The bounds of the Mordell-Weil ranks in cyclotomic towers of function fields

Journal of Number Theory 202 (2019) 332–346

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Journal of Number Theory www.elsevier.com/locate/jnt

General Section

The bounds of the Mordell-Weil ranks in cyclotomic towers of function fields Yusuke Aikawa Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

a r t i c l e

i n f o

Article history: Received 15 March 2018 Received in revised form 11 January 2019 Accepted 12 January 2019 Available online 18 February 2019 Communicated by A. Pal

a b s t r a c t We present new examples of elliptic curves having the bounded rank in cyclotomic towers of function fields over C. Our key method is to utilize the monodromy of the Gaussian hypergeometric functions for the computation of the MordellWeil groups. © 2019 Elsevier Inc. All rights reserved.

Keywords: Elliptic curves Mordell-Weil rank Elliptic surfaces Function fields

1. Introduction Let K be a field and E an elliptic curve over K. As is well known, the set of rational points E(K) carries a structure of abelian group, which is called the Mordell-Weil group. The Mordell-Weil theorem states that this group is finitely generated in the arithmetic situation, i.e. K is a number field or the function field of an algebraic curve over a finite field. The rank of E(K) as a finitely generated abelian group is called the Mordell-Weil rank of E. E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2019.01.003 0022-314X/© 2019 Elsevier Inc. All rights reserved.

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In this paper we study the Mordell-Weil group E(K) when K is the function field of an algebraic curve over C. This is a finitely generated group if E/K satisfies certain condition [12]. Then we shall discuss a function field analogue of Mazur’s conjecture [8], namely the rank growth of E(Kn ) for Kn /K a cyclotomic tower. In this situation, there are several previous results. Silverman gave a conditional upper bound for the rank E(K  ) where K  /K a finite unramified extension in [15]. In [14], [4] the upper bounds were discussed and recently Pál [9] improved those results by using Hodge theory. Stiller [16] and Fastenberg [5], [6] constructed examples of elliptic curves over C(t) whose ranks 1 are bounded for C(t n )/C(t) independently on n. On the other hand, Shioda [11] and Ulmer [17] discovered elliptic curves over Fp (t) having arbitrary large rank. The aim of this paper is to present new examples of elliptic curves E over C(t) having 1 the bounded ranks for the cyclotomic extensions C(t n )/C(t). The main theorem is stated in Theorem 3.1. We take a complex geometric method to prove the main theorem. In particular, the Gaussian hypergeometric function plays a central role. To be more precise, let f : X → P1 be an elliptic surface degenerating at three points {0, 1, ∞} and E/C(t) be the generic fiber of E. We assume that the singular fiber at ∞ is additive and others are multiplicative. Such elliptic surfaces are classified in [10]. Let f1 : X1 → P1 be the base change of f by the morphism P1 → P1 ; t → α − t and E1 /C(t) the generic fiber. 1 Then the main theorem asserts that the rank of E1 (C(t n )) is bounded independently on n. The proof goes in the following way. Firstly, let fn : Xn → P1 be the elliptic surface obtained by the base change of f1 : X1 → P1 by P1 → P1 ; t → tn and the minimal 1 desingularization. Then the Mordell-Weil group E(C(t n )) is translated into the quotient N S(Xn )/Tn by the subgroup Tn generated by fibral divisors and the zero section, and this agree with the Hodge (1,1)-part of Mn := H 2 (Xn )/Tn . The surface Xn is endowed with the automorphism σ given by (x, y, t) → (x, y, ζn t). One has the eigen decomposition  d d Mn = d|n Ln where Ln are Q(ζd )-vector spaces (see §3.2). It is not hard to show dimQ(ζd ) Ldn = 2. The crucial step is computing dimQ(ζd ) Ldn ∩ L1,1 , that is either 0, 1 or 2. This step is a new encounter compared with the previous results by Silverman etc. In order to compute dimQ(ζd ) Ldn ∩ L1,1 we employ the period formula [1] Theorem 4.1 and apply the monodromy theorem of Gaussian hypergeometric functions [3]. 2. Preliminaries In this section, we will fix the setting and prepare some notation. Moreover we will explain how to translate the computation of the Mordell-Weil groups into the computation of the cohomology of the elliptic surfaces. 2.1. The setting In this paper, we mean by an elliptic surface a surjective morphism f : E → C from a smooth projective surface to a smooth projective curve with a section such that its generic fiber is an elliptic curve over the function field of C. We often denote by E

