3.3 (local field) F ( ) dx a F F x ax F dx d(ax) F a F := d(ax)/dx F : F R + a F a 0 F : F a 0 a = 0 R 0 1 { nx n N} R

16 1 3 F (module) F [Wei95, I.3] (i) (non-archimedean) (ultrametric inequality) x + y F max( x F, y F ), x, y F F F R + (a) 0 F p p Q p (b) p F p F q (q p ) Laurent F q ((T )). (ii) (archimedean) R, C R + R z C = z z 3.4 F F D D dx a D a D := d(ax)/dx R + 0 D = 0 D (0) D (1) O D := {x D x D 1} F (D = F (integer ring) (order) ). O D F A D a D > 1 A O D O D () p D := {x D x D < 1} O D O D = {x D x D = 1}. O D O D = {x O D x 1 D = x 1 D 1} 1 p D O D O D p D = O D p D B r := {x F x F r}, (r > 0) F 0

17 3.5. Hensel 13 (3) p D = O D ϖ D = ϖ D O D ϖ D F O D (prime element) (uniformizer). D D R + q D > 1 D D = qd Z ϖ D D = qd 1 ϖ O (4) D (residue field) k D := O D /p D p ( ) p D (residual characteristic). O D p D k D (5) k D q D D D = q Z D. val D : D x log q x D Z D (valuation) O. ϖ D D = D d(ϖ D x) p = D dx O D dx O D dx = k D Hensel F O p = (ϖ) val f(x) O[X], x O m, r > 0, Z f(x) p m+r, f (x) / p m+1 t f(x) + pm+r+1 f (x) p m+r val(f (x)) x := x + t x x + p m+r, (val(f (x)) m ). f(x ) p m+r+1. f(x) X x n f(x + t) = f(x) + f (x)t + α k t k, k= α k O t p m+r n k= α k t k p m+r f(x) + f (x)t p m+r+1

18 (Hensel ). f(x) O[X], x O m > 0, Z f(x) p m+1, f (x) / p m+1 y x + p m+1 f(y) = 0. {x r } r N x r+1 x r p m+r ; f(x r ) p m+r p m+1 Cauchy y p m+1 f(x) F f(y) = lim r f(x r ) = a 1 + p val()+1 f(x) := X a x = 1 f(1) = 1 a p val()+1, f (1) = / p val()+1 Hensel f(b) = 0, b = a b 1 + p val()+1 (F ) := {b b F } F 1 + p val(f )+1

19 D F O D k D 4.1. (i) ε O p q D 1 (ii) Z/(q D 1)Z E (p) D E (p) {0} k D O D (iii) Z/(q D 1)Z E (p) 1 D Ad(ϖ D )E (p) = E (p) 1 ϖ D n N (1 + p D ) pn 1 + p n+1 D.. n = 0 n (1 + p D ) pn+1 = Ä ä p 1 p (1 + p D ) pn (1 + p n+1 D ) p = p n+ D 1 + pp n+1 D = 1 + p p(n+1) D k=1 0 p (a) µ : O D x lim x qn O D n (b) x O D k D µ(x) ( ) p p k(n+1) D + p p(n+1) D k. (a) x p D lim n x qn D = 0 k D F qd X q D X x O D xqd 1 1+p D q D = p f D x qn D (qd 1) (1 + p D ) pnf D 1 + p nf D+1 D, x qn+1 D x qd n p nf D +1 D x qn D = x + (x q D 1) + (x q D x q D ) + + (x qn D x q n 1 D ) (4.1)

20 16 4 Cauchy µ (b) µ(1 + p D ) = 1 (4.1) µ(x) x (mod p D ) µ 1 (1) = 1 + p D µ(x n ) = µ(x) n (i) ε O n p n q D q D m 1 (mod n) m N ε qkm D = ε, k N µ ε 4.1. (b) (ii) k D k D x O D 4.1. E (p) := µ(x) Z/(q D 1)Z. (i) E (p) k D (iii) D Ad(a) : D x axa 1 D O D p D k D O D k D D /O D a Ä Ad(a) OD mod p D ä Aut(kD ) ϖ D Aut(k D) = Gal(k D /F p ) f D r Ad(ϖ D) x = x pr, x k D (4.) E (p) ε ε k D ε 1 (4.) Ad(ϖ D)ε ε pr 1 (mod p D ), ϖ D ε pr 1 ϖ Dε 1 (mod p D), ε E (p) ε 1 E (p) 1 ϖ D := ε E (p) εpr 1 ϖ Dε 1 p D ϖ D ϖ D (1 q D )ϖ D ϖ D (mod p D) ϖ D γ E (p) ϖ D γ = ε E (p) ε pr 1 ϖ Dε 1 γ = Ad(ϖ D )E (p) = E (p) 1 ε E (p) (γ 1 ε 1 ) pr ϖ Dε 1 = γ pr 1 ϖ D 6. p Q p Q(x, y) = x + y p 1 (mod 4) (Q p 1 4 ) z D = z dim F D F, (z F ) O F = O E F, p F = p E F k F (z mod p F ) (z mod p D ) k D

21 k D k F f D/F D/F (modular degree) E/F (ramification index) e E/F := val D (ϖ F ) q dim F D F = ϖ F dim F D F = ϖ F D = q val D(ϖ F ) D = (q f E/F F ) e E/F dim F D = e E/F f E/F e D/F = 1 D F (unramified) E/F (1) ε E 1 q E 1 E = F (ε). (F (ε) E [F (ε) : F ] f F (ε)/f = f E/F = [E : F ] ) E X q E X F [X] E/F Galois F [E : F ] O F O E () Gal(E/F ) E (p) Gal(E/F ) τ (τ OE mod p E ) Gal(k E /k F ) Gal(E/F ) Φ E/F (x) x 1/q F ( Frobenius ) (mod p), (x O E ) F D F F (central division algebra) 4.. D F (i) e D/F = f D/F. n ( dim F D = n ) (ii) D F n E (iii) O D ϖ D (a) ϖ n D O F, O E (b) {1, ϖ D,..., ϖd n 1 } O D O E (c) ϖ D E Ad(ϖ D ) E Gal(E/F ) Z/nZ. E (p) O D 4.1 ϖ D E := F (E (p) ) (1) E/F [E : F ] = log qf q D = f D/F n := f D/F (ii) D = N Z ϖ N DO D x ϖ N DO D x = ε n ϖd, N ε n E (p) {0} (4.3) n=n

