2., D, Thm log Hodge 2.1 Hodge Hodge Kähler Definition 2.1. Z H H C := H C F = {F p } p (H, F ) w, Hodge {h p,q } p,q Hodge (1) dim C F p = Σ r

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1 log Hodge 1 Introduction 0 f : E. t 0 f 1 (t), γ t, δ t 1 ω t, H 1 H 1 ; t γ t ω t / δ t ω t (γ t δ t = 1). t 0,., Γ, ϕ : Γ\H 1.,, Γ\H 1 =., Γ\H 1 = 0, ϕ., C C, [ ]. f 1 (t),. log, log, ϕ,. [KU], (Γ\H 1 ) {0}, log Hodge..,, ϕ,,. D log Hodge. D,. 1 (cf. [AMRT], [H]): Step1. D, F. D, F, W (F ). ( 3 Siegel ) Step2. Z- Γ(F ) Γ(F )\D F W (F ) ( Γ(F ) C /Γ(F )). Lie (Γ(F ) R ), [ ] Γ(F )\D. D,. [KU], F, Lie Lie (Γ(F ) R ), Γ(F )\D (cf. 2.3)., log Hodge ( D ), D,. D K3, Calabi-Yau t-hayama@cr.math.sci.osaka-u.ac.jp 1, Aut(D) Γ, Step F, Γ\D.. 1

2 2., D, Thm log Hodge 2.1 Hodge Hodge Kähler Definition 2.1. Z H H C := H C F = {F p } p (H, F ) w, Hodge {h p,q } p,q Hodge (1) dim C F p = Σ r p h r,w r for all p. (2) H C = p F p F w p. Hodge Kähler X H = H w (X, Z), F p = q<p Hw q (X, Ω q ) (H, F ) w Hodge H w (X, C) (1 w dim X) Hodge-Riemann Definition 2.2. w, Hodge {h p,q } p,q Hodge (H, F ) H Q := H Q ψ(, ) ψ (H, F ) (H, F, ψ) Hodge (1) w ψ w ψ (2) ψ(f p, F q ) = 0 for p + q > w. (3) ( 1) p q ψ(v, v) > 0 for all v F p F q, p + q = w. ( ψ H C ) w, Hodge {h p,q } p,q Hodge (H 0, F 0, ψ 0 ) Hodge D D := {F H C (H 0, F, ψ 0 ) w, Hodge {h p,q } p,q Hodge } ({w, {h p,q } p,q, H 0, ψ 0 } ) Hodge D D Ď D open Ď := {F H C dim C F p = p <p h p,w p, ψ 0 (F p, F q ) = 0 for p + q > w}. D(resp. Ď) G R := Aut(H 0,R, q 0 ) = {g AutH 0,R q 0 (gx, gy) = q 0 (x, y)}(resp. G C ) π : X S, Hodge φ : S Γ\D (Γ ). Example 2.3. w = 1, {h 1,0 = h 0,1 = g, 0 otherwise}, H 0 rank 2g Z- H 0 H 0 ( ) ψ 0 (u, v) = t 0 I g u v (u, v H 0 ) I g 0

3 , D = { {( ) } } F 0 = H 0,C F 1 τ = span C F 2 = {0} τ H g I g = H g (g Siegel ). {( ) } ) F = (F 0 = H 0,C F 1 τ = span C F 2 = {0} I ( ) g τ g = 1 I g { ( ) τ Ď = 1 τ C } ( ) 1 = P 1 (C) log Hodge Hodge,. Griffiths : D, Hodge. Schmid {0} = {0} φ : Γ\D Γ T Γ = Z T φ H 1 φ : H 1 D τ(w) := exp ( wn) φ(w) (N = log T/2π 1) τ τ : Ď τ(0) Ď H 1 Ď ; w exp (wn)τ(0) Im w 0 φ ( ) exp (CN)τ(0) 2 {w, {h p,q } p,q, H 0, ψ 0 } Definition 2.4. g R = Lie(G R ) (G R = Aut(H 0,R, ψ 0 )) 2 σ (1) N σ End(H 0,R ). (2) N, N σ NN = N N. Definition 2.5. σ = j R 0N j, Z = exp (σ C )F Ď (F Ď) Z σ- F Z (1) NF p F p 1 for all p Z, N σ. (2) exp ( j 1yj N j )F D for y j 0. 2

4 4 Σ D Σ := {(σ, Z) σ Σ, Z is a σ-nilpotent orbit} G Z neat 3 Γ Γ\D Σ log Hodge ([KU] B) ( ) 0 1 Example 2.6 (Exm.2.3 ). g = 1 N =, σ = R 0 N 0 0 { ( ( )) } ( ( )) z C D σ = {0}, z H 1 σ, Γ\D Σ Γ\D Σ Definition 2.7. g R Σ G Z Γ Γ Σ σ Σ (1) γσγ 1 Σ for all γ Γ, (2) σ = i R 0 log (γ i ) for some γ i Γ exp (σ). Σ Γ σ Σ Γ(σ) := Γ exp (σ) fs Γ(σ) toric σ := Spec(C[Γ(σ) ]) an, torus σ := Spec(C[Γ(σ) gp ]) an toric σ Γ Σ σ = i R 0 log (γ i ) σ C := σ C torus σ = C Γ(σ) gp e : σ C torus σ = C Γ(σ) gp ; z log (γ) exp (2π 1z) γ toric σ = τ σ torus σ 1 τ toric σ e(z) 1 τ (τ σ) z σ C log (Γ(σ) gp ) + τ C Ě σ := toric σ Ď, E σ := {(e(z) 1 τ, F ) Ěσ exp (τ C ) exp (z)f τ } E σ well-defined toric σ log Ěσ toric σ Ěσ log Ěσ log E σ [KU] strong topology E σ E σ Ěσ E σ log E σ π σ :E σ Γ(σ) gp \D σ Γ\D Σ, (e(z) 1 τ, F ) (τ, exp (τ C ) exp (z)f ) mod Γ(σ) gp (τ, exp (τ C ) exp (z)f ) mod Γ. 3 Γ γ C Γ neat

5 σ Σ, π σ Γ\D Σ σ Σ π σ Γ\D Σ U Γ\D Σ O Γ\DΣ (U) (resp. M Γ\DΣ (U)) := {map f : U C f π σ O Eσ (π 1 (U)) (resp. M Eσ (π 1 (U))) ( σ Σ)} O Γ\DΣ, M Γ\DΣ Γ\D Σ log [KU] A E σ log Γ neat Γ\D Σ log E σ Γ(σ) gp \D σ log σ C -torsor 4 { ( ) 1 s Example 2.8 (Exm.2.6 ). Γ = SL(2, Z) Γ(σ) = 0 1 s N } torus σ = C 0, toric σ = C, Ě σ = C { ( Ď, E σ = exp (2π ( )) } { ( ( )) } z 2 1z 1 ), C 1 Ď z z 1 + z 2 H 1 0, {0} 1 Ď z C. 5 3 [AMRT]p.238 Thm, D, σ exp (σ C ) D exp (σ C )\(exp (σ C ) D).. Theorem 3.1. D, σ. E σ π σ trivial bundle toric σ Y (toric σ Y ) E σ = Γ(σ) gp \D σ σ C Y torus σ Y (Γ(σ) gp \D) ) = = exp (σ C ) D Γ(σ) gp \(exp (σ C ) D) Γ(σ) gp \D ρ σ trivial bundle exp (σ C )\(exp (σ C ) D), Y ρ σ, (Γ(σ) gp \D) ) toric σ Y Γ(σ) gp \D. ( ) Example 3.2 (Exm.2.8 ). Y = {p} where p Ď 1. 0 [AMRT] A. Ash, D. Mumford, M. Rapoport and Y. S. Tai, Smooth compactification of locally symmetric varieties, Math. Sci. Press, Brookline, [H], Siegel log Hodge, 4,

6 6 [KU] K. Kato and S. Usui, Classifying space of degenerating polarized Hodge structures, to appear in Ann. Math. Studies, Princeton University Press.

7 Non-gap sequence C g, P C. f j (P ) P j, C, f j (P ) j. 1. j P gap f j (P ) h 0 (C, jp ) = h 0 (C, (j 1)P )), j P non-gap value f j (P ) h 0 (C, jp ) = h 0 (C, (j 1)P )) + 1., non-gap sequence (= non-gap value ). non-gap sequence.,. 2. g 1 gap value g, 1 2g 1., C S. P 2 a Σ a blowing up,. S P C P 2 Σ a C 1 ϕ blowing up Q P n Q ϕ 1 (Q) = {P 1,... P n }, C P 1,... P n non-gap sequence. 1,., S.