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an elliptic surface f : E → C for simplicity. Moreover, for the singular fiber of elliptic surfaces, we use Kodaira’s notation [7], Theorem 6.2. Let f : X → P1u be an elliptic surface over C having multiplicative reduction at two points and additive reduction at a point. Here, we denote by P1u the projective line with inhomogeneous coordinate u. We may assume that X degenerates at three points {0, 1, ∞} ⊂ P1u and has the singular fibers of type Ia (resp. Ib ) at 0 (resp. 1) and the fiber of ∞ is additive fiber by the assumption. The possible types of singular fibers of such elliptic surfaces are classified in [10] and summarized in the following Table 2.1. Table 2.1 Possible combinations of singular fibers in this situation.

Type.1 Type.2 Type.3 Type.4 Type.5 Type.6

Multiplicative fiber 1

Multiplicative fiber 2

Additive fiber

I1 I1 I1 I1 I1 I2

I1 I2 I3 I4 I1 I2

II∗ III∗ IV∗ I∗ 1 I∗ 4 I∗ 2

For α ∈ P1 \ {0, 1, ∞} and n ∈ N, let fα,n : X ×P1u P1un → P1u be the base change of f by the morphism gα,n : P1un → P1u ; u → α − un , where P1un denotes the source of gα,n with inhomogeneous coordinate un to distinguish P1u . We define an elliptic surface fα,n : Xα,n → P1un by the following diagram; Xα,n

i

fα,n

X ×P1u P1un f α,n

P1un

pr

2

gα,n

X f

P1u

where i is the minimal desingularization and pr is the first projection. Then Xα,n has √ the singular fibers at (2n + 1)-points. The Ia -type appears at un = ζnk n α and the √ Ib -type appears at un = ζnk n α − 1 for k = 0, 1, · · · , n − 1, where we fix ζn := exp( 2πi n ) throughout this paper. The following Tables 2.2–2.5 collect the variation of singular fibers at ∞ depending on the index n of the above base change. These follow from the computation of local monodromy matrices in [7], Theorem 9.1, Table.I or [2], Chapter V, Table 6.  −1 −1 For further discussion, we prepare several notations. Let Zα,n := fα,n (0) + fα,n (∞) +  √ n−1 −1 k √ n −1 k n (f (ζ α) + f (ζ α − 1)) be the reduced divisor on X and U α,n α,n ⊂ α,n n k=1 α,n n red 1 1 Xα,n the inverse image of Pu \{0, 1, α, ∞} via gα,n ◦fα,n . Moreover, set Sα,n := Pun \{unn = 0, α, α−1, ∞}. We denote by Eα,n the generic fiber of fα,n . According to the Mordell-Weil 1

theorem for function fields ([13], III, Theorem 6.1), the Mordell-Weil group Eα,1 (C(u1n )) is a finitely generated abelian group. We will study the rank of the finitely generated 1 abelian groups Eα,1 (C(u1n )).

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Table 2.2 −1 The type of fα,n (∞) in Xα,n of Type.1. n mod 6