22 18 4 F = {z D Ad(ϖ)z = Ad(ε)z = z, ε E (p) } Ad(ϖ D ) E (p), E Gal(E/F ) E Ad(ϖ D) F Galois Ad(ϖ D ) E Gal(E/F ) (iii) (c) E E (p) ϖ D ϖ n D E = ϖ n D, E (p) (4.3) (iii) (a), (b) dim E D = n e D/F f D/F = dim F D = dim E D[E : F ] = n (i) 4.3. F n (Z/nZ). F n E σ := Φ E/F 4. F n D E E ϖ D Ad(ϖ D ) E Gal(E/F ) Z/nZ σ = Ad(ϖ D ) k E k Z/nZ σ Gal(E/F ) k (Z/nZ) (E, ϖ D ) D k D k (Z/nZ) km 1 (mod n) m Z/nZ â ì z â ì ϖf σ m (z)... σ m(n 1) (z) z E E, ϖ D := M n (E) F n σ = Ad(ϖ D ) k E 4.4. (i) D k/n 1 n Z/Z D inv F (D) (ii) D M n (E) Gal(E/F ) σ m D := Ad(ϖ D ) σ m D M n (F ) (inner form) (iii) M n (F ) dm = n d, m N F d D M m (D) inv F (M m (D)) := inv F (D) 4.3 inv F : {M n (F ) } Z/nZ

23 F D D opp

24 Hilbert F a F F E a := F [X]/(X a) x E a E a F det(x Ea ) F x (E a /F ) N Ea /F (x) N Ea /F : E a F F (E a, N Ea /F ) (F, 1) (F, a) F Hilbert (Hilbert symbol) (, ) F : F F {±1} (a, b) F = 1 ax + by = 1, x, y F b N Ea /F (E a ) a N Eb /F (E b ) 5.1. E a E b N Ea /F (E a ) = N Eb /F (E b ). N Ea /F (E a ) = N Eb /F (E b ) (x, y) F x ay = x by (x, y ) F (V, Q) := (F, 1) (F, a) (F, 1) (F, b) F (x, y) (x, y, x, y ) V.4 (V, Q) (H, Q H ) a = b mod (F ) a, b F F B = B a,b := F 1 F i F j F ij i = a 1, j = b 1, ij = ji F (a, b) (quaternion algebra) Hamilton R ( 1, 1) B ι : B x 0 + x 1 i + x j + x 3 ij x 0 x 1 i x j x 3 ij B opp B (main involution) ν B : B b b ι b F

25 5.1. Hilbert 1 B (reduced norm) b = x 0 +x 1 i+x j +x 3 ij ν B (b) = x 0 ax 1 bx + abx 3 (B, ν B) 5.. (i) B a,b a, b F /(F ) (ii) B = B a,b (a) B (b) (B, ν B ) (c) (a, b) F = 1.. (i) a ac i c i (ii) (a) (b) b B ν B (b) 0 ν B (b) 1ι b B b (a) ν B (b) = 0 b = 0 (b) (b) (c) (a, b) F = 1 1 ax by = 0, ν B (1+xi+yj) = 0 x, y F ν B (c) (b) (B, ν B ) (F, 1) (F, a) (F, b) (F, ab) (1, 1) a, b, ab 1 ( ) a b 1 (a, b) F = 1 ab = 1 ax + by xy (a, b) F = F (i) (a, b) F = (b, a) F (a, ) F : F /N Ea/F (E a ) {±1} (ii) a F (F ) (a, ) F : F /N Ea/F (E a ) {±1}. (i) F C F /N Ea/F (E a ) Z/Z F = R F b, c F N Ea /F (E a ) bc N Ea /F (E a ) (a, b) F = (a, c) F = 1 5. B a,b, B a,c F 4 (F ) 4.3 F B a,b F B a,c ν Ba,b, ν Ba,c (F, b) (F, ab) (F, c) (F, ac) b = c mod (F ) bc (F ) N Ea /F (E a ) ab = c mod (F ) bc = a mod (F ) bc N Ea /F (E a )

26 5 5. (V, Q) r i=1 (F, q i ) n i=r+1 (F, 0), (q i F /(F ) ) Hasse ε(v ) = ε(q) := 1 i<j n (q i, q j ) F {±1} 5.4. F = R (V, Q) sgn Q = (p, q) ε(q) = ( 1, 1) q(q 1)/ R = ( 1) q(q 1)/ (V, Q) F α F Q(v) = α v V (i) rk Q = 1 det Q = α mod (F ). (ii) rk Q = (α, det Q) F = ε(q). (iii) rk Q = 3 (a) det Q α mod (F ) (b) det Q = α mod (F ) ε(q) = ( 1, det Q) F. (iv) rk Q 4.. (i) (ii) (V, Q) (F, a) (F, b) Q(x, y) = ax + by = α (x, y) V (aα, bα) F = 1 (aα, bα) F = (a, α) F (α, b) F (α, α) F (a, b) F = ( det Q, α) F ε(q) (iii) (V, Q) (F, a) (F, b) (F, c) Q(x, y, z) = ax + by + cz = α (x, y, z) V ax + by = β = αw cz, x, y, z, w F β F (ii) (β, ab) F = (a, b) F, (β, αc) F = (α, c) F (5.1) β F F F /(F ) β dim F F /(F ) 1 (5.1) ab = αc mod (F ) (a, b) F (α, c) F 1 1 q F 1 ε ϖ F F

27 5.. 3 det Q = α mod (F ) (a, b) F ( abc, c) F = ( abc, 1) F (a, c) F (b, c) F ( c, c) F =( det Q, 1) F (a, c) F (b, c) F, ε(q) ( det Q, 1) F (iv) Q(x, y, z, w) = u(x ay bz + bcw ), (u, a, b, c F ) a c mod (F ) N Ea /F (E a ), N Ec /F (E c ) F 5.1 N Ea/F (E a )N Ec/F (E c ) = F. b = x ay z cw x ay bz + bcw = 0 (x, y, z, w) V Q Q(V ) = F a = c mod (F ) Q = u ν Ba,b B = B a,b ν B Q(V ) = F. B D D 4. E ε /F (ε 1 q F 1 ) E ϖ := F (ϖ D )/F ν D/F Ea = N Ea /F F ν D/F (D ) N Eε/F (E ε )N Eϖ/F (E ϖ) = F. (Hamilton H : ν H/R (H ) = R + ) 5.6. F 5 (V, Q). (V, Q) 5 (V, Q) = (V 1, Q 1 ) (F, a), (dim V 1 = 4) 5.5 (iv) Q 1 (v 1 ) = a v 1, (V, Q) (v 1, 1).4 8. F C (i) F /(F ) F (ii) Hilbert (, ) F F F /(F ) D 4.3 ν D/F det

28 F Hasse. (V, Q), (V, Q ) rk Q = rk Q = n, det Q = det Q mod (F ), ε(q) = ε(q ) n = 1 n 5.5 Q(v 1 ) = α = Q (v 1) v 1 V, v 1 V, α F V 1 := (F v 1 ), V 1 := (F v 1), Q 1 := Q V1, Q 1 := Q V 1 (V, Q) (F, α) (V 1, Q 1 ), (V, Q ) (F, α) (V 1, Q 1) rk Q 1 = rk Q 1 = n 1, det Q 1 = det Q/α = det Q /α = det Q 1, ε(q 1 ) = ε(q)(α, det Q 1 ) F = ε(q )(α, det Q 1) F = ε(q 1) (V 1, Q 1 ) (V 1, Q 1) (V, Q) (V, Q ) 5.5 ( ) rk Q = 1 ε(q) = 1, rk Q = det Q = ε(q) = 1