8 1.1 S,. 1 S, 2 D 2..., D d 1 S. S,., 2 R 2,. S S D d σ(d d ) D 1 σ(d 1 ) : D 1 σ(d 2 ) D 2 σ(d 3 ) 1 D 3, S K S = d i=1 D i D S. 1.1 S D 2..., D d 1, D = d 1 i=2 q id i (q i Z). q 1 = q d = 0. D D,. 3. σ(d i ) Z (x i, y i ). D = { (z, w) R 2 x i w y i z q i (1 i d) } D. 4. h 0 (S, D) = { (z, w) Z 2 x i w y i z q i (1 i d) } = { D Z }., K S = d i=1 D i h 0 (S, K S + D) = { (z, w) Z 2 x i w y i z q i 1 (1 i d) } = { D Z }. S g C, g = { C Z }.

9 2 6. S, C = d 1 i=2 p id i (p i Z) S, i = min{i 2 C.D i 1}, i = max{i d C.D i 1}. C (i) C.D 1 = 1, (ii) x i y i y i x i 0. P = D 1 C, j. h 0 (C, jp ) = { (z, w) Z 2 x 1 w y 1 z j, 0 x i w y i z p i 1, i + 1 i i s.t. x i w y i z p i }. 6, j h 0 (C, jp )., C P non-gap sequence., C : x 5 +x 4 y 2 +xy 4 +y 2 = 0, (0 : 0 : 1) blowing up ϕ : C C. C, 2 C = D 2 +2D 3 +5D 4 +8D 5 +4D 6 +8D 7 +5D 8 +2D 9, ϕ 1 ((0 : 0 : 1)) = D 1 C. D 1 C = P, h 0 (C, jp ) 3 x 1 X y 1 Y j Z. non-gap sequence, j k 1 k, Z k, {5, 8, 10, 11, 12, 13,...}. x 1 X y 1 Y = j O 3 6 3,. C g, 6 Z j. j, H 1 (S, jd 1 ) H 1 (C, jd 1 C ) H 2 (S, jd 1 C) H 2 (S, jd 1 ). 5, h 2 (S, jd 1 ) = h 0 (S, K S jd 1 ) = 0., h 0 (C, jp ) = h 0 (C, jd 1 C ) = h 1 (C, jd 1 C ) + deg jd 1 C + 1 g h 2 (S, jd 1 C) + j + 1 g = h 0 (S, K S + C jd 1 ) + j + 1 g. (1)

10 , C C 4, 5 g C Z., 5 h 0 (S, K S + C jd 1 ) = { (z, w) Z 2 x 1 w y 1 z j 1, x i w y i z p i 1 (2 i d) }., = { 5 a Z } j + 1 = { (z, w) Z 2 j x 1 w y 1 z 0, 0 x i w y i z p i 1 }, b Z. C O 4 x 1 X y 1 Y = j a b O 5 x 1 X y 1 Y = j c O (1), h 0 (C, jp ) 6 c Z., c 6, h 0 (C, jp ) Z j. 6, l j 1 : x 1 X y 1 Y = j, l j 1 C Z., S I = jd 1 + d i=2 r id i. (I) h 1 (S, I) = 0, (II) h 0 (S, K S + C I + D 1 ) = h 0 (S, K S + C I) + 1. H 1 (S, I) H 1 (C, I C ) H 2 (S, I C) H 2 (S, I) (I) h 1 (S, I) = 0, 5 h 2 (S, I) = h 0 (S, K S I) = 0, h 1 (C, I C ) = h 2 (S, I C) = h 0 (S, K S + C I). h 0( C, jp + d i=2 r id i C ) = h 0 (C, I C ) = h 0 (S, K S +C I)+j+ d i=2 r ic.d i +1 g., H 1 (S, I D 1 ) H 1 (C, I D 1 C ) H 2 (S, I D 1 C) H 2 (S, I D 1 )

11 , h 0( C, (j 1)P + d i=2 r id i C ) h 0 (S, K S +C I+D 1 )+(j 1)+ d i=2 r ic.d i +1 g. (II) h 0( C, jp + d i=2 r id i C ) h 0 ( C, (j 1)P + d i=2 r id i C ),, h 0( C, jp + d i=2 r id i C ) = h 0 ( C, (j 1)P + d i=2 r id i C ). h 0 (C, jp ) = h 0 (C, (j 1)P ).. j = 1. h 0 (C, (j 1)P ) = Z j 1. l j 1 C Z. l j 1 6 c Z. h 0 (C, jp ) = h 0 (C, (j 1)P ). h 0 (C, jp ) = Z j 1., x 1 X y 1 Y = (j 1) x 1 X y 1 Y = j c Z, Z j = Z j 1. h 0 (C, jp ) = Z j., l j 1 C Z. l j 1 c Z. x 1 X y 1 Y = (j 1) x 1 X y 1 Y = j, c Z 1. Z j = Z j h 0 (C, jp ) h 0 (C, (j 1)P ) + 1 = Z j = Z j. h 0 (C, jp ) Z j, h 0 (C, jp ) = Z j. 3 6 non-gap sequence,. P 2 (X 0 : X 1 : X 2 ) x = X 0 X 2, y = X 1 X 2, C : x p + y q + x a y b = 0. p > q > a + b, a, b 1. C Q 1 = (0 : 0 : 1) Q 2 = (0 : 1 : 0). blowing up ϕ : C C C, Q 1 2, Q 2 1, ϕ 1 (Q 1 ) = {P 1, P 2 }, ϕ 1 (Q 2 ) = P 3. 7 S P C P 2 C 1 P 2 ϕ Q 1 Q 2 P 3 7

12 , C g = 1 (pq aq bp 1), 2 6.,. (i) b p a h 0 (C, jp 1 ) = { (z, w) Z 2 (p a b)w bz j, 0 (q p)w + qz 2g, (ii) a q b h 0 (C, jp 2 ) = { (z, w) Z 2 (a + b q)w (q b)z j, (ii) p q h 0 (C, jp 3 ) = { (z, w) Z 2 (q p)w + qz j, (a + b q)w (q b)z 0 }. 0 (p a b)w bz 2g, (q p)w + qz 0 }. 0 (a + b q)w (q b)z 2g, (p a b)w bz 0 }. [1] S. J. Kim, On the existence of Weierstrass gap sequences on trigonal curves, J. Pure Appl. Algebra, 63 (1990), [2] J. Komeda, On the existence of Weierstrass gap sequences on curves of genus 8, J. Pure Appl. Algebra, 97 (1994), [3] J. Komeda, Existence of the primitive Weierstrass gap sequence on curves of genus 9, Bol. Soc. Brasil. Mat., 30 (1999), [4] T. Oda, Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 15 (1988). [5] Perez del Pozo, Angel L. Gap sequences on Klein surfaces, J. Pure Appl. Algebra, 195 (2005),

13 ON QUOTIENT CURVES OF THE FERMAT CURVE OF DEGREE TWELVE ATTAINING THE SERRE BOUND Goppa,.. Serre., p, q p. F q g C, Hasse Weil #C(F q ) q g q Serre, [9] Serre #C(F q ) q g 2 q ( ). L- (1 + 2 q t + qt 2 ) g. Lauter [5], Serre g (q 2 q)/( 2 q + 2 q 2 2q). Hasse Weil, Stichtenoth Hermit. Garcia, Stichtenoth, Hermit., Serre,. 3, Serre F 2 3 Klein x 3 y + y 3 + x = 0, Lachaud [4] p 1 mod 4, a > 0, a 3 mod 4, b 2 a a, b p = a 2 + b 2, x 4 + y 4 = (2/p) F p Serre. (/) Legendre. Auer, Top [1], 4 Serre. 4,. 1 ([6]). F p y 12 = x 4 (1 x) Serre, p 1 mod 12, 2 p 1 mod 3, n p = p n 2. 11, [8], F 2 11 y 23 = x 4 (1 x)..