The type of fiber

1 2 3 4 5 0

II∗ IV∗ I∗ 0 IV II I0

Table 2.3 −1 The type of fα,n (∞) in Xα,n of Type.2. n mod 4

The type of fiber

1 2 3 0

III∗ I∗ 0 III I0

Table 2.4 −1 The type of fα,n (∞) in Xα,n of Type.3. n mod 3

The type of fiber

1 2 0

IV∗ IV I0

Table 2.5 −1 The type of fα,n (∞) in Xα,n of Type.4.5.6. n mod 2

the Type of fiber

1 0

I∗ nm Inm

2.2. Mordell-Weil groups and Néron-Severi groups From now on, we denote u1 by t and un by s. In the above setting, we have    1  Eα,1 C(t n ) ∼ = Eα,n C(s) . To translate the problem of elliptic curves over function fields into the problem of elliptic surfaces, we recall the relationship between Néron-Severi groups of an elliptic surface and the Mordell-Weil groups of its generic fiber. But we do not recall all of this material here; for detail, see [12]. Let N S(Xα,n ) be the Néron-Severi group of Xα,n :  N S(Xα,n ) := divisors on Xα,n / ∼ .   Here ∼ denotes algebraic equivalence. A rational point (x(s), y(s)) ∈ Eα,n C(s) corresponds to a section P1 → Xα,n ; t → (x(t), y(t), t). This is one-to-one correspondence. By

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regarding sections as divisors on Xα,n , rational points of Eα,n map to divisors on Xα,n . Let Tα,n denote the subgroup of N S(Xα,n ) generated by fibral divisors of fα,n and the zero section. Then the above correspondence gives an isomorphism:   ∼ Eα,n C(s) −→ N S(Xα,n )/Tα,n . Hence the Lefschetz’ theorem on (1,1) classes implies an isomorphism:     Eα,n C(s) ⊗ Q ∼ = H 2 (Xα,n , Q) ∩ H 1,1 /Tα,n,Q where Tα,n,Q := Tα,n ⊗ Q. 3. Computations of the cohomology The elliptic surface Xα,n has an automorphism σ given by (x, y, t) → (x, y, ζn t). In this section, we study the structure of the cohomology of elliptic surface Xα,n as Q[σ]-module. This is the first step toward the main theorem, that is, the bound of the rank of Mordell-Weil group of elliptic curve Eα,1 . 3.1. The dimension of the cohomology Set Mα,n := H 2 (Xα,n , Q)/Tα,n,Q ∼ = W2 H 1 (Sα,n , j ∗ R1 (fα,n )∗ Q) where j : Sα,n → P1s is the embedding. In this section, we study the structure of this module. Note that Mα,n is a Q-Hodge structure on account of the inclusion Tα,n,Q ⊂ H 2 (Xα,n , Q) ∩ H 1,1 endowed with multiplication by Q[σ]. Proposition 3.1. We have

dimQ Mα,n

⎧ ⎪ ⎪ ⎨2n − 2 = 2n − 3 ⎪ ⎪ ⎩2n − 4

−1 if fα,n (∞) is additive; −1 if fα,n (∞) is multiplicative; −1 if fα,n (∞) is smooth.

Note that the terms on the right hand side are non-negative since n ≥ 1. Proof. We have an exact sequence 0 → H 1 (Sα,n , j ∗ R1 (fα,n )∗ Q) → H 2 (Uα,n , Q) → H 2 (Xα,n,s , Q)

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where Xα,n,s is a smooth general fiber of fα,n . By taking the graded piece of weight 2, we have an isomorphism   W2 H 1 (Sα,n , j ∗ R1 (fα,n )∗ Q) ∼ = Ker W2 H 2 (Uα,n , Q) → H 2 (Xα,n,s , Q) .

(3.1)

Note that H 2 (Xα,n,s , Q) Q and the arrow in the right hand side is surjective. Hence dimQ Mα,n = dimQ W2 H 2 (Uα,n , Q) − 1.

(3.2)

The localization exact sequence induces the following: 0

Coker(HZ2 α,n (Xα,n ) → H 2 (Xα,n ))

H 2 (Uα,n )

HZ3 α,n (Xα,n )

H 3 (Xα,n )

H1 (Zα,n )

0.