29 : (V, Q) Hasse 1 (F, α) α 1 (H, Q H ) 1 1 (E α, N Eα /F ), (α F (F ) α 1 ( ) α, β F (E α, βn Eα/F ), α 1 (α, β) F = 1 m + 1 (F, α) (H, Q H ) m ( 1) m α (( 1) m(m 1)/ α m, 1) F (m 1) (D 0, ν D/F D 0) (H, Q H ) m 1 ( 1) m 1 ( 1, 1) m(m+1)/ F (D 0, αν D/F D 0) (H, Q H ) m 1 ( 1) m α (( 1) m(m 1)/ α m, 1) F (α F (F ) ) m (H, Q H ) m ( 1) m ( 1, 1) m(m 1)/ F (m ) (E α, N Eα/F ) (H, Q H ) m 1 ( 1) m α (( 1) m(m 1)/ α m 1, 1) F (α F (F ) ) (E α, βn Eα /F ) (H, Q H ) m 1 ( 1) m α (( 1) m(m 1)/ α m 1, 1) F (α, β F, (α, β) F = 1) (D, ν D/F ) (H, Q H ) m ( 1) m ( 1, 1) m(m 1)/ F 6. (global field) (algebraic number field). Q 1 F p (X) F q (T ). F (completion) K ι : F K ι(f ) K ι : F K, j : F L φ : K L F ι K F j φ L F F (place) F v F F v F v v 6.. Q

30 6 6 (i) : Q R R Q (ii) p p : Q Q p 0 Q p Q Q p 6.3. E/F w : E L (i) w(f ) K L/K w F : F K F v (ii) L = K(E) (K E ). (i) K L a L < 1 a K {a n } n N K K L = {1}, K O L O L K F K L K K [L : K] (ii) K(E) K L K(E) = (K(E) ) (E ) = L E w F v w v F (i) F 0 (a) F Q R R r 1 C r r 1 p i : R r 1 C r R v i : F F Q R p i R F r q i : R r 1 C r C u i : F F Q R q i C (b) p F Q Q p r i=1 K i i p i : F Q Q p K i v i : F F Q Q p K i Q p F (ii) F p > 0 F = F q (T ) (a) v : F q (T ) F q ((T 1 )). (b) ϖ F q [T ] Å ã v ϖ : F q (T ) lim F q [T ]/(ϖ n ) Fq[T ] F q (T ) F q deg ϖ((x)) n

31 6.. 7 F S A(S) := v S F v v / S O v O v = O Fv F v v / S O v Tychonov A(S) S T A(S) A(T ) A = A F := A(S) = lim A(S) S S F (adele ring) 6.4. (i) F V V A := V F A A (ii) V V ξ (ξ) v A V A V A /V. (i) A V A A (ii) F = V = Q, F p (T ) Tychonov Q Z := p Z p (p ) (1) Q A () (0, 1) Z A Q\A. (1) 0 Q A A ( 1, 1) Z Q ( 1, 1) 0 () Q(p) := {a/p n a Z, n N} x p n Z p p x = x k p k, x k {0, 1,..., p 1} k= n Q p = Q(p) + Z p x = (x v ) v A(S) p S x p = ξ(p) + z p, (ξ(p) Q(p), z p Z p ) x = p S ξ(p) + (x, (z p ) p S, (x p ) p/ S ) Q + A( ) Q+A( ) = A A ξ+x = η+y, (ξ, η Q, x, y A( )) x y Q A( ) = Q Z = Z

32 F F 6.5. F ι : F K, j : F L α F α K < 1 α L < 1. α K < 1 {α n } n N K 0 γ L > 1 F γ K > 1 γ L = γ λ K λ > 0 F L = λ K α F α K = γ ν K, (ν R) α L = γ ν L ν a/b α K = γ ν K < γ a/b K αb /γ a K < 1 α b /γ a L < 1 α L < γ a/b L ν a/b α L γ ν L ν a/b α K > γ a/b K α L γ ν L 6.6 ( ). F S F F S := v S F v. S = {v 1,..., v n } γ v1 > 1, γ vi < 1, ( i n) γ F. n n = α v < 1 α v1 < v 1 = v n α v1 > 1, α v,..., α vn 1 < 1 α F n = β v1 > 1, β vn < 1 β F α vn 1 α m β vi < 1, ( i n 1) m N γ := α m β lim m α m 1 + α m = 1 (F v1, F vn ) 0 (F v,..., F vn 1 ) γ = α m β/(1 + α m ) m N ε 1,..., ε n > 0 δ 1 v1 < ε 1, δ vi < ε i, ( i n) δ F

33 γ δ m := γ m /(1 + γ m ) {δ m } m N F v1 1, F v,..., F vn 0 (x v ) v S F S ε > 0 F F vi x vi ξ i vi < ε 3, 1 i n ξ 1,..., ξ n F j n δ j 1 vj < ε 3 ξ j vj, δ j vi < ε 3(n 1) ξ j vi, (i j) δ i F ξ := n j=1 δ j ξ j F x vi ξ vi x vi ξ i vi + ξ i ξ vi ε 3 + ξ i δ i ξ i vi + j i δ j ξ j vi v i S = ε 3 + ε 3 + j i ε 3(n 1) = ε

34 30 7 Hasse 7.1 ( ) A M (simple) (irreducible) 0 A A A (semisimple) (completely reducible) A A M M A (0) A A F F 7.1 (Schur ). A M End A (M). φ End A (M) 0 M 0 M M φ M A 7.. A F M F A End A (M) End F (M) A. B := End A (M) A Cent (B, End F (M)) = End B (M) φ End B (M) A x M φ(x) A.x.. A M A A.x A p : M A.x p B φ(p(x)) = p(φ(x)) im p = A.x M F {v 1,..., v n } A M n End A (M n ) = M n (B) φ n Cent (M n (B), End F (M n )) 7..1 Ä φ(v1 ),..., φ(v n ) ä = Ä a.v 1,..., a.v n ä, a A. φ = a A

35 ( ) F F D M n (D). A F F A M ( A ) 7.1 D opp := End A (M) F 7. A = End D opp(m) M D opp {v 1,..., v n } D opp D End D opp(m) M n (D) ( 10 ) F (central simple algebra) F A Z(A) F 7.3 F 9. A F M F A End A (M) F D i M ni (D i ) 10. D F M F D (i) M D (ii) M D n End D (M) M n (D opp ) 7. F F A F A v := A F F v F v F v D v A v M nv (D v ) dim Fv D v = d v dim F A = dim Fv A v = (d v n v ) n := d v n v v D v ( 4.4 ) inv Fv (D v ) (Z/d v Z) A v inv(a v ) Z/nZ 7.4. K n Br n (K) F Br n (K) A (inv(a v )) Z/nZ v v Z/nZ κ v v κ v Z/nZ 7.5. F F F E A E Ā := lim A E Galois E 0 H (F, F ) H (F, Ā ) H (F, Ā / F ) Q/Z