14 , 1, 12 Fermat, Jacobian,. Serre,., [7]. 1. Fermat 12 Fermat, Fermat Fermat F 4 : x 4 + y 4 = 1 F p Serre, 2 p 6 mod 8, n p = p n 2.,. p 977, 2617, 3041, 5641, 5689, 6257, 8297, 9041, 9817, 13241,. 3. 2, 4 Fermat F 4 Jacobian. J F4 Γ 1 Γ 2 2,, Γ 1 : y 2 = 1 + x 4, Γ 2 : y 2 = 1 x 4. k 2, 3, E 1 : y 2 = x 3 + 1, E 2 : y 2 = x 3 1, E 3 : y 2 = x 3 + 4, E 4 : y 2 = x 3 4, E 5 : y 2 = x k Fermat F 6 : x 6 + y 6 = 1 F p Serre, p 1 mod 12, 2 p 1 mod 3, n p = p n 2. p 15733, 24133, 26029, 27997, 38917, 43789, 51637, 60133, 72469, 93133,. C k 6 Fermat F 6 Jacobian. J F6 E 4 1 E 2 2 E 2 3 E 4 E 5. 6 Fermat. 6. D 1 : y 6 = x 2 (1 x 2 ) F p Serre p 1 mod 12, 2 p 1 mod 3, n p = p n k D 1 Jacobian,. J D1 E 2 3 E 4 E 5

15 Fermat, y 12 = x n (1 x). C 1 : y 12 = x 4 (1 x) F p Serre 1, k Jacobian, J C1 E 3 E 2 5 E 6., E 6 : y 2 = x C 2 : y 12 = x 5 (1 x) F p Serre, p 1 mod 12, 2 p 6 mod 8, n p = p n 2. p, 3061, 3517, 5077, 6277, 8317, 14197, 24061, 24169, 26713, 30661,. 9. C Jacobian. J C2 Γ 3 Γ 2 4,, Γ 3 : y 2 = x 3 + x, Γ 4 : y 2 = x 3 3x. 10. C 3 : y 12 = x 6 (1 x) F p Serre, p 1 mod 12, p p 0 mod 8, n p = p n 2. p, 2437, 4261, 15661, 21061, 24169, 26713, 28597, 32797, 34261, 42061,. 11. C Jacobian. J C3 Γ 5 Γ 2 6,, Γ 5 : y 2 = x 3 x, Γ 6 : y 2 = x 3 + 3x. 3.,. 12. [2] C 1 : y 12 = x 4 (x+16) F p Serre, p 1 mod 12, p 2 mod 3, 2 p 1 mod 3, n p = p 2 + 3n [2] k C 1 Jacobian,. J C 1 E 3 1 E 2

16 14. F 4 : x 4 + y 4 = 1 F p Serre, 2 p 6 mod 8, n p = p 2 + 4n , F 4 Jacobian. J F 4 Γ 3 7,, Γ 7 : y 2 = x D 1 : y 6 = x 2 (4 4x 2 ) F p Serre, p 1 mod 12, p 2 mod 3, 2 p 1 mod 3, n p = p 2 + 3n k D 1 : y 6 = x 2 (4 4x 2 ) Jacobian,. J D 1 E 3 1 E 2. 1, 2, 6, 12, 14, 16,,. 4.. Buniakowski. a, b, c, a, gcd(a, b, c) = 1, a + b c, b 2 4ac, am 2 + bm + c. 18. Buniakowski, 1, 2, 4, References [1] R. Auer, J. Top, Some genus 3 curves with many points, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369, Springer-Verlag 2002, [2] B. Brock, M. Q. Kawakita, M. Zieve, Private communication, [3] K. Ireland, M. Rosen, A classical introduction to modern number theory, Second edition, Graduate Texts in Mathematics 84, Springer-Verlag [4] K. Lachaud, Sommes d Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, C. R. Acad. Sci. Paris, Sér. I Math. 305(16)(1987), [5] K. Lauter, Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, with appendix by J.-P. Serre, J. Algebraic Geom. 10(1)(2001), [6] M. Q. Kawakita, A quotient curve of the Fermat curve of degree twelve attaining the Serre bound, J. Algebra Appl. 4(2005), no. 2, [7] M. Q. Kawakita, On quotient curves of the Fermat curve of degree twelve attaining the Serre bound, submitted. [8] S. Miura, Algebraic geometric codes on certain plane curves (in Japanese), IEICE Trans. Fundamental J75-A(11)(1992),

17 [9] J. -P. Serre, Sur le nombre des points rationnels d une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér. I Math. 296(1983), no. 9, address: kawakita@belle.shiga-med.ac.jp

18 2 1, E 8, [5]., II. 2,,,., 2 [9],., X/C, f : X P 1 2.., f, (O)., f X. NS(X), (O) T. ( 1), NS(X), T. [10] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0., :. {f λ : X λ P 1 } λ Λ NS(X λ ). λ 0 Λ, λ Λ T λ0 T λ, MWL(f λ0 ), {f λ } λ Λ. {f λ : X λ P 1 } λ Λ, λ Λ f λ, NS(X λ ). f. Supported by The 21st Century COE Program named Towards a new basic science: depth and synthesis 1

19 1. i = 1, 2 D i f : X P 1 1 ( ). NS(X), D i T i. NS(X)/T 1 NS(X)/T 2,, (T1 ) NS(X) (T 2 ) NS(X). ( 1) ( 1) (1) (5)., 4 2K 2 X, : (1) K 2 X = 4 : (2) K 2 X = 3 : 3 1 (3) K 2 X = 2 f ( 1) : 1 1 (4) K 2 X = 2 f ( 1) : (5) K 2 X = 1 : 1 1 1,,. 2. ( k), k. 2 (1) [8], [4]. (4), 2007.,,,. 3. MWG(f). f, 1, f (O). f 2 (1), 1 :

20 f 2 (2), 1 4 4, 2 : f 2 (4), 6 : (4-i) (4-ii) (4-iii) (4-iv) (4-v) (4-vi) f,., 2, 3, 4 (4-i) 3

21 (4-ii)., 2 (3) (5), MWG(f) f. : f (3) : f (5) : , [2],.., f. f Φ KX +F : X Σ 0 := P 1 P 1 P(f O X (K X/P 1)) B. Σ 0 Γ, 0. B, (k 1), k, k., Σ 0, B. 4. k 1. Γ B : (0). (I k ) B Γ, B Γ., (2k 1) (2k). (II k ) B Γ, B., (2k) (2k + 1). (III k ) B Γ, B Γ Γ (4k 2) (4k 1). 4

22 (IV k ) B Γ, B Γ (4k) (4k + 1). (V) B Γ, B Γ Γ. B Γ, Γ.. f f Φ KX +F 4, f (I k ), (II k ), (III k ), (IV k ), (V). f (0)., Γ, B B, (0). 5. ( ) #{ }. : ( ) B (#{I k } + #{III k }) + #{V} + K 2 X + 6 Γ (1) K X = #{V} + k k ((2k 1)(#{I k } + #{III k }) + 2k(#{II k } + #{IV k })). (2) (2) H f := K 2 X + 4 ( )., ( ) 1 : (V) (I k ) (III k ) (II k ) (IV k ) 3 2k 2k + 1 2k + 1 2k + 2 ρ(x) (10 K 2 X ),, rkmwl(f) = ρ(x) 2 (f 1 (t) 1) t P H f. (3) σ : Σ 0 Σ 0, B 2H f :, 4 (0). (I k ), (2k 2). (II k ) (2k 1), (III k ) (4k 3), (IV k ) (4k 1). (V) B Γ, B Γ Γ. B,., σ ( 2), B B. Σ B, 5

23 . X. Σ 0, H f ( 1), 2. H f ( 1) f : X P 1.., p 2 (X) = 0. h 0 ( K Σ 0 + K X 2 (σ 0) Γ) = 0 (4) P 1 P 1 B, 4 (1) (4), (2) 2.,,.. 2, K 2 X = 4 MWG(f), X P 2.,. K 2 X = 4 f, f. 6. K 2 X = 4 MWG(f). ν 0 : X P 2, f F : (1) 4l 2e 1 13 e i i=2 8 (2) 6l 2 e i i=1 (3) 7l 3e 1 2 (4) 9l 3 12 e i i=9 11 e i i=2 8 e i 2e 9 2e 10 e 11 i=1 10 (5) 13l 5e 1 4 i=2 e i (K 2 X = 4 ), (K 2 X = 3 ), (K 2 X = 2 ), (K 2 X = 2 ), (K 2 X = 1 ). l = ν0 O P2(1) e i ν 0 ( 1) X., ν : X P 2 deg ν 0 (F ) deg ν(f ).. (X, F ) #- (cf. [1], [3]), f (cf. [7]). K 2 X = 1, (5) [2]., 6 ν 0. (2) e 9, e 10, e 11, e 12 (4) e 11, ν 0, (cf. [6, 1]). (4) e 9 e 10 6