∼ =

W2 H 2 (Uα,n )

(3.3) Here, all objects in the above diagram are with rational coefficient. Recall that Zα,n =   −1 √ n−1 −1 k √ −1 −1 −1 (∞) + k=1 (fα,n (ζn n α) + fα,n (ζnk n α − 1)) red where fα,n (0) is smooth, fα,n (0) + fα,n √ √ −1 k n −1 k n fα,n (ζn α) and fα,n (ζn α − 1) are multiplicative for k = 1, · · · , n − 1. The fiber −1 fα,n (∞) depends on n, according to the table in §2. Thus we obtain ⎧ −1 ⎪ ⎪ ⎨2n + 2 if fα,n (∞) is additive; −1 dimQ H1 (Zα,n , Q) = 2n + 3 if fα,n (∞) is multiplicative; ⎪ ⎪ ⎩2n + 4 if f −1 (∞) is smooth. α,n Moreover, the Leray spectral sequence gives an exact sequence 0 → H 1 (Sα,n , j ∗ R1 (fα,n )∗ Q) → H 2 (Uα,n , Q) → H 0 (San , j ∗ R2 (fα,n )∗ Q) ∼ = H 2 (Xα,n,t , Q)π1 (Sα,n ) .

(3.4)

Note that the last term is one dimensional and the last arrow is surjection. Employing the formula χ(Sα,n , j ∗ R1 (fα,n )∗ Q) = χ(Sα,n , Q) × rankj ∗ R1 (fα,n )∗ Q, we have dimH 1 (Sα,n , j ∗ R1 (fα,n )∗ Q) = 4n. Hence by (3.4) dimH 2 (Uα,n , Q) = 4n + 1

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and so by (3.3) ⎧ −1 ⎪ ⎪ ⎨2n − 1 if fα,n (∞) is additive; −1 dimQ W2 H 2 (Uα,n , Q) = 2n − 2 if fα,n (∞) is multiplicative; ⎪ ⎪ ⎩2n − 3 if f −1 (∞) is smooth. α,n We reach a conclusion by (3.2). 2 3.2. The automorphism of the elliptic surface Let σ : Xα,n → Xα,n be an automorphism given by (x, y, s) → (x, y, ζn s). Then, Q[σ] acts on Mα,n . We will determine the structure of Mα,n as Q[σ]-module. For a positive integer d which divides n, we set   Ldα,n := Ker Φd (σ) : Mα,n → Mα,n where Φd (X) is the minimal polynomial of ζd over Q. We have a decomposition Mα,n =



Ldα,n

d|n

of the Hodge structures. Then, we have    1  rankEα,1 C(t n ) = rankEα,n C(s) = dimQ Mα,n ∩ H 1,1 = dimQ Ldα,n ∩ (Ldα,n )1,1 .

(3.5)

d|n −1 Proposition 3.2. Let dmin be the minimal integer such that the fiber fα,d (∞) in Xα,dmin min is smooth or multiplicative. According to the Tables 2.2–2.5, if the elliptic surface X is Type.1 (resp. 2, 3, otherwise) in Table 2.1, then dmin = 6 (resp. 4, 3, 2). Then, we have

Ldα,n

∼ =

⎧ ⎪ ⎪ ⎨0





Q[σ]/ Φd (σ) ⎪ ⎪ ⎩Q[σ]/Φ (σ)⊕2 d

if d = 1 if d = dmin if d = 1, dmin

as Q[σ]-module. Proof. We use induction on n. Write n = ml where m, l are positive integers. Then we have an unramified cyclic covering πl : Uα,n → Uα,m ; (x, y, un ) → (x, y, uln )

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and this induces an injection πl∗ : Mα,m → Mα,n . Via the above injection, since Un / σ m  ∼ = Um , we obtain an isomorphism ∼ =

m

σ =1 Mα,m −→ Mα,n

(3.6)

m

σ =1 where Mα,n denotes the subspace of Mα,n consisting of elements on which σ m acts trivially. Moreover, πl∗ (Ldα,m ) ⊂ Ldα,n for d|m|n. By the isomorphism (3.6), we have m Ldα,m ∼ = (Ldα,n )σ =1 = Ldα,n for d|m|n.