36 3 7 Hasse [CF86, VII ] 7.6. α, β F v (α, β) Fv = a, b F v (a, b) F v = ( 1) inv(b a,b) 7.4 (ii) v inv(b α,β ) v = 0 1 Z/Z 7.3 Hasse F 7.4 F 7.7 (Hasse-Minkowski). (V, Q) (i) dim F V 3 v (V v := V F F v, Q v := Q F F v ) (ii) dim F V (V, Q) v (V v, Q v ). (i) Q V = F 3, Q(x, y, z) = x αy βz, (α, β F ) F B = B α,β 7.4 v B v M (F v ) 5. (ii) dim F V dim F V = 1 Q Q v dim F V = Q(x, y) = x αy, (α F ) v (α, x) Fv = 1, x F v β F (α, β) Fv = 1, v inv(b α,β ) v = 0, v 7.4 B α,β M (F ) x αy β(z αw ) β Q dim F V = 3 Q(x, y, z) = x αy βz, (α, β F ) v (α, β) Fv = 1, inv(b α,β ) v = 0 B α,β M (F ) 5. (α, β) F = 1, Q

37 7.3. Hasse 33 dim F V = 4 Q(x, y, z, w) = x αy βz +βγw, (α, β, γ F ) Q (x, y, z, w) 0, F 4 Q(x, y, z, w) = 0, β = x αy z γw N E α /F (E α )N Eγ /F (E γ ) = N Eα.E γ /E αγ ((E α.e γ ) ) F ( 11 ) α γ F F E αγ dim F V = 3 ( E αγ w (α, β) Eαγ,w = 1 (α, β) Eαγ = 1, β N Eα.E γ/e αγ ((E α.e γ ) ) ) dim F V 5 (V, Q) = (F, P ) (V, Q ), dim F V 3 (i) F S v / S Q v S P (x v, y v ) = c v = Q (x v ), (x v, y v ; x v ) 0 (F v ) F v (v 3. ) 6.6 P (ξ, η) F P (ξ, η) =: γ c v mod (F v ), v S (V 1, Q 1 ) := (V, Q ) (F, γ) v / S v S F Q (ξ ) = γ ξ 0, V Q(ξ, η; ξ ) = F α, γ F N Eα.E γ/e αγ ((E α.e γ ) ) F = N Eα/F (E α )N Eγ/F (E γ )

38 { 8.1. F } F n (i) (V, Q) (V v, Q v ) v (ii) (a) (det Q(v)) v F (mod v (F v ) ), { Fv n } (b) v ε(q(v)) = 1 v ε(q(v)) = 1 Ä V (v), Q(v) ä v. (i) F n (V, Q), (V, Q ) (V v, Q v ) (V v, Q v), v (V, Q) (V, Q ) n n = 0 n v (V v V v, Q v Q v) Hasse-Minkowski 7.7 (V V, Q Q ) Q(ξ) = α = Q(ξ ), ξ 0, V, ξ 0, V, α F V 1 V, V 1 V ξ, ξ Q, Q Q 1, Q 1 F v (V v, Q v ) = (F v, α) (V 1,v, Q 1,v ) (V v, Q v) = (F v, α) (V 1,v, Q 1,v) ( 1.4) (V 1,v, Q 1,v ) (V 1,v, Q 1,v) (V 1, Q 1 ) (V 1, Q 1) (ii) (a), (b) Ä V (v), Q(v) ä v (V v, Q v ) (V (v), Q(v)), v F (V, Q) n Q(v) det Q(v) v δ F n = 1

39 n = Q(v)(x, y) = a v x + a v δy, (a v F v ) δ (F ) v Q(v) (V, Q) = (H, Q H ) 1 = v ε(q(v)) = v (a v, a v δ) Fv = v (a v, δ) Fv v inv(b av, δ) = 0 1 Z/Z ( 7.6 ) 7.4 α, β F (B α,β ) v B av, δ, v ν Bα,β δ F ( 7.7) δ 1 (mod (F ) ) α, β, αβ δ = β (mod (F ) ) (V, Q) := (F, α) (F, αδ) det Q = δ (mod (F ) ) v ( 1, 1) Fv ε(q v ) = ( 1, 1) Fv (α, δ) Fv = ε(ν (Bα, δ ) v ) = ε(ν Bav, δ ) = ( 1, 1) Fv (a v, δ) Fv = ( 1, 1) Fv ε(q(v)) Hasse n = 3 S := {v ε(q(v)) ( δ, 1) Fv } (B δ, 1 ) v v M (F v ) (b) S v (F v ) F S ( 3.) 6.6 αδ / (F v ), v S α F v S αδ 1 F (V 1 (v), Q 1 (v)) det(q 1 (v)) = αδ, ε(q 1 (v)) = (α, αδ) Fv ε(q(v)) (8.1) (5 6.1 ) v / S αδ = 1 S (α, αδ) Fv ε(q(v)) = (α, 1) Fv ( δ, 1) Fv = (1, 1) F = 1 (8.1) F v (V 1 (v), Q 1 (v)) (a), (b) F (V 1, Q 1 ) (V 1,v, Q 1,v ) (V 1 (v), Q 1 (v)), v (V, Q) := (V 1, Q 1 ) (F, α) det Q = α det Q 1 = δ (mod (F ) ) v ε(q v ) = (α, det Q 1 ) Fv ε(q 1,v ) = ε(q(v)) 3 Hasse

40 36 8 n 4 v sgn Q(v) = (p v, q v ) F v F v F v +, Fv 6.6 α F F v + v p v 0 α v p v = 0 F v n 4 Q(v)(x v ) = α x v V (v) ( 5.5) α v (V (v), Q(v)) (V 1 (v), Q 1 (v)) (F v, α) 3 det Q 1 (v) = αδ F, ε(q 1 (v)) = (α, αδ) Fv ε(q(v)) (a), (b) F n 1 (V 1, Q 1 ) det Q 1 = αδ, ε(q 1,v ) = (α, αδ) Fv ε(q(v)), v; (p v 1, q v ) v p v 0 sgn (Q 1,v ) = (p v, q v 1) v p v = 0 (V, Q) := (V 1, Q 1 ) (F, α) 8.. R (V, Q) sgn Q = (p, q) det Q = ( 1) q, ε(q) = ( 1, 1) q(q 1)/ R = 1 q 0, 1 (mod 4) 1 q, 3 (mod 4) F {((V (v), Q(v))} v F v (p v, q v ) (p v + 4k v, q v 4k v ), (k v Z) F