24 ., i < j ν 0 (e j ) ν 0 (e i ). (1) e 13, (2) e 12, (4) e 11, f ( 1). (4) e 11 ( 1), (3) (5) f ( 1), 6. 2, 6, 2., 2 ν 0 l e i NS(X), T. 2.1 K X 2 = 1 rkmwl(f) = 6 K 2 X = 1 f : X P 1. (3). K 2 X = 1 2,. P 1 P 1 t, x. f B00 = 32(t 1) 2 t 3 +(x 1)(7x x 4 134x x 2 109x 5)t 5 +4(x 1)(73x 5 125x x 3 102x x + 11)t 4 4(x 1)(160x 5 256x x 3 115x x + 7)t 3 +8x(x 1)(67x 4 110x x 2 30x + 3)t 2 16x 2 (x 1) 2 (13x 2 10x + 3)t +32x 3 (x 1) 3., t Γ 0, x f B00 B 0. B 0 (0, 1) (3, 10), (0, 0)., B 0. Γ 0 B 0 4 (I 2 )., (3t 2)x 2 + ( 2t + 2)x + t Γ, B 0 (0, 1), (0, 0). (4)., B 0 Γ 0, B 0 Γ 5 12,., P 1 P 1 (B 0 + Γ 0 ), 2., 01 B 0 (1, 1), 01, ( 2)., T., 6 e 2, e 3,..., e 10, Φ F +4KX : X P 1 ( 1),. f B00 7

25 t = 0 t = 1 x = 1 x = , e 8 e 9, e 9 e 10, 2K X + e 10, F, (O) T. T. rel. min. e 13 e 12 e 11 e 10 ψ X X 2 : 1 can. res. π Σ σ 0 f Φ KX +F f Σ 0 σ Φ Γ Σ 0 σ e 8. e 13 e 9 e 8 ν 7 X 7 ν 0 P 2 Σ 1 P 1 Σ Γ Γ Φ KX7 P 2 6. f 2 (3), 6 e 11 e 10 Φ KX +e 10 +e 11 Φ F +2KX : X P 1 P 1,. e 10 e 11, K X + e 11, F, (O) T 8

26 2.2 K X 2 = 3 X e 12, e 11, e 10, e 9 X 8. K X + F = K X + e 12 + e 11 + e 10 + e 9, e 12, e 11, e 10, e 9 Φ KX8. K X8 ɛ : S P 1, K X + e 11 + e 12, F, (O) T. : 7. f : X P 1 (V) K 2 X = 3 2. MWG(f) MWG(ɛ) MWL(f) MWL(ɛ).., (2), K 2 X = 3 f (I 1 ) (III 1 ), (V)., (I 1 ) (III 1 ) f,. 7, MWG(f),. P 1 P 1 t, x. (α, β, γ, δ) (C 4 \{γ = 0})\{α 2 4β = βδ + γ = 0} x 5 + γt 3 + βt 2 x + αtx(x δt) 2 δtx 2 (3x 2 3δtx + δ 2 t 2 ) (5) B 0. Γ 0 B 0 (0, 0). Γ 0 B 0., γ 0 B 0 {x = 0}, (0, 0). (, ) B 0 (3, 5). (0, 0), (0, 0) P 1 P 1 B 0 Γ 0. β α 2 /4 γ α 2 δ/4,. B 0. B 0., β = α 2 /4 γ = α 2 δ/4, (5) (x δt)(2x 2 2δtx + αt) 2 / β = 0 δ = 0 β 0,, α 2 4β 0,, α 2 4β = 0., β = 0 9

27 , ( 3), ( 2), α 0, α = 0. f 2 (4), (2). [1] Hartshorne, R.: Curves with high self-intersection on algebraic surfaces. Publ. Math. IHES 36, (1969) [2] Horikawa, E.: On algebraic surfaces with pencils of curves of genus 2, in Complex Analysis and Algebraic Geometry, a Volume Dedicated to Kodaira K., pp (Cambridge, 1977) [3] Iitaka, S.: On irreducible plane curves. Saitama Math. J. 1, (1983) [4] Nguyen, K. V.: On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces. J. Math. Sci. (New York), 102, (2000) [5] Oguiso, K., Shioda, T.: The Mordell-Weil lattice of a rational elliptic surface. Comment. Math. Univ. St. Paul. 40, (1991) [6] Kitagawa, S., Konno, K.: Fibred rational surfaces with extremal Mordell-Weil lattices. Math. Z. 251, (2005) [7] Kitagawa, S.: Maximal Mordell-Weil lattices of fibred surfaces with p g = q = 0. Rend. Sem. Mat. Univ. Padova, 117, (2007) [8] Saito, M.-H., Sakakibara, K.: On Mordell-Weil lattices of higher genus fibrations on rational surfaces. J. Math. Kyoto Univ. 34, (1994) [9] Shioda, T.: Mordell-Weil lattices of type E 8 and deformation of singularities. Lecture Notes in Math., 1468, Springer, Berlin, (1991) [10] Shioda, T.: Mordell-Weil lattices for higher genus fibration over a curve. New trends in algebraic geometry (Warwick, 1996), , London Math. Soc. Lecture Note Ser., 264, Cambridge Univ. Press, Cambridge,

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47 Mordell-Weil lattices of maximal rank on elliptic K3 surfaces X = (X, Φ) : proj. elliptic K3 surface over C i.e., X : proj. ( smooth ) K3 surface over C ( i.e., K X = O X and dim H 1 (X, O X ) = 0. ) Φ:X P 1 : elliptic fibration with a global section O MW (Φ) := {sections of Φ} : Mordell-Weil group ( free part, ) lattice Mordell-Weil lattice ρ := rank NS(X) = rank S X 0 rank MW (Φ) ρ 2 r := rank MW (Φ), ρ : fix min{det MW (Φ)} Theorem 1. r = 1 min{det MW (Φ)} = 11/420. ρ = 19 or 20 (X, Φ) r = ρ 2 ( rank MW (Φ) ) 1 Lattice r MW L T X det 18 M A 2 (10) 300 Shioda primitive sublattice of M 16 M 1 H(5) = 625 Shioda primitive sublattice of M primitive sublattice of N primitive sublattice of L primitive sublattice of L primitive sublattice of L primitive sublattice of L Λ 10? 768 primitive sublattice of M 1 9 Λ 9? 512 primitive sublattice of M 1 8 E 8 (2) 256 primitive sublattice of M 1 By a lattice (L, b) we mean a finitely generated free Z-module L, endowed with a nondegenerate symmetric bilinear form b : L L Z. An even lattice is a lattice whose associated quadratic form x 2 := b(x, x) takes even values. Simply, we say a lattice L instead of (L, b) when there is no fear of confusion. If {e 1, e 2,..., e n } is a Z-basis for a lattice L, we define a non-degenerate symmetric matrix I = (b(e j, e k )) 1 j,k n. Then the determinant and the signature of a lattice L are defined as det L := det I > 0 and sgn L := sgn I. A lattice L is unimodular if det L = 1. We define the positive- (negative-) definiteness of a lattice L by that of the matrix I. Frequently, a lattice L is expressed by the matrix I. 1

48 The hyperbolic lattice H is defined by H = ( If (L, b) is a lattice, then (L, mb) (m is a non-zero integer) is also a lattice. We denote this lattice by L(m). Put L := L( 1). A sublattice T of L is a submodule of L such that (T, b T T ) is a lattice. By S T, we denote the orthogonal direct sum of lattices S and T. The orthogonal complement T of T is defined as Λ : unimodular lattice ). T = {x L b(x, y) = 0 for all y T }. L M in Λ def L = M and M = L ( in Λ ). A sublattice T of L is said to be primitive if the quotient L/T is torsion-free. Lemma 2. T L : primitive and L Λ : primitive = T Λ : primitive Let L be a positive-definite even lattice. We call e L a root if e 2 = 2. Put (L) := {e L e 2 = 2}. Then the sublattice of L spanned by (L) is called the root type of L and is denoted by L root. The lattice E 8 is the positive-definite even unimodular lattice corresponding to the Cartan matrix of type E 8. E 8 Dynkin diagram 2 elliptic K3 surfaces X : K3 surface over C (H 2 (X, Z), the cup product) : an even unimodular lattice of signature (3, 19) H 2 (X, Z) = H 3 ( E 8 ) 2 Λ (3,19) := H 2 (X, Z) K3 surface ( the Néron-Severi group the Picard group ) NS(X) S X := H 1,1 (X, R) H 2 (X, Z) T X := SX in H2 (X, Z) : transcendental lattice of X T X S X in Λ (3,19). 2

