A bound of the number of reduced Arakelov divisors of a number field (joint work with Ryusuke Yoshimitsu) Takao Watanabe Department of Mathematics Osa

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1 A bound of the number of reduced Arakelov divisors of a number field (joint work with Ryusuke Yoshimitsu) Takao Watanabe Department of Mathematics Osaka University

2 , Schoof Algorithmic Number Theory, MSRI Publications Vol.44, 2006 Arakelov.

3 r F - F Arakelov - Arakelov - Schoof - Vol(Ω a ) - ω R n 1(K ρ H e ) slicing theorem

4 F : n, F O F F : F R := F Q R = F σ = R r 1 C r 2 σ V (n = r 1 + 2r 2 ) Id F F. a Id F,. 1 a, O F a. a a σ(a) < 1 ( σ V ), a = 0. a F R., 1 F R max-norm a. Red F.

5 a Red F, a 1 O F Nr(a 1 ) ( 2 π ) r2 F (=: F ) a 1 O F. F R C := {(x σ ) σ V F R : x σ (Nr(a) F ) 1/n }. a F R, C Minkowski, 0 (x σ ) C a 1 a, σ V, s.t. 1 x σ Nr(a 1 ) F. (qed) r F := Red F <.

6 F = Q( ), F = > 0. Z- q(x, y) = [a, b, c] := ax 2 + bxy + cy 2 (a, b, c Z, b 2 4ac = ). q 2 a < b <,. [a, b, c] [ a, b, c] F := Z- RF := F, F SL 2 (Z) F = SL 2 (Z) RF.

7 = b 2 4 a c a b b a

8 F = SL 2 (Z) RF q = [a, b, c] F, q SL 2 (Z)-., s(q) Z s(q) := sgn(c) b 2 c ( c ) sgn(c) +b ( c < ) 2 c ( x x ). ρ(q) := [c, b+2c s(q), c s(q) 2 b s(q)+a]., 0 m Z, ρ m (q) RF. ρ. q = [ , , ] F 5, ρ 5 (q) = [ 1, 1, 1] RF 5.

9 q = ρ(q), ρ RF. RF / < ρ >:= ρ-, F = (RF / < ρ >) F q = [a, b, c] a > 0., a q := Z + b + D Z 2a, q a q {[a, b, c] RF : a > 0} = RedF. r F = (RF ) 2

10 F = Q( 5), RF 5 = {[1, 1, 1], [ 1, 1, 1]} a [1,1, 1] = Z Z = O F Q( 5) R O F -

11 r F F (r 1 + r 2 1)!R F h F (log F ) r 1+r 2 1 r F r F πr F h F 2 r 2w F ρ n 1 2 r1 + πr 2 R F h F J r1,r 2 w F ρ n 1 R F := F h F := F w F := F 1 ( ) 2 r2 F := F π ρ := 1 2 log 4 3 J r1,r 2 := ( ) ( ) sin t r1 r2 J1 (2t) dt 0 t 2t (J 1 (t) Bessel )

12 F = Q( F ), 2R F h F log F r F 2R F h F log(4/3) 2 2RF h F log(4/3).. F LB r F UB

13 (r 1 + r 2 1)!R F h F (log F ) r 1+r 2 1 r F, r F πr F h F 2 r 2w F ρ n 1 2 r1 + πr 2 R F h F J r1,r 2 w F ρ n 1. Arakelov Schoof Vaaler cube slicing theorem.

14 - F Arakelov Arakelov,. D = λ σ σ + n p p σ V p V f, λ σ F σ, n p Z (n p = 0 a.a. p V f ). Arakelov Div F := σ V F σ σ Zp. p V f.. div : Id F F R Div F div (a, u) := u 1 σ σ V σ + p V f n p p a = p np Id F, u = (u σ ) F R.

15 - Arakelov D deg(d ) deg(d ) := σ V d σ log λ σ + p V f d p n p (d σ = [F σ : R], d p = (O F /p)). 0 Div 0 F := {D : deg(d ) = 0}. a F, pd (a) := div (a 1 O F, (σ(a)) σ V ), Div 0 F. Pic 0 F := Div 0 F /pd (F ) F Arakelov.