41 37 9 Weil Weil [Wei64] 9.1 Weil F ψ : F C F ψ a (x) := ψ(ax), a F F ψ R (x) := e πix, ψ C (z) := ψ R (z + z) F ψ p n n Z ψ (order) ord ψ F (V, Q) Schwartz-Bruhat S(V ) S(V ) F V V dv Fourier F Q φ(v) := φ(v )ψ((v, v ) Q ) dv, φ S(V ) Fourier F Q φ(v )ψ((v, v ) Q ) dv = φ(v) V V dv V (self-dual measure) dv 9.1. (i) F (V, Q) 1 γ ψ (V ) = γ ψ (Q) Å Q(v) ã Å φ(v)ψ dv = γ ψ (Q) F Q φ(v)ψ Q(v) ã dv V V (ii) Q γ ψ (Q) Witt γ ψ : Witt(F ) C 1. (i) (V, Q) ( 1.4) Fubini (V, Q) = (F, a) F φ(x)ψ a Å x ã dx = γ ψ (a) F F a φ(x)ψ a Å x ã dx, γ ψ (a) C 1 (9.1)

42 38 9 Weil f(x) := ψ a ( x /), ξ φ (x) := φ( x)ψ a (x /), (φ S(F )) Å Å f ξ φ (x) = ψ a (x + y) y ãφ(y)ψ a ã dy F Å =ψ a x ã Å φ(y)ψ a ( xy) dy = ψ a F φ 0 S(F ) F F f ξ φ0 (x) dx = γ ψ (a) := F F x Å ψ a x ã F a φ 0 (x) dx 0 Å ξ φ0 (x) dx F ã 1 f ξ φ0 (x) dx ã F a φ( x). Å x f ξ φ0 (x) dx φ(x)ψ a ã dx = f ξ φ0 (x + y)ξ φ ( y) dy dx F F F = f ξ φ0 ξ φ (x) dx = ξ φ0 ( y)f ξ φ (x + y) dy dx F F F = ξ φ0 (x) dx f ξ φ (x) dx (9.) γ ψ (a) F =γ ψ (a) F F f ξ φ0 (x) dx F F a φ(x)ψ a Å x ã dx. (9.) F f ξ φ 0 (x) dx (9.1) (9.1) φ 1 S(F ) Fourier F a φ(x) = F a φ(x) = Fa 1 φ(x) (9.1) F Å x φ(x)ψ a ã Å dx = γ ψ (a) F a φ(x)ψ a x ã dx F Å x ã Å =γ ψ (a) Fa 1 φ(x)ψ a dx = γ ψ (a)γ ψ (a) φ(x)ψ F a F Å x =γ ψ (a)γ ψ (a) φ(x)ψ a ã dx. F x ã dx γ ψ (a) C 1 (ii) γ ψ (Q) (V, Q) ( 1 ) F Q φ(v) = F Q φ(v) γ ψ (Q) V Å φ(v)ψ Q(v) ã dv = γ ψ (Q) V Å Q(v) F Q φ(v)ψ ã dv γ ψ ( Q) γ ψ ( Q) = γ ψ (Q) = γ ψ (Q) 1 γ ψ Witt(F )

43 9.1. Weil 39 γ ψ (Q) (V, Q) Weil 1. (i) a, c F γ ψ (ac ) = γ ψ (a) (ii) γ ψr (a) = e sgn (a)πi/4, (a R ) γ ψc (c) = 1, ( c C ) (Gauss e πx, e πz z Fourier ) 9.. (i) F = C γ ψ (Q) = 1. (ii) F = R sgn Q = (p, q) γ ψ a R (Q) = e sgn (a)(p q)πi/4. (iii) F ord ψ = 0 γ ψ (a) = 1, a O. (i), (ii) 1 (iii) a O ψ a O 1 φ S(F ) O F (9.1) F C D F γ ψ (ν D ) = 1.. F = R 9. (ii) F ν D F ν D γ ψ (ν D ) ψ ord ψ = 0 O D 4. ϖ D 1 q F 1 ε D D O D = O Oε D ϖ D (O Oε D ) ν D (x, y) D = τ D (xy) O D ϖ D = ϖ F OD :={x F τ D (xy) O, y O D } x 1 + x ε D = x 1 y 1 + x y ε D + x 3 y 3 ϖd, x 4 y 4 ε DϖD O +x 3 ϖ D + x 4 ε D ϖ D y 1 + y ε D + y 3 ϖ D + y 4 ε D ϖ D O D =O Oε D p 1 ϖ D p 1 ε D ϖ D = ϖ 1 D (Oϖ D Oε D ϖ D O D O D ε D ) =p 1 D φ S(D) p val() D F D φ(x) = φ(x)ψ D Å τd (xy) ã dy = p val() D meas p val() D = 0 Å τd (xy) ψ τ D ( 1 xpval() D ) O ã dy

44 40 9 Weil {x D 1 ϖ val() D x O D} = p val() 1 D c := meas p val() D ( 4. val D () = val() ) D d D x := d D x/ x D D Å F D φ(x)ψ ν D(x) ã d D x = c n=val() 1 ϖ n D O D Å ψ ν D(x) ã x D d D x D = ν D ( ) F D1 := ker ν D O D ν D : O D/D 1 O =c meas D 1 n=val() 1 ϖ n O Å ψ x ã x F dx. d D x /d D 1x = dx := dx/ x F D 1 d D 1x D 1 meas D 1 Å ψ x ã Å x ϖ n O F dx = ψ x ã x F dx p n p n+1 q n (1 q 1 ) n val() = q 1 val() O ψ( x/) dx = q1 val() n = val() 1 D Å F D φ(x)ψ ν D(x) D ã d D x =c meas D 1 Å q 1 val() + = c meas D 1 q1 val() 1 + q 1 Å ã νd (x) φ(x)ψ d D x = p val() D n=val() d D x > 0 ã q n (1 q 1 ) Weil ( 9.1) γ ψ (ν D ) C 1 R <0 = { 1} 9.4. (i) a, b F γ ψ(1)γ ψ (ab) γ ψ (a)γ ψ (b) = (a, b) F. Weil γ ψ (Q) 1 8 (ii) γ ψ (Q) = γ ψ (1) rkq 1 γ ψ (det Q)ε(Q).. (i) γ ψ (ν Ba,b ) = (a, b) F (B a,b, ν Ba,b ) (F, 1) (F, a) (F, b) (F, ab) 9.1 γ ψ (ν Ba,b ) = γ ψ(1)γ ψ (ab) γ ψ (a)γ ψ (b)

45 9.1. Weil 41 c F γ ψ (c) 4 = γ ψ c(1) 4 = γ ψ c(1)γ ψ c(1) γ ψ c( 1)γ ψ c( 1) = ( 1, 1) F {±1} (ii) rkq 1 Q Q 1 q ( ) =γ ψ (1) rkq 1 γ ψ (q det Q 1 )ε(q 1 )(q, det Q 1 ) F (i) =γ ψ (1) rkq 1 γ ψ (q det Q 1 )ε(q 1 ) γ ψ(q)γ ψ (det Q 1 ) γ ψ (1)γ ψ (q det Q 1 ) =γ ψ (1) rkq 1 1 γ ψ (det Q 1 )ε(q 1 )γ ψ (q) =γ ψ (Q 1 )γ ψ (q) = γ ψ (Q).