49 Theorem 3. (i) ( Shioda-Inose ) Let Q be a positive-definite even lattice of rank 2. Then there exists a singular K3 surface X with T X = Q. (ii) Suppose Q Λ (3,19) is a primitive sublattice of signature (2, 20 ρ). Then there exists an algebraic K3 surface X with T X = Q. (iii) Suppose Q Λ (3,19) is a primitive sublattice of signature (1, ρ 1). Then there exists an algebraic K3 surface X with S X = Q. Φ:X P 1 : elliptic fibration with a section O U Φ := O, F ( F : the fibre of Φ ) ( O 2 = 2, O F = 1, F 2 = 0 ) ( ) 0 1 U Φ O + F, F = = H. 1 0 H = U Φ S X : primitive. Theorem 4. (Shioda) MW (Φ) S X /T. T fibre O lattice U Φ T Φ : singular fibres T = U Φ H. sgn (U Φ in S X) = (0, ρ 2) W := (U Φ in S X ) ( S X = U Φ ( W ) ) W positive-definite rank = ρ 2. the Riemann-Roch theorem W root T Φ : singular fibres W root MW (Φ) = W (X, Φ) : elliptic K3 surface and rank MW (Φ) = ρ 2 T X S X in H 3 ( E 8 ) 2 U Φ ( W ) H T X ( W ) in H 2 ( E 8 ) 2. ( T X ) W in H 2 E8 2 W root Λ (18,2) := H 2 E 2 8 3

50 Lemma 5. (Kondo) If there exists an embedding of H into S X, then X has a Jacobian fibration Φ such that c 1 (F ) / c 1 (F ) = H in S X where F is the fibre of Φ. Lemma Lemma 6. L W in Λ (18,2) and W root positive-definite lattice W, L T X L ( S X H ( W ) ) K3 surface X MW (Φ) W X elliptic fibration Φ Lemma 7. W root positive-definite lattice and W Λ (18,2) : primitive S X H ( W ) K3 surface X MW (Φ) W X elliptic fibration Φ r : fix { min det MW (Φ) Φ : ρ = r + 2 K3 surface singular fibres elliptic fibration } { min det W W : root positive-definite lattice and W Λ (18,2) : primitive and rank W = r } lattice 3 Lemma 8. W : root positive-definite lattice and W Λ (18,2) : primitive W 0 W : primitive (X 0, Φ 0 ) : elliptic K3 surface st. MW (Φ 0 ) = W 0. W 0 W : primitive and W Λ (18,2) : primitive Lemma 2 W 0 Λ (18,2) : primitive. W : root positive-definite lattice W 0 : root positive-definite lattice Lemma 7 ok. q.e.d. Lemma 9. (X, Φ) : elliptic K3 surface and Φ : singular fibres W 0 MW (Φ) : primitive (X 0, Φ 0 ) : elliptic K3 surface st. MW (Φ 0 ) = W 0. 4

51 elliptic K3 surfaces Mordell-Weil lattices primitive sublattices ( ) 1. ( Shioda ) M : The Mordell-Weil lattice of y 2 = x 3 + t 5 1/t ( M 1 : M Mordell-Weil lattice ) 2. ( Usui ) L 12 : The Mordell-Weil lattice of y 2 = x 3 + t ( Nishiyama ) N 399 : (X, Φ) ( : elliptic ) K3 surface st. 2 1 T X = and rank MW (Φ) = Theorem 10. (Nikulin) Let there be given two pairs of nonnegative integers, (t (+), t ( ) ) and (l (+), l ( ) ). The following properties are equivalent: a) Every even lattice of signature (t (+), t ( ) ) admits a primitive embedding into some even unimodular lattice of signature (l (+), l ( ) ). b) l (+) l ( ) 0 (mod 8), t (+) l (+), t ( ) l ( ) and t (+) + t ( ) (l (+) + l ( ) )/2. Theorem 11. (Serre) Let L be even unimodular lattice. If L is neither positive- nor negative-definite, then it is determined by its signature up to isometries. In particular, L is isometric to H j, H k E8 k or H l ( E 8 ) l. l (+) = 18, l ( ) = 2, t (+) = r = rank W, t ( ) = 0 Λ (18,2) = H 2 E 2 8 : unimodular r = t (+) + t ( ) (l (+) + l ( ) )/2 = 10 Lemma 12. W : positive-definite lattice st. rank W 10 W Λ (18,2) : primitive r 10 5

52 3 1 C S B f : S B g g g [3] 2 [4] 120 [2] 2 (0) (V) f : S B S K 2 S [2] [5] 3 2 ϕ : W B B P 1 R ϕ 2g + 2 W L [R] = 2L W [R] R L W B π : X W X R X R x 1 R m 1 x 1 W ψ 1 : W 1 W E 1, R 1, L 1 E 1 := ψ 1 1 (x 1 ) R 1 := ψ 1R 2[ m 1 2 ]E 1 L 1 := ψ L [ m 1 2 ][E 1] ( x [x] x ) 1

53 [R 1 ] = 2L 1 L 1 W 1 R 1 X 1 x i R i 1 m 1 x i W i 1 ψ i : W i W i 1 E i, R i, L i R X r = Ŝ ϕ : W B Ŝ g ˆf = ϕ π ψ 1 ψ r ˆf : Ŝ B g X Ŝ ˆf ( 1) g g (W, R, L) ([5]) 3 (W, R, L) x R m x ϕ Γ σ + : W W W x Γ σ + Γ Γ ( 1) σ : W W y = σ (Γ ) E R L E := σ 1 + (x) R := σ +R 2[ m 2 ]E L := σ +L [ m 2 ][E] [R ] = 2L R := (σ ) R R y n R = σ R 2[ n 2 ]Γ R h R h R R m R h x n Rh y n = 2g 2 m m g + 2 n g Pic(W ) σ Pic(W ) Z[Γ ] 2Γ (L + [n/2]γ ) = Γ σ R = 0 2

54 L + [n/2]γ σ (Pic(W )) W L σ L = L + [n/2]γ [R ] = 2L σ L (W, R, L ) L W R X X X (W, R, L) (W, R, L ) g f : S B (W, R, L) (1) R h (g + 1) (2) (1) R 2 (W, R, L) k 1 k [ g+1 2 ] x i R i 1 2k + 1 x i+1 E i R i 2k + 2 (x i, x i+1 ) (2k + 1 2k + 1) 4.2. Ŝ ( 1) ˆR = R r ( 2) ( 2) (2k + 1 2k + 1) (g + 2 g + 2) Γ R 4.3. F f : S B F ϕ : W B Γ F i s i (F ) i 4 Γ i i+1 (i 1 i 1) (i + 1 i + 1) s i (F ) 3

55 i 3 Γ (i i) s i (F ) s i (f) := p B s i (f 1 p) s i (f) f i 4.4. Ŝ S ( 1) [ g 2 ] 2s g+2 (f) + s 2k+1 (f) k= F f : S B g+1 Ind h (F ) := 2 [ 2 ] g ( (2gk 2k 2 g 2 )s 2k+1(F ) + k=1 [ g+1 2 ] Ind h (F ) F (k 1)(g k)s 2k (F )) + s g+2 (F ) 4.6. K f = K S f K B f χ f = χ(θ S ) χ(θ F )χ(θ B ) k 2 f = (4 4 g )χ f + p B k=2 Ind h (f 1 p) 5 f : S B g (W, R, L) ϕ : W B Γ f : S B F Γ Ŝ ( 1) F Γ R f : S B Γ R F F x 1 R m x 1 R m x 2 x 2 m x 3 m (x 1, x 2,...x k ) 5.1. k = 1 x 1 simple m k 2 x 1 k-fold m 4