16 Pic 0 F, F Cl F,+. Div 0 F Id F : div (a, u) a,. 0 T 0 Pic 0 F Cl F,+ 0 T 0 := FR,1 0 /O F F R,1 0 n 1, FR,1 0 := {(u σ) F R,1 : Nr FR (u) = 1} F R,1 := {(u σ ) F R : u σ > 0 if F σ = R}. F R,1 F R 1.

17 0 T 0 Pic 0 F Cl F,+ 0, Pic 0 F Lie. T 0, π 0 ( Pic 0 F ) = Cl F,+., Pic 0 F.. Vol( Pic 0 F ) = 2r 1(2π) r 2 nrf h F. w F

18 - Schoof a = p n p Red F, div (a, Nr(a) 1 n) = σ V Nr(a) 1 nσ + p V f n p p Arakelov Arakelov., Ω a Div 0 F Ω a := div (a, Nr(a) 1 nu) : u = (u σ ) F 0 R,1 log u σ < ρ ( σ V ). [Schoof, 2006] Ω a, a Red F, disjoint, a Red F Ω a, Div 0 F Pic 0 F Pic 0 F.

19 Ω a Pic 0 F a Red F a Red F Vol(Ω a ) Vol( Pic 0 F ).., Vol(Ω a ) a. a Red F r F Vol(Ω a ) Vol( Pic 0 F ).

20 - Vol(Ω a ) F R R n, ω R n R n Lebesgue. Pic 0 F ω R n. Vol(Ω a ) = 2 r 2 n r 1 + 4r 2 ω R n 1(K ρ H e ), e := 1 r1 + 4r 2 (1,, 1, 2, 0, 2, 0,, 2, 0) R n ( ), H e :=< e > R n. K ρ := I r 1 ρ D r 2 ρ R n, n I ρ := ( ρ, ρ), D ρ := {(y, z) y 2 +z 2 < ρ 2 }. ρ = 1 2 log 3 4.

21 Vol(Ω a ) = 2 r 2 n r 1 + 4r 2 ω R n 1(K ρ H e ) ω R n 1(K ρ H e ), K ρ H e

22 Fourier. t R f(t) f(t) := ω R n 1(K ρ (H e + te))

23 χ Kρ K ρ, f Fourier f(r) = = f(t)e rt 1 dt R n χ K ρ (u)e r(u,e) = 2 r 1+r 2 π r 2ρ n r 1 i=1 1 du sin(ρar) ρar r 2 i=1 J 1 (2ρar) 2ρar. a := 1/ r 1 + 4r 2. Fourier ω R n 1(K ρ H e ) = f(0) = 1 2π f(r)dr = 2r 1+r 2 π r 2ρ n 1 r 1 + 4r 2 J r1,r π 2. J r1,r 2 = 0 ( ) ( ) sin t r1 r2 J1 (2t) dt. t 2t

24 r F Vol( Pic 0 F ) r2 2r1π r1 + 4r 2 R F h F = Vol(Ω a ) w F ω R n 1(K ρ H e ) πr F h F = 2 r 2w F ρ n 1 J r1,r 2 - ω R n 1(K ρ H e ) J r1,r 2. Vaaler cube slicing theorem J r1,r J 1,1 = 3 3+2π 24

25 [Vaaler, 1979] n n = n 1 + n n r, B ni R n i 1, n Q n := B n1 B n2 B nr R n.., n m A, rank(a) = m < n. 1 ω R m(q n Im(A)) det(a t A). ( He A := n (n 1) ( (2ρ) R := 1 ) E r1 0 0 ( πρ) 1 E 2r2 ) 1 ω R n 1(K ρ H e ) det( t A t RRA)

26 A, 2 r 1 1 π r 2ρ n 1 r1 + 4r 2 r 1 + πr 2 ω R n 1(K ρ H e ). r F 2 r 1 + πr 2 R F h F w F ρ n 1 Vaaler Siegel. Bombieri and Gubler, Heights in Diophantine Geometry, Cambridge, 2006 Appendix C3

27 slicing theorem n = p + 2q, R n K p,q = I p D q I = [ 1 2, 1 ], D = {z C : z 1} 2. R n e = (u 1,, u p, w 1,, w q ), (w i C), H e f(t) = ω R n 1(K p,q (H e + te)), Fourier f(r) = (2π) q p i=1 sin(u i r/2) u i r/2. f(t) = (2π)q π p i=1 cos(rt) 0 sin(u i r/2) u i r/2 q j=1 q j=1 J 1 ( w j r) w j r J 1 ( w j r) dr w j r