Here the second equality follows from the fact that Φd (σ) divides σ m − 1. We sum up the above discussion in the following diagram: Mα,n 

Ldα,n = (Ldα,n )σ

m

m

=1

σ =1 Mα,n

∼ =

∼ =

Ldα,m

πl∗

Mα,m

 d d d Put lα,n := dimQ Ldα,n , then dimQ Mα,n = d|n lα,n . The above diagram yields lα,n = d 1 1 for d|m|n. In particular, lα,n = lα,1 = 0 by Proposition 3.1. And for n = dmin , we lα,m have by induction 



dmin d lα,d = lα,d +2 min min

d|dmin

=

 2n − 3 2n − 4

φ(d)

d|dmin ,d=1,dmin −1 (∞) is multiplicative i.e. dmin = 2; if fα,n −1 (∞) is smooth i.e. dmin =3 or 4 or 6. if fα,n

dmin Thus we obtain lα,d = φ(dmin ). Here, φ denotes the Euler function. min For general n, we similarly have

 d|n,d=1

d lα,n =

⎧ n ⎪ ⎪ ⎨lα,n + 2



φ(d)

d|n,d=1,n

n ⎪ ⎪ ⎩lα,n + φ(dmin ) + 2

if dmin does not divide n; 

d|n,d=1,dmin ,n

φ(d)

if dmin divides n.

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−1 If dmin not divides n, the fiber fα,n (∞) is additive. Then, we have



n dimQ Mα,n = lα,n +2

φ(d) = 2n − 2.

d|n,d=1,n n −1 Hence lα,n = 2φ(n). If dmin divides n and fα,n (∞) is multiplicative, then φ(dmin ) = 1 and



n dimQ Mα,n = lα,n + φ(2) + 2

φ(d) = 2n − 3,

d|n,d=1,dmin ,n n −1 and hence lα,n = 2φ(n). Finally, if dmin divides n and fα,n (∞) is smooth, then φ(dmin ) = n 2 and we conclude lα,n = 2φ(n) in the same way. We finish the proof. 2

3.3. The dimension of the subspace Ldα,n ∩ (Ldα,n )1,1 By the Proposition 3.2, the dimensions of the subspaces Ldα,n ∩ (Ldα,n )1,1 over Q(ζd ) are at most 2. In the following, we will determine the dimension completely under the assumption that α is a transcendental number. Firstly, we treat the case that d ≤ dmin . This case can be computed as follows. Proposition 3.3. Let dmin be an integer as in Proposition 3.2. Then, if d ≤ dmin , we have Ldα,n ∩ (Ldα,n )1,1 = Ldα,n . d )2,0 := dimQ (Ldα,n )2,0 = 0. We denote the numProof. The assertion is equivalent to (lα,n  −1  −1 (s) by ms . The Euler number e fα,n (s) ber of irreducible components of the fiber fα,n −1 of the fiber fα,n (s) are given by

⎧ −1 ⎪ (s) is smooth; if fα,n ⎪0  −1  ⎨ −1 e fα,n (s) = ms if fα,n (s) is multiplicative; ⎪ ⎪ ⎩m + 1 if f −1 (s) is additive. s α,n

(3.7)

The Hodge number of (2,0)-part is given by the Euler numbers of fibers: 2,0 h2,0 (Xα,n ) = −1 + n := dimQ H

1   −1  e fα,n (s) . 12 1

(3.8)

s∈Ps

By the Tables 2.1–2.5 and (3.7), (3.8), h2,0 n =

n−1  dmin

(3.9)

where, for a real number r, r denotes the maximum of integers which are smaller than or equal to r.

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d d If d ≤ dmin , we have h2,0 = 0 by (3.9). Thus, since (lα,d )2,0 = (lα,n )2,0 , we have d d (lα,n )2,0 = 0. 2

Secondly, we will prove that Ldα,n ∩ (Ldα,n )1,1 = 0 for d > dmin . In order to prove this, we set S := P1 \ {0, 1, ∞} and consider a smooth fibration: X → S such that the fiber of α ∈ S is the elliptic surface Xα,n . Since π1 (S, α)-action commutes with σ-action, the monodromy action on the cohomology of Xα,n induces π1 (S, α)-action on Ldα,n . Proposition 3.4. Suppose that α is a transcendental number. If dimQ(ζd ) Ldα,n ∩(Ldα,n )1,1 = 0, then π1 (S, α)-action on Ldα,n factors trough a finite quotient. In other wards,   Im π1 (S, α) → Aut(Ldα,n ) is a finite group. Proof. Throughout this proof, all of the fundamental groups are considered with fixed base point α ∈ S and we omit to write the base point. Take a model XQ of X over Q. Consider the following cartesian diagram XQ