46 4 10 Weil F 10.1 F ( ) W, : W W F w, w = w, w, w, w W ; W w [w w, w ] W (W,, ) F Y W, Y Y 0 Lagrange (Lagrangian subspace) W Lagrange Ω(W ) Y, Y W Y y [y y, y ] Y (W,, ) F (i) (Witt ) Y W {e 1,..., e r } {e 1,..., e r} W e i, e j = δ i,j, e i, e j = 0, (1 i, j r) (ii) Y, Y W Y Ω(W ) Y Ω(W ) Y = Y W = Y Y (W,, ) (polarization) Y Ω(W ) Y := {w W w, y = 0, y Y } = Y..

47 10.. Leray 43 ( 10.. ) F n 0n 1 n 1 n 0 n (y 1, y 1 ), (y, y ) = y 1 t y y 1 t y, y i, y i F n (W,, ) Sp(W ) := {g End F (W ) w.g, w.g = w, w, w, w W } W Y Ω(W ) P Y := Stab(Y, Sp(W )) Sp(W ) Siegel U Y := {g P Y g Y = id Y } Y Y = {0} ( Y ) Y Ω(W ) P Y Levi M Y := Stab(Y, P Y ) 1 n (W,, ) = (F n, Ä ä 0 n 1 n 0 n ) Y = {(0,..., 0; x1,..., x n ) W }, Y := {(x 1,..., x n, 0,..., 0) W } P Y = M Y U Y ( ) a M Y = m Y (a) := t a 1 a GL(n, F ), ( ) U Y = u 1n b Y (b) := 1 b = t b M n (F ) n 10. Leray Lagrange (Y 1, Y, Y 3 ) Ω(W ) 3 Y 1 Y Y 3 Q Y1,Y,Y 3 Q Y1,Y,Y 3 (y 1, y, y 3 ) := y 1, y + y, y 3 + y 3, y 1 (Y 1 Y Y 3, Q Y1,Y,Y 3 ) Witt(F ) (Y 1, Y, Y 3 ) Leray L(Y 1, Y, Y 3 ) [Ler74] (i) σ S 3 L(Y σ(1), Y σ(), Y σ(3) ) = sgn (σ)l(y 1, Y, Y 3 ). (ii) (Y 1, Y, Y 3 ), (Y 1, Y, Y 3) Ω(W ) 3 Y i Y j = Y i Y j = {0}, (1 i < j 3) g.(y 1, Y, Y 3 ) = (Y 1, Y, Y 3) g Sp(W ) L(Y 1, Y, Y 3 ) = L(Y 1, Y, Y 3)

48 44 10 Weil. (i) (ii) L(Y 1, Y, Y 3 ) = L(Y 1, Y, Y 3) (Y 1, Y ).g = (Y 1, Y ) g Sp(W ) (Y 1, Y, Y 3) (Y 1, Y, Y 3).g (Y 1, Y ) = (Y 1, Y ) Q Y1,Y,Y 3, Q Y1,Y,Y 3 a i Aut F (Y i ), (i = 1, ) h : Y 3 Y 3 (y 1, y, y 3 ) Y 1 Y Y 3 y 1, y + y, y 3 + y 3, y 1 = y 1.a 1, y.a + y.a, y 3.h + y 3.h, y 1.a 1 y 3 = 0 y 1, y = y 1.a 1, y.a m Y (a 1 ) := a 1 a Sp(W ) Siegel Levi M Y := Stab (Y 1, P Y ) y 1, y 0 y 3, y i = y 3.h, y i.a i, (i = 1, ), y 3, w = y 3.h, w.m Y (a 1 ) = y 3.hm Y (a 1 ) 1, w, y 3 Y 3, w W y 3.h = y 3.m Y (a 1 ) (Y 1, Y, Y 3 ).m Y (a 1 ) = (Y 1, Y, Y 3) 10.3 Weil F (W,, ) W F (w 1 ; z 1 )(w ; z ) := Ä w 1 + w ; z 1 + z + w 1, w ä, wi W, z i F H(W ) W Heisenberg H(W ) F W = Y Y Y F H(W ) F ψ ψ : Y F (y; z) ψ(z) C H(W ) L φ : H(W ) C H(Y ) := (i) (ii) φ((y; z)h) = ψ(z)φ(h), y Y, z F, h H(W ) Y φ(y ; 0) dy < H(W ) ρ ψ H(W ) ψ-schrödinger H(Y ) L (Y ) ρ ψ Å ρ ψ (y, y; z)φ(x ) = ψ z + x + y, y ã φ(x + y ), y, x Y, y Y, z F 10.5 ( Schur ). G G (π, V ) End G (π) C

49 10.3. Weil (Stone-von-Neumann ). (ρ ψ, L (Y )) F ψ H(W ) ( ) Sp(W ) H(W ) (w; z).g := (w.g; z), w W, z F, g Sp(W ) J (W ) := H(W ) Sp(W ) W Jacobi g Sp(W ) g.ρ ψ : h ρ ψ (h.g) F ψ H(W ) 10.6 H(W ) ω g : g.ρ ψ ρ ψ C 1 ω g1 ω g, ω g1 g g 1 g.ρ ψ = g 1.(g.ρ ψ ) ρ ψ H(Y ) U(H(Y )) Sp(W ) g (ω g mod C 1 ) U(H(Y ))/C 1 g ω g Sp(W ) H(Y ) (projective representation) Y 1, Y Ω(W ) A Y,Y 1 : Y 1 /Y 1 Y y + Y 1 Y y, (Y /Y 1 Y ) g Sp(W ) ρ ψ (g) : H(Y ) H(Y ) Ä ρψ (g)φ ä (h) := A Y.g,Y 1/ F φ((y; 0)h.g) dȳ Y.g/Y.g Y Y dy Y.g g Y.g Y du Y/Y.g Y dȳ = dy/du Y.g/Y.g Y d(ȳ.g) = d(y.g)/du (Y.g/Y.g Y ) d(ȳ.g) ψ d(ȳ.g) dȳ A Y.g,Y A Y.g,Y F := da Y.g,Y (ȳ) d(ȳ.g) ρ ψ (g) du H(W ) ρ ψ (g) : g.ρ ψ ρ ψ g 1, g Sp(W ) ρ ψ (g 1 ) ρ ψ (g ) = γ ψ (L(Y, Y.g 1, Y.g 1 ))ρ ψ (g 1 g ). Weil 13. Y Y.g = Y.g Y.g 1 g = Y Y.g 1 g = {0} 10.7 ρ ψ (g 1 g ) = γ ψ (L(Y, Y.g, Y.g 1 g ))ρ ψ (g 1 ) ρ ψ (g )