28 ω R n 1(K p,q H e ) = (2π)q π 0 p i=1 sin(u i r/2) u i r/2 q j=1 J 1 ( w j r) dr w j r (star body),. [Koldobsky, 1998] Ω R n, φ Ω (x) := min{r 0 : x rω}, (x R n ) Ω Minkowski., e R n ω R n 1(Ω H e ) =. 1 (n 1)π (φ n+1 Ω ) (e) φ n+1 Ω R n, Fourier.

29 ω R n 1(Ω H e ) =,. 1 (n 1)π (φ n+1 Ω ) (e) [Minkowski ] Ω 1, Ω 2 R n. e R n ω R n 1(Ω 1 H e ) = ω R n 1(Ω 2 H e ), Ω 1 = Ω 2. Koldobsky, Fourier Analysis in Convex Geometry, AMS, Koldobsky and Yaskin, The Interface between Convex Geometry and Harmonic Analysis, AMS, 2008.

30 I n = K n,0, ω R n 1(I n H e ) = 2 π n 0 i=1.. sin(u i r) dr u i r [Ball, 1986] e 1 ω R n 1(I n H e ) 2. e = (1, 0,, 0), e = (1/ 2, 1/ 2, 0,, 0). Hörder. [Ball, 1986] 2 s R sin(πx) s 2 dx πx s. s = 2.

31 , Marichal - Mossinghoff ω R n 1(I n H e ). [M-M, 2006] a = (a 1,, a n ), Π a := a 1 a n 0 ω R n 1(I n H a ) = a a + a n 2 n 2 (n 1)!Π a. ϵ V n Π ϵ (a, ϵ) n 1 + V n := {ϵ = (ϵ 1,, ϵ n ) : ϵ i {±1}} Vn := {ϵ V n : ϵ n = 1} (a, ϵ) + := max{0, (a, ϵ)}. [M-M] J.-L. Marichal and M. J. Mossinghoff, Slices, slabs and sections of the unit hypercube, arxiv MG/

32 [M-M]. a R n 0 < θ R, S a,θ := {x R n : (a, x) θ 2 } R n (slab ) a = ( 1, 2, 2), θ = 2/3 S a,θ

33 ω R n(i n S a,θ ). [Pólya, 1912] a = (a 1,, a n ) Π a = a 1 a n 0 = 2 π. ω R n(i n S a,θ ) 0 sin(θr) r n i=1 sin(a i r) dr a i r f(t) = ω R n 1(K p,q (H e + te)) θ/2 t θ/2. [M-M] [M-M, 2006] a, â = (a 1,, a n, θ) R n+1 ω R n(i n S a,θ ) 1 = n 1 n!π a ϵ V n+1 Π ϵ (â, ϵ) n +.

34 n 2 sin(θr) sin(a i r) dr π 0 r i=1 a i r 1 = n 1 Π ϵ (â, ϵ) n + n!π a ϵ V n+1 p 1 = 3, p 2 = 5, p 8 = 23,. a = ( 1 p 1,, 1 p 8 ), θ = = (â, ϵ) + = max{0, 1+ ϵ 1 p ϵ n p 8 } = 0 ( ϵ), m 8 0 sin r r m i=1 sin(r/p i ) r/p i dr = π 2

35 D q = K 0,q (2q = n), ω R n 1(D q H e ) = (2π)q π 0 q j=1 J 1 ( w j r) dr w j r. (Ball ),., D q R n = C q,. e C q Hermite H e,c. Oleszkiewicz - Pelczyński, Polydisc slicing in C n, Studia Math. 142, 2000.

36 [O-P, 2000] e = (w 1,, w q ) C q. ω R 2q 2(D q H e,c ) = πq 1 2,. q 0 r j=1 π q 1 ω R 2q 2(D q H e,c ) 2π q 1 J 1 ( w j r) dr w j r.. [O-P, 2000] 2 s R r 2J 1 (r) s dr 4 0 r s. s = 2,.

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi) 0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()