1 SQ := SpecQ[T, T1 , 1−T ]

Y := Xα,n ×C SpecQ(T )

SpecQ(T )

Xα,n

SpecC

where the morphism SpecC → SpecQ(T ) is induced by the morphism Q(T ) → C; T → α (here we use the assumption that α is a transcendental number). Let YQ(T ) := Y ×Q(T ) SpecQ(T ). Since N S(Xα,n ) ∼ ), one has the isomorphism = N S(Y Q(T )

    Ldα,n ∩ (Ldα,n )1,1 ∼ = N S(YQ(T ) )/Tα,n ∩ Ker Φd (σ) : Mα,n → Mα,n ,   and hence the Galois group Gal Q(T )/Q(T ) acts on this. Since the Néron-Severi group of Xα,n is finitely generated and the action of the Galois group on each cycles factors through a finite quotient, we have   Gal K/Q(T ) → Aut(Ldα,n ∩ (Ldα,n )1,1 ) for some finite extension K of Q(T ). This completes the proof in the case Ldα,n ∩ (Ldα,n )1,1 = Ldα,n .

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Suppose Ldα,n ∩ (Ldα,n )1,1 = Ldα,n , namely dimQ(ζd ) (Ldα,n ∩ (Ldα,n )1,1 ) = 1. Then there is the orthogonal decomposition Ldα,n = (Ldα,n ∩ (Ldα,n )1,1 ) ⊕ (Ldα,n ∩ (Ldα,n )1,1 )⊥ with respect to polarization on Ldα,n and π1 (S  ) acts on each component, where S  is a smooth model of K. There is a Z-lattice in each component induced from the Z-lattice H 2 (Xα,n , Z) in Mα,n , and π1 (S  ) acts on it. Therefore, the image of π1 (S  ) to Aut(Ldα,n ) is contained in Z× × Z× = {±1} × {±1}. In particular, it is finite. Thus a diagram π1 (S  )

Aut(Ldα,n )

π1 (S) concludes the desired assumption. 2 We postpone the proof of the following proposition to the next section. Proposition 3.5. For d > dmin ,   Im π1 (S, α) → Aut(Ldα,n ) is an infinite group. By Proposition 3.3, 3.4 and 3.5, we have the main theorem of this paper, the explicit ranks of the Mordell-Weil group. Theorem 3.1. Suppose that α is a transcendental number. We have 

1

rankEα,1 (C(t n )) =

dimQ Ldα,n ∩ (Ldα,n )1,1

1
⎧ ⎪ ⎪ ⎨

=



2φ(d)

if dmin does not divide n.

2φ(d) + φ(dmin )

if dmin divides n,

1


⎪ ⎪ ⎩

1
where φ denotes the Euler function. 4. Proof of Proposition 3.5 In this section, we will give the proof of Proposition 3.5. We assume that d > dmin throughout this section.

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4.1. Eigendecomposition and the structure of eigenspaces Recall Mα,n =



Ldα,n

d=1,d|n

and ⊕2  Ldα,n ∼ = Q[σ]/ Φd (σ)

if d = 1, dmin

(4.1)

as Q[σ]-modules. If we write Ldα,n,C := Ldα,n ⊗ C, we have Ldα,n,C =



Ldα,n (χ)