56 g (W, R, L) R (W, R, L ) F g 2 g 1 Γ W ( 1 ) R Γ Γ R g simple (g + 1) s g+2 (F ) = 0 s g+1 (F ) = 0or1 ( 2 ) R Γ Γ R g simple (g + 1) s g+2 (F ) = 0 s g+1 (F ) = 0 (I), (II) R Γ Γ R 0-fold simple (g + 1) l-fold (g + 1) (l 2) s g+2 (F ) = 0 s g+1 (F ) = [ l 2 ] (III), (IV) R Γ Γ R (l)-fold (g + 1) (l 2) s g+2 (F ) = 0 s g+1 (F ) = [ l 2 ] Γ R (g + 1) 5

57 5.3. (0 1 ) (0 2 ) g 2 (0 1 ) s g+1 (F ) = 1 (V) s g+1 (F ) = 0 (0 1 ) (0 2 ) (0) g 2 s 3 (F ) + s 4 (F ) s 4 (F ) = 0 Ind h (F ) s 3 (F ) 5.4. (I, II) (III, IV) A B (I, II) (III, IV) A B A 2 (I, II) ( 2) [ l 2 ], (III, IV) 2 2 ( 2) [ l 2] 2 6

58 g (W, R, L) R (W, R, L ) F g 2 g 1 Γ W ( 1 ) R Γ Γ R simple (g + 1) g s g+2 (F ) = 0 ( 2 ) R Γ Γ R simple (g + 1) g s g+2 (F ) = 0 (I) R Γ Γ R 0-fold (g + 1) l-fold (g + 1) (l 2) s g+2 (F ) = 0 s g+1 (F ) l (II) R Γ Γ R 0-fold (g + 1) l-fold (g + 1) (l 2) s g+2 (F ) = 0 s g+1 (F ) = l (III) R Γ Γ R (l)-fold (g +1) (l 2) s g+2 (F ) = 1 s g+1 (F ) = l 2 (IV) R Γ Γ R (l)-fold (g + 1) (l 2) 7

59 s g+2 (F ) = 0 s g+1 (F ) = l R g (I) (IV) A B 8

60 (I) A B 2 (II) A B (III) A (IV) A

61 ( 2) (I) (2l 1) (II) (l 1) (III) (2l 5) ((III) (l 3) ) (IV) l (III) Ind h (F ) = 5 3 s 3(F ) s 4(F ) s 5(F ) F s 3 (F ),s 4 (F ),s 5 (F ) s 5 (F ) (II) (III) (IV) l s 5 (F ) (0 1 ) (0 2 ) (I) s 3 (F ),s 4 (F ) (II) (III) (IV) s 3 (F ) 3 10

62 [1] E. Horikawa On Deformations of Quintic Surfaces Invent. Math. 31(1975) [2] E. Horikawa, On algebraic surfaces with pencils of curves of genus 2, Complex Analysis and Algebraic Geometry, a volume dedicated to K. Kodaira(1977), 79-90, Tokyo and Cambridge, Iwanami Shoten Publishers and Cambridge University Press [3] K. Kodaira On compact complex analytic surfaces II Annals of Math. 77(1963) [4] K. Ueno and Y. Namikawa The complete classification of fibres in pencils of curves of genus two Manuscripta. Math 9(1973) [5] G. Xiao π 1 for ellirtic and hyperelliptic surfaces Intern. J. Math. 2(1991)

63 The abc-theorem, Davenport s Inequality and Elliptic Surfaces Tetsuji Shioda January 18, 2008 Abstract We consider the application of the abc-theorem and Davenport s inequality to elliptic surfaces over the projective line P 1, with special attention to the case of equality in the abc-theorem. Some existence theorem and the finiteness results will be given for certain type of elliptic surfaces. 1 The abc-theorem and Davenport s inequality 1.1 Statements Let k = C be the field of complex numbers. For a polynomial f = f(t) k[t], let N 0 (f) denote the number of distinct zeroes of f(t); thus we always have N 0 (f) deg(f). Theorem 1.1 (abc) Let a, b, c be relatively prime polynomials such that a+ b + c = 0 and deg(abc) > 0. Then we have max(deg(a), deg(b), deg(c)) N 0 (abc) 1. (1.1) This theorem is known as the abc-theorem for k[t] following the analogous ABC-conjecture for the integer ring Z formulated in middle 1980 s. But the above geometric version is in fact proven earlier by Stothers [22] (in a more general situation of Riemann surfaces) and also by Mason [11]. For the sake of completeness, we recall in 1.2 the proof based on the Riemann-Hurwitz relation, which gives further information for the case of equality. 1

64 Theorem 1.2 (Davenport) Let f, g be relatively prime polynomials and let h := f 3 g 2. Then we have deg(f) 2(N 0 (h) 1), deg(g) 3(N 0 (h) 1). (1.2) Proof Applying the abc-theorem to f 3 g 2 h = 0, we have deg(f 3 ) N 0 (f 3 g 2 h) 1 N 0 (fgh) 1 deg(f) + deg(g) + N 0 (h) 1, which implies Similarly we have 3 deg(f) deg(f) + deg(g) + N 0 (h) 1. 2 deg(g) deg(f) + deg(g) + N 0 (h) 1, and these two inequalities imply the claim. q.e.d. Remark The original Davenport s inequality [5] (cf. [21]) says that, for any non-constant complex polynomials f(t) and g(t) such that h = f 3 g 2 0, we have deg(f) 2(deg(h) 1), deg(g) 3(deg(h) 1), (1.3) without assuming the condition (f, g) = Proof of the abc-theorem There are several proofs for this. Given complex polynomials a, b, c satisfying a + b + c = 0, (a, b) = 1, (a, c) = 1, (1.4) consider the rational function φ = a/c. The most elementary one is to look at the behavior of the logarithmic differential d log φ (cf. [10]). But the best proof perhaps is the one using the Riemann-Hurwitz relation viewing φ as the covering map of Riemann spheres, because it makes clear the case of equality in the abc-theorem (see below 1.3). So let us outline it here. Without loss of generality, we may assume that n = deg(a) = deg(b) deg(c). (1.5) 2

65 First factorize the complex polynomials a, b, c into distinct linear factors: a(t) = a r1 0 i=1(t α i ) e i, b(t) = b r2 0 j=1(t β j ) e j, c(t) = c r3 0 k=1 (t γ k) e k, By assumption, we have r1 i=1 e i = deg(a) = n j=1 e j = deg(b) = n k = deg(c) n. r2 r3 k=1 e (1.6) N 0 (abc) = r 1 + r 2 + r 3. (1.7) Then the degree n map w = φ(t) = a(t)/c(t) from P 1 t to P 1 w is ramified as follows: (i) over w = 0, at the r 1 points t = α i with (ramification) index e i ; (ii) over w = 1, at the r 2 points t = β j with index e j; (iii) over w =, at the r 3 points t = γ k with index e k and at t = with index n deg(c) in case deg(c) < n; and (iv) possibly over some other points w 0, 1,. Applying the Riemann-Hurwitz formula to this situation, we have, in case deg(c) < n, 2n 2 = i (e i 1) + j (e j 1) + k (e k 1) + (n deg(c) 1) + V (1.8) where V 0 is the contribution from ramification points of (iv) above. Rewriting this relation, we have n = (r 1 + r 2 + r 3 ) 1 V = N 0 (abc) 1 V N 0 (abc) 1. (1.9) In case deg(c) = n, we have similarly n = N 0 (abc) 2 V < N 0 (abc) 1. (1.10) Thus we have proven the abc-theorem. q.e.d. 1.3 Case of equality In the above proof, assume that equality n = N 0 (abc) 1 holds. This is the case if and only if we have deg(c) < n and V = 0; the latter condition says that the map φ : P 1 t P 1 w is unramified outside {0, 1, }. As is wellknown, such a map, called a Belyi map, has an amazing property that φ is a rational function of t with coefficients in Q (up to a change of the coordinate t) (see [2], [9, Ch.2]). Here Q denotes the field of algebraic numbers. Then the above argument implies: 3