50 46 10 Weil 10.7 c ψ (g 1, g ) := γ ψ (L(Y, Y.g 1, Y.g 1 )) Sp(W ) C 1 ( 1 8 ) : c ψ (g, g 3 )c ψ (g 1 g, g 3 ) 1 = c ψ (g 1, g g 3 ) 1 c ψ (g 1, g ), g 1, g, g 3 Sp(W ). Sp(W ) C 1 Mp ψ (W ) = Sp(W ) C 1 : (g 1, z 1 )(g, z ) := (g 1 g, z 1 z c ψ (g 1, g )), (g i Sp(W ), z i C 1 ) W 10.7 ω ψ : Mp ψ (W ) (g, z) zρ ψ (g) U(H(Y )) Mp(W ) Mp ψ (W ) Weil (oscillator representation) 10.8 (ω ψ ). W = Y Y Witt {e 1,..., e n; e 1,..., e n } H(Y ) = L (Y ) = L (F n ) ω ψ ω ψ (m Y (a), z)φ(y ) =z det a 1/ F φ(y.a), (a GL(n, F )) (10.1) Å y ω ψ (u Y (b), z)φ(y b t y ã ) =zψ φ(y ), (b = t b M n (F )) (10.) Å ( ) ã 0 n 1 n ω ψ, z φ(y ) = φ(y)ψ( y t y) dy (10.3) 1 n 0 n F n

51 47 11 SL(, F ) F 11.1 O(E) SL(, F ) Weil F γe := (E, γn E/F ), γ F mod N E/F (E ) ( 6.1 ) E F F F Aut F (E) Z/Z σ N E/F : E z zσ(z) F Tr E/F : E z z + σ(z) F (z, z ) γe := γtr E/F (zσ(z )), z, z E γe γ SO(E) = {z E N E/F (z) = 1} O(E) = SO(E) σ E = E α α F ω E/F : F x (x, α) F {±1} ω E/F : F /N E/F (E ) {±1} Langlands λ λ(e/f, ψ) := γ ψ (N E/F ) = γ ψ (1)/γ ψ (α) (W = F,, = ( )) Sp(W ) = SL(, F ) (W := E F W,, := (, ) γe, ) F 4 ι γe,w = ι W ι γe : O(E) SL(, F ) (g, g ) g g Sp(W) W = Y Y, Y := {(0, y) x F }, Y := {(y, 0) y F } W = Y Y Y := γe F Y, Y := γe F Y γ ψ (L(Y, Y.g 1, Y.g 1 )), (g 1, g Sp(W)) 1 C 1 Mp(W) Sp(W) 1 (10.3 )

52 48 11 SL(, F ) ( ) a b 11.1 ([Kud94] 3.1). g = SL(, F ) c d β γe (g λ(e/f, ψ)ω E/F (γc) c 0 ) := ω E/F (d) c = 0 ι γe,w : O(E) SL(, F ) (g, g ) (ι γe,w (g, g ), β γe (g )) Mp(W) Mp(W) Weil (ω ψ, L (Y ) = L (E)) O(E) SL(, F ) Weil ω γe,w = ω W ω γe : O(E) SL(, F ) ι γe,w Mp(W) ω ψ U(L (Y )) (ω γe,w, L (E)) ω W (g)φ(z) =φ(g 1.z), g O(E) (11.1) Å ( ) ã a 0 ω γe φ(z) =ω 0 a 1 E/F (a) a F φ(az), a F (11.) Å ( ) ã Ç å 1 b ω γe φ(z) =ψ γ NE/F (z)b φ(z), b F (11.3) 0 1 Å ( ) ã Ç å 0 1 ω γe φ(z) =λ(e/f, ψ γ ) φ(z )ψ γ E zσ(z ) dz. (11.4) 1 0 E 11. O(E) SL(, F ) θ O(E) = SO(E) σ η : E /F C η u : SO(E) z/σ(z) η(z) C (Hilbert 90 g SO(E) g = z/σ(z), (z E ) ) Mackey

53 11.. O(E) SL(, F ) θ (i) η : E /F C η 1 τ η := ind O(E) SO(E) η u O(E) τ η τ η η = η ±1 (ii) η = 1 F E η = ω K/E, (K := E F E ) ind O(E) SO(E) η u τ K ± τ K(σ) ± = ±1 τ K ± = τ ± K K E K K F E τ K + = 1 O(E), τk = sgn O(E) (iii) O(E) (i), (ii) L (E) ω γe,w S(E) O(E) τ S(E) S(E, τ) SL(, F ) θ ψ γ(τ, W ) E/F SO(E) η : E /F C p η : S(E) φ(z) φ(g.z)η u (g) dg S(E) SO(E) η : E /F C (i) 11.3 S(E, τ η ) p η : S(E) S(E, η u ) := S(E, τ K) + S(E, τk) S(E, 1 O(E) ) η 1 η = ω K/E 1 η = 1 O(E) SL(, F ) (ii) η 1 θ ψ γ(τ η, W ) := (ω γe, S(E, η u )) SL(, F ) a (iii) η = ω K/E 1 θ ψ γ(τ ± K, W ) := (ω γe, S(E, τ ± K)) SL(, F ) b (iv) η = 1 θ ψ γ(1, W ) := (ω γe, S(E, 1 O(E) )) SL(, F ) θ ψ γ(sgn O(E), W ) = 0 a SL(, F ) b SL(, F ). (i) (ii) (iv) (i) θ [Kud86], [MVW87, 3 ] SO(E) S(E) = η im p η ω W (g)p η (φ) = η u (g)p η (φ),

54 50 11 SL(, F ) (g SO(E)) η 1 im p η = S(E, τ η ) η = ω K/E 1 ω W (σ)p η (φ)(z) = φ(g.σ(z))η u (g) dg = φ((gσ(z)/z).z)η u (g) dg SO(E) SO(E) = φ(g.z)η u (g z/σ(z)) dg = ω K/E (z)p η (φ)(z) SO(E) im p η S(N K/E (K )) = S(E, τ + K), im p η S(E N K/E (K )) = S(E, τ K) η im p η 0 N K/E (K) = E im p η = S(E, 1 u ) E = F γe = (H, Q H ) O(E) = Ä a 0 0 a 1 ä a F ( ) η : F F / F C η u : F x η(x, 1) C F = R C (1) (ω γe,w, S(E)) Fourier F X : S(E) φ F X φ(x, x) := F ( ) x Å xy φ ψ γ ã dy S(W ) y (x, y) ψ γ (xy/) Å ( ) ã a 0 ω W ϕ(w) = a 1 0 a 1 F ϕ(a 1.w), a F (11.5) ) ã ϕ(w) = ω W Å ( ω γe,w W Å w ϕ(w )ψ γ ã, w dw (11.6) ω γe (g )ϕ(w) =ϕ(w.g ), g SL(, F ). (11.7) () SO(E) SL(, F ) W SO(E) SL(, F ) {0} 0 W (a) {0} S({0}) = Cδ 0 (Dirac ) 1 F (b) W (0, 1) ϕ S(W ) p η (ϕ)(g ) := F ω W 1 SL(,F ) Å ( ) ã t 0 ϕ((0, 1).g )η 0 t 1 u (t) 1 dt = ϕ((0, t).g )η u (t) t F dt F