66 Theorem 1.3 Assume (1.1) holds with equality, i.e. max(deg(a), deg(b), deg(c)) = N 0 (abc) 1. (1.11) Then, by replacing a, b, c C[t] (if necessary) by la, lb, lc for some l 0 k, we have a, b, c Q[t] up to a change of the coordinate t. The corresponding result for Theorem 1.2 is the following: Theorem 1.4 Suppose that we have, for some integer m, deg(f) = 2m, deg(g) = 3m, N 0 (h) = m + 1. (1.12) Then, by replacing if neccesary f, g by c 2 f, c 3 g for some c k, f, g belong to Q[t] provided that t is suitably chosen. A triple {f, g, h} such that f 3 g 2 = h is called a Davenport-Stothers triple (DS-triple in short) of order m if it satisfies the condition: deg(f) = 2m, deg(g) = 3m, deg(h) = m + 1. (1.13) By [21, Lemma 3.1], we have then (f, g) = 1 and N 0 (h) = h. Hence we have: Corollary 1.5 Any DS-triple {f, g, h} have algebraic integers as coefficients, provided that t is suitably chosen. Example 1.6 (see [21, 5]) Here is an example of DS-triple of order m = 1: f = t 2 1, g = t t, h = 3 4 t2 1. (1.14) For m 4, there is essentially a unique DS-triple of order m, and f, g, h Q[t] in such cases. For m = 5, there are four (essentially distinct) DS-triples of order m, of which the two are realized by f, g, h Q[t] but the other two are realized by f, g, h Q( 3)[t]. Example 1.7 Here is an example of non DS-triple satisfying (1.12). Start from the Legendre cubic x(x 1)(x t) and transform it into the Weierstrass cubic x 3 3f(t)x 2g(t). For this {f, g}, h = f 3 g 2 is a constant multiple of the discriminant (t(t 2)) 2. This gives an example of (1.12) for m = 1. This level 2 case can be extended to the case of elliptic modular curves (or surfaces [17]) of level n = 3, 4, 5, which satisfy (1.12) with h = h n 0. 4

67 2 Elliptic surfaces over P Elliptic surfaces S f,g and S h Given a triple {f, g, h} such that f 3 g 2 = h, let E f,g and E h be the elliptic curves over k(t), defined by and E f,g : y 2 = x 3 3f(t)x 2g(t) (2.1) E h : Y 2 = X 3 h(t). (2.2) Then P = (f, g) is an integral point of E h (in the sense that both coordinates belong to k[t]), while the discriminant of E f,g is equal to h up to a constant. We have proposed in [21] to call E f,g and E h as Shafarevich partner, since this interplay of E f,g and E h reflects the famous Shafarevich theorem ([16]) for the finiteness of elliptic curves over Q with good reduction outside a given finite set of primes. We denote by S f,g the elliptic surface over P 1 defined by (2.1), i.e. the Kodaira-Néron model of E f,g /k(t). Similarly, we denote by S h the elliptic surface defined by (2.2). Now let S be any elliptic surface over P 1 with a section. Then S is isomorphic to S f,g for some f, g k[t] which can be so chosen that if f is divisible by l 4 and g is divisible by l 6 for some l k[t], then l must be a constant in k. In the following, we exclude the case where both f, g are in k. The smallest integer n such that deg(f) 4n, deg(g) 6n (2.3) is called the arithmetic genus of S, and it is denoted by χ = χ(s). This is the main numerical invariant of an elliptic surface over P 1 with a section. For instance, we know from surface theory the following facts: χ(s) = 1 S is a rational elliptic surface (2.4) χ(s) = 2 S is a K3 elliptic surface. (2.5) The topological Euler number e(s) of S is equal to: e(s) = 12χ(S). (2.6) 5

68 2.2 The number of singular fibres Let S = S f,g and let Φ : S P 1 be the given elliptic fibration. We freely use the known facts on singular fibres which can be found in [8], [13] or [23]. Denote by Σ P 1 the set of points supporting the singular fibres of Φ, and let N = #Σ be the number of singular fibres of S. We assume that Σ, i.e. there is a singular fibre at t =. Then we have N = N 0 (h) + 1 (2.7) since the discriminant of E f,g is equal to h up to a constant. (Here N 0 (h) denotes as before the number of distinct zeroes of h.) To simplify the statement below, the condition (ss-1) ( semistable minus one ) will mean that the elliptic fibration Φ : S P 1 is semistable outside t =, i.e. all singular fibres at t are of Kodaira type I n for some n = 1, 2,.... Theorem 2.1 Let S = S f,g and N = #Σ. Assume Σ and the condition (ss-1). Then we have deg(f) 2(N 2), deg(g) 3(N 2). (2.8) Proof Suppose that (f, g) = 1. Then the assertion follows from Theorem 1.2 and (2.7). Hence it is enough to prove the following lemma: Lemma 2.2 The condition (ss-1) is equivalent to (f, g) = 1. Proof If we let d = GCD(f, g), then d divides h, i.e. the discriminant of E f,g. Assume that d 1, and suppose that t α is a factor of d. Then we have f(α) = g(α) = 0, and the equation (2.1) becomes y 2 = x 3 at t = α. Hence the singular fibre at t = α cannot be semistable. The converse is shown by reversing the argument. q.e.d. This proves Theorem 2.1. q.e.d. Corollary 2.3 Under the same assumption as in Theorem 2.1, we have { 2χ + 2 if N : even N 2χ + 1 if N : odd. (2.9) Proof This follows immediately from (2.8) and the definition of χ in (2.3). q.e.d. 6

69 Theorem 2.4 Suppose that Φ : S P 1 is an elliptic surface with a section and let N be the number of singular fibres. Assume the condition (ss-1). Then (i) we have N 2χ + 1. Moreover, if N = 2χ + 1 holds, then the singular fibre at is not semistable. (ii) If Φ is a semistable elliptic fibration, then we have N 2χ + 2. Proof (i) It follows from the above Corollary that N 2χ + 1. Moreover, if N = 2χ + 1, then we have deg(f) 4χ 2, deg(g) 6χ 3. In terms of the coodinates at t =, the Weierstrass equation (2.1) becomes a cuspidal curve: y 2 = x 3 at t =. Hence the fibre at is additive, i.e. it is not semistable. (ii) This is clear from (i). q.e.d. Thus we recover Beauville s results [1]: Corollary 2.5 For any semistable elliptic surface S over P 1 with a section, the number N of singular fibres is at least 4. Further, if N = 4, then S is a rational elliptic surface with 4 fibres of type I a, I b, I c, I d where [a, b, c, d] is one of the following six cases: [1, 1, 1, 9], [1, 1, 2, 8], [1, 1, 5, 5], [1, 2, 3, 6], [2, 2, 4, 4], [3, 3, 3, 3]. (2.10) Proof The first part is an immediate consequence of Theorem 2.4, since χ 1. If N=4, then we have χ = 1, hence by (2.4), S is a rational elliptic surface. Suppose the 4 fibres are of type I a, I b, I c, I d. Then, looking at its topological Euler number, we have a + b + c + d = 12. (2.11) On the other hand, the trivial lattice T (in the sense of Mordell-Weil lattices [18]) is isomorphic to the direct sum of root lattices ([4, Ch.4]) A a 1,..., A d 1, and its rank and determinant are given by rkt = (a 1) + + (d 1) = 8, det T = abcd. (2.12) Hence T is a sublattice of E 8 of finite index, say ν. Since E 8 is unimodular, we have det T = ν 2, which implies that abcd = ν 2. (2.13) 7

70 The integer solutions of (2.11) and (2.13) are easily determined, and they are those listed in (2.10). q.e.d. The above case of N = 4 is the first example of more general results related to the case of equality in Theorem 1.1 or 1.2 (see 1.3). Let us consider this subject in the next section. 3 Existence and finiteness theorems Theorem 3.1 For any positive integer χ, (i) there exist semistable elliptic surfaces S with a section over P 1 with χ(s) = χ which have the minimal number of singular fibres N = 2χ+2. (ii) The number of isomorphism classes of such elliptic surfaces S is finite. (iii) In each isomorphism class, there exists some S defined over Q. (iv) Any elliptic surface in (i) is extremal in the sense that the Mordell-Weil rank r = 0 and the Picard number is maximal: ρ = h 1,1. A variant is: Theorem 3.2 For any positive integer χ, (i) there exist elliptic surfaces S with χ(s) = χ satisfying the condition (ss-1) which have the minimal number of singular fibres N = 2χ + 1. (ii), (iii), (iv): the same assertion as in Theorem 3.1. Proof We prove both theorems together. Let S = S f,g. Then any S attaining the minimal number N = 2χ + 2 (or N = 2χ + 1) corresponds to the triple {f, g, h} satisfying (1.12) in Theorem 1.4. As for the existence (i), we have a more precise statement that there exists such an S among S f,g corresponding to the Davenport-Stother triples of order m {f, g, h} with m = 2χ (or m = 2χ 1) (see 1.3 and [9], [21], [22]). (ii) This is a consequence of the finiteness of Belyi maps of bounded degree. Namely, for S = S f,g, the absolute invariant J is equal to φ = a/c in the proof of abc-theorem ( 1.2), i.e. J = f 3 /h up to constants, and it is a Belyi map of degree 6m = 12χ (or 12χ 6). The number of such maps is obviouly finite, since the mode of ramification at the three points {0, 1, } has only a finite number of possibility once the degree is fixed. [In the case of DS-triples, this number is enumerated as a function of m in [22]. 8