55 11.. O(E) SL(, F ) θ 51 S(W ) I(η u ) := φ : SL(, F ) C (i) φ( Ä ä a b 0 a g) = 1 ηu (a) a F φ(g) (ii) φ SL(, F ) SL(, F ) η u (i) η u 1 θ ψ (τ η, W ) I(η u ) SL(, F ) (iii) η u = ω E /F 1 I(η u ) θ ψ (τ ± E, W ) (iv) η u = 1 θ ψ (1 O(E), W ) I(1) θ ψ (sgn O(E), W ) = (11.5), (11.6), (11.7)

56 5 1 SL(, A) 1.1 SL(, A) SL(, A) F A (6. ) F S A(S) := v S F v O v A = S A(S) = lim S A(S) F 3.4 () A : A (x v ) v v x v v R + A 1 F A A = R + R F q (T ) A A = q Z Z 1.1. F A 1 (Artin ) A 1 /F SL() ( ) a b SL(, A) := c d a, b, c, d A ad bc = 1 SL() Borel B = T U T := {( a a 1 ) SL() }, U := {( ) } 1 b SL() 1 K = v K v SO(, R) v K v := SU(, R) v SL(, O v ) v

57 1.1. SL(, A) 53 ( Langlands) SL(, A) = B(A)K = U(A)T (A) 1 AK (1.1) ( 1 ) ( ) ( a T (A) 1 := a A1, A := a a 1 ) a 1 a R + R + F = v F v 1. (Mikowski ). U(A) = U(F )Ω U, T (A) 1 = T (F )Ω T Ω U U(A), Ω T T (A) 1 ( 6.4 (ii), 1.1) Ω := Ω U Ω T B(A) c > 0 SL(, A) = SL(, F )ΩA B (c)k A B (c) := { Ä a 0 0 a 1 ä A a > c } 1.3. SL(, F )\SL(, A) SL(, A). SL(, A) ΩA B (c)k f(g) dg = f(utak) du dt a da a dk SL(,A) K A T (A) 1 U(A) 1.4. SL(, A) (R, L(SL())): L(SL()) := φ : SL(, A) C [R(g)φ](x) := φ(xg), (i) φ(γg) = φ(g), γ SL(, F ) (ii) SL(,F )\SL(,A) φ(g) dg < + g SL(, A), φ L(SL()) SL(, A) ( ) (π, V ) R R (R 0, L 0 (SL())): Å L 0 (SL()) := φ L(SL()) ( ) ã 1 x φ g dx = 0, g SL(, A) F \A 0 1 (π, V ) ( )

58 54 1 SL(, A) 1. O(E A ) E F ( ) E/F Galois σ (E, N E/F ) O(E) A O(E A ) Å O(E A ) = lim O(E v ) ã K(E v ) S v S v / S E v /F v K(E v ) := O(E v ) E v F v F v O(E v ) = F v σ, (σ(x) = 1/x) K(E v ) := {x F v x v = 1} σ O(E A ) SL() φ : O(E A ) C (i) φ(γg) = φ(g), γ O(E) L(O(E)) := (ii) O(E)\O(E A ) φ(g) dg < + O(E A ) 1.5. η : A E/E A C η v = 1 Σ 11.3 τ η (Σ) := v Σ τ K v v / Σ η v=1 τ + K v v;η v 1 τ ηv ηv = 1 η v = ω Kv/Ev Hilbert L(O(E)) τ η (Σ) η:a E /E A C mod σ Σ; Σ. ( ) SO(E) ( Langlands ) L(SO(E)) 11.3 η:a E /E A C η u ind O(E A) SO(E A ) η u Σ τ η (Σ) σ O(E) O(E A ) Σ

59 γ F F γe = (E, γn E/F ) A/F ψ = v ψ v v O(E v ) SL(, F v ) Weil (ω γev,w v, S(E v )) 1.6. F v E v /F v γ O v, ψ v O v ϖv 1 O v O Ev 1 S(E OE v v) ω Wv (K(E v )) ω γev (K v ). O Ev K(E v ) ω Wv (K(E v )) ω γev (SL(, O v )) OEv ψ γ E-Fourier Å ω γe,w := ω γev,w v, S(E A ) := lim S(E v ) ã 1 OE S v v v S v / S O(E A ) SL(, A) ( ) Weil Φ S(E A ) θ Φ (g, g ) := ξ E ω γe,w (g, g )Φ(ξ), g O(E A ), g SL(, A) ω γe,w (g, g )Φ 1.5 L(O(E)) τ L(O(E)) ( ) L(τ), SO(E A ) A(τ) f L(τ) Φ S(E A ) Θ Φ (f, g ) := O(E)\O(E A ) f(g)θ Φ (g, g ) dg 1.7. τ O(E A ) (i) f L(τ), Φ S(E A ) Θ Φ (f) L 0(SL()). (ii) τ = τ η (Σ) v Σ η v 1 Θ(τ η (Σ), W ) := cl.span {Θ Φ (f) Φ S(E A ), f L(τ)} SL(, A) Θ(τ η (Σ), W ) = 0.

60

61 57 [CF86] [Kud86] J. W. S. Cassels and A. Fröhlich, editors. Algebraic number theory, London, Academic Press Inc. [Harcourt Brace Jovanovich Publishers]. Reprint of the 1967 original. S. S. Kudla. On the local theta correspondence. Invent. Math., Vol. 83, pp. 9 55, [Kud94] S.S. Kudla. Splitting metaplectic covers of dual reductive pairs. Israel J. Math., Vol. 87, pp , [Ler74] Jean Leray. Complèment à la théorie d Arnold de líndice de Maslov. In Proc. Sympos. Pure Math., Vol. XXVI, pp Amer. Math. Soc., Providence, R.I., [MVW87] C. Mœglin, M.-F. Vignéras, and J.-L. Waldspurger. Correspondences de Howe sur un corps p-adique. Springer Verlag, Lecture Notes in Math [Wei64] A. Weil. Sur certains groupes d opérateurs unitaires. Acta Math., Vol. 111, pp , [Wei95] André Weil. Basic number theory. Springer-Verlag, Berlin, Reprint of the second (1973) edition.