71 In [9], this number is nicely represented as the number of certain graphs on a sphere, drawn in Grothendieck s way of dessins d enfants. For instance, for m = 5(χ = 3), the four DS-triples of order m = 5 (cf. Example 1.6 in 1.3) are represented by the four graphs in Fig of [9, p.128].] (iii) This is immediate from Theorem 1.4. (iv) By the standard Picard number formula and inequality, we have ρ = r rk T h 1,1. (3.1) The trivial lattice T (in case Theorem 3.1) is the direct sum of the root lattices A ni if S has singular fibres of type I ni (1 i N). Hence we have ρ = r i n i N = r χ (2χ + 2) = r + 10χ, since i n i is equal to the Euler number ([8]) which is 12χ by (2.6). On the other hand, the Hodge number h 1,1 is equal to 10χ, as is easily seen. Therefore we conclude that r = 0 and ρ = h 1,1. (A slight modification should be made in case Theorem 3.2 (i), but it is easy.) q.e.d. Example 3.2 The case χ = 1 in Theorem 3.1 is already mentioned in Corollary 2.5. In case χ = 2, by (2.5), S is a K3 surface. The semistable elliptic K3 surfaces with N = 6 singular fibres are classified by Miranda- Persson [12]. The first member in their list is the K3 surface with maximal singular fibre I 19, i.e. with total fibre type [1, 1, 1, 1, 1, 19]. We gave the defining equation for this surface in [20] by relating it to the DS-triple of order 4 discovered by Hall [6] in The defining equations (or the J-invariant as the Belyi map) for all surfaces in the list of [12] are recently treated by [3]. For χ 3, Theorem 3.1 and 3.2 are new, as far as we know. 4 Toward classification of elliptic surfaces with N singular fibres over P 1 Problem 4.1 Given a positive integer N, classify all elliptic surfaces with a section over P 1 with N singular fibres. 9

72 The answer is known only for N 4. For N = 1, there is none. For N = 2, there are four types of elliptic surfaces with two singular fibres: {II, II }, {III, III }, {IV, IV } or {I 0, I 0} in Kodaira s notation [8]. The J-invariant is constant for all cases. For the first nontrivial case N = 3, the classification has been carried out by Schmickler-Hirzebruch [15], and for the case N = 4, by Herfurtner [7]. Around the beginning of the millenium, we reconsidered this problem from the viewpoint of integral points and Mordell-Weil lattices (cf. [18], [19]). Our approach was reported at the conference at Tokyo Univ. 2000, with a new (purely algebraic) proof for N = 3 (unpublished). Let us indicate below our method which should work for any N in principle. (In carrying out the computation, a computer should be helpful, and indispensable for N > 4.) (I) For a given N, the most essential case is when J is nonconstant and all singular fibres are reduced except possibly one. This means that, except at one place t =, all singular fibres are either semistable (type I n for some n) or of type II, III, IV. Let us say that the condition (red-1) ( reduced minus one ) holds if this is the case. In particular, the previous condition (ss-1) implies (red-1). The following result generalizes Theorem 2.1: Theorem 4.2 Assume S = S f,g has N singular fibres. Assume Σ and the condition (red-1). Then we have deg(f) 2(N 2), deg(g) 3(N 2). (4.1) Proof (Outline) Let d = GCD(f 3, g 2 ) and set F = f 3 /d, G = g 2 /d, H = h/d. (4.2) Then apply the abc-theorem to F G = H, and the result will follow. q.e.d. As a consequence, we obtain (in the same way as Cor.2.3): { 2χ + 2 if N : even N 2χ + 1 if N : odd. (4.3) 10

73 This bounds the arithmetic genus χ(s) of S in terms of N, when S has N singular fibres and satisfies the condition (red-1). (II) Take any polynomials h k[t] such that N 0 (h) = N 1. Then the question is to determine f, g satifying f 3 g 2 = h and (4.1). This is equivalent to the question: to find all integral points P = (x, y) of the elliptic curve E h defined by (2.2) (integral in the sense x, y k[t]) such that the height P, P is at most N 2 or N 1. This is the hardest and at the same time most interesting part of the problem. Very precise result can be expected (cf. [18], [19], [21, 8]). (III) If any S with N singular fibres (J Const) is given and if it does not satisfy the condition (red-1), then one can pass from S to another surface S, by an elementary twisting (cf. [7]), such that S has N N singular fibres and satisfies (red-1). This is to replace a pair of non-deduced fibres to a pair of reduced fibres by the rule: I n I n (n = 0, 1,...), II IV, III III, IV II, (4.4) and to continue this until one has at most one non-reduced fibre. The choice of S is not unique from S, but it does not matter. Exercise For N = 3 or N = 4, try the above method and compare the results with [15] or [7]. Then try the open case N = 5 or 6. (This is of some interest from the viewpoint of integral points on elliptic K3 surfaces.) References [1] Beauville, A.: Les familles stables de courbes elliptiques sur P 1, C. R. Acad. Sci. Paris, 294 (I), (1982). [2] Belyi,G.V.: On Galois extensions of a maximal cyclotomic field, Math. USSR Izvestija 14, no.2, (1980). [3] Beukres, F., Montanus, H.: Fibrations of K3 surfaces and Belyi maps, Preprint, [4] Conway, J., Sloane, N.: Sphere Packings, Lattices and Groups, Springer- Verlag, 2nd ed.(1993); 3rd ed.(1999). 11

74 [5] Davenport, H.: On f 3 (t) g 2 (t), Norske Vid. Selsk. Forh. (Trondheim) 38, (1965). [6] Hall, M.: The Diophantine equation x 3 y 2 = k, in: Computers in Number Theory, Academic Press, (1971). [7] Herfurtner, S.: Elliptic surfaces with four singular fibres, Math. Ann. 291, (1991). [8] Kodaira, K.: On compact analytic surfaces II-III, Ann. of Math. 77, (1963); 78, 1 40(1963); Collected Works, III, , Iwanami and Princeton Univ. Press (1975). [9] Lando, S.K., Zvonkin, A.K.: Graphs on Surfaces and Their Applications, Encyclopedia of Math. Sci., Low-Dim. Topology II, Springer (2004). [10] Lang, S.: Old and new conjectured diophantine inequalities, Bull. AMS 23, (1990). [11] Mason, R. C.: Diophantine equations over function fields, London Math. Soc. Lect. Note Series 96 (1984). [12] Miranda, R., Persson, U.: Configurations of I n fibers on elliptic K3 surfaces, Math. Z. 201, (1989). [13] Néron, A.: Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math. IHES 21 (1964). [14] Oguiso, K., Shioda, T.: The Mordell Weil lattice of a rational elliptic surface, Comment. Math. Univ. St. Pauli 40, (1991). [15] Schmickler-Hirzebruch, U.: Elliptische Fläche über P 1 C mit drei Ausnahmefasern und die Hypergeometrische Differentialgleichung, Diplomarbeit, Univ. Bonn (1978). [16] Shafarevich, I.R. : Algebraic number fields, Proc. Int. Congr. Math., Stockholm 1962, (1963); Collected Math. Papers, , Springer (1989). [17] Shioda, T.: On elliptic modular surfaces, J. Math. Soc. Japan 24, (1972). 12

75 [18] : On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39, (1990). [19] : Integral points and Mordell-Weil lattices, in: A Panorama in Number Theory or The View from Baker s Garden, Cambridge Univ. Press, (2002). [20] : The elliptic K3 surfaces with a maximal singular fibre, C. R. Acad. Sci. Paris, Ser. I 336, (2003). [21] : Elliptic surfaces and Davenport-Stothers triples, Comment. Math. Univ. St. Pauli 54, (2005). [22] Stothers, W. W.: Polynomial identities and Hauptmoduln, Quart. J. Math. Oxford (2), 32, (1981). [23] Tate, J: Algorithm for determining the type of a singular fiber in an elliptic pencil, SLN 476, (1975). Department of Mathematics Rikkyo University Nishi-Ikebukuro, Toshima-ku, Tokyo Japan [ ] ( abc, D DS-triple 13