χ

where χ runs through the set of homomorphisms from Q(ζd ) to Q and Ldα,n (χ) are the spaces of eigenvectors of σ with π1 (S, α)-action. By (4.1), the spaces Ldα,n (χ) are two-dimensional over C. To prove Proposition 3.5, it suffices to find an eigen component whose monodromy group is infinite. Hereafter we fix χ to be the homomorphism χ(ζd ) = ζd . Moreover we fix rational numbers λ1 , λ2 ∈ Q such that exp(2πiλj )(j = 1, 2) and λ1 ≤ λ2 are eigenvalues of the local monodromy on Q ⊗χ,Q(ζd ) R1 f∗ Q where f : X → P1 as in the beginning of § 2.1. Since the local monodromy of elliptic surfaces over C is completely classified, see [7] or [2], from the Table 2.1, we can list all of pairs (λ1 , λ2 ) as in Table 4.1. Table 4.1 Type of X and dmin , the pair (λ1 , λ2 ). Type of X in Table 2.1

dmin

Type.1

6

Type.2

4

Type.3

3

Type.4

2

Type.5

2

Type.6

2

(λ1 , λ2 ) 1 5 ,  61 63  ,  41 42  ,  31 31  2, 2 1 1 ,  21 21  2, 2

In order to study the structure of Ldα,n (χ) as C[π1 (S, α)]-module, we make use of the Gaussian hypergeometric function.   1 1 λ −d ,λ2 − d Lemma 4.1. Put 2 F1 (x) := 2 F1 1 1− ; x the Gaussian hypergeometric functions. 1 d

Let Vα to be two dimensional vector space over C spanned by 2 F1 (α) and 2 F1 (1 − α), on which π1 (S, α) acts in the natural way. Then there is an isomorphism Vα ∼ = Ldα,n (χ)∨ of C[π1 (S, α)]-modules where Ldα,n (χ)∨ denotes the dual space.

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344

Proof. Recall Uα,n = (gα,n ◦ fα,n )−1 (P1 \{0, 1, ∞}) as in § 2.1. Put H 2 (Uα,n )0 :=   Ker H 2 (Uα,n ) → H 2 (Xα,n,s ) where Xα,n,s is a general smooth fiber. There is a natural isomorphism W2 H 2 (Uα,n )0 ∼ = H 2 (Xα,n )/Tα,n = Mα,n and this yields Q ⊗χ,Q(ζd ) W2 H 2 (Uα,n )0 ∼ = Ldα,n (χ). We employ [1] Theorem 4.1 (period formula). There are two homology cycles Γ1 , Γ2 ∈ H2 (Uα,n , Q) which form a basis of Q ⊗χ,Q(ζd ) H2 (Uα,n , Q), and two global rational forms 2 ω1 , ω2 ∈ Γ(Uα,n , Ω2 ) which form a basis of Q⊗χ,Q(ζd ) W2 HdR (Uα,n /Q)0 , and a differential operator Θ on Q[α] such that  Γ1 ω1 ω Γ1 2

   Γ2 ω1 = a1 G1 (α) a1 G1 (α) ω Γ2 2

a2 G2 (α) a2 G2 (α)



   for some constants ai ∈ C× where we put G1 (α) := Θ 2 F1 (α) and G2 (α) := Θ 2 F1 (1 −  α) . Note that the fact that the space Ldα,n (χ) is two-dimensional implies that the spaces 2 Q ⊗χ,Q(ζd ) H2 (Uα,n , Q) and Q ⊗χ,Q(ζd ) W2 HdR (Uα,n /Q)0 are two-dimensional. Moreover, the above matrix is invertible. This means that the composition 2 CΓ1 ⊕ CΓ2 −→ H2 (Uα,n , C) −→ W2 HdR (Uα,n /C)0 (χ)∨ = Ldα,n (χ)∨

is bijection, and the image is spanned by two column vectors 

G(α) G (α)



 ,

G(1 − α) G( 1 − α)



with respect to the dual basis of ω1 , ω2 . Note that the action of π1 (S, α) on Ldα,n (χ)∨ is compatible with that on H2 (Uα,n , C). Therefore, we have an isomorphism Ldα,n (χ)∨ ∼ = G(α), G(1 − α) of C[π1 (S, α)]-modules. The right hand side is isomorphic to Vα = 2 F1 (α), 2 F1 (1 − α) as C[π1 (S, α)]-module, so we are done. 2 4.2. Monodromy of the Gaussian hypergeometric functions d Let us put D := x dx and consider the following differential equation, so-called the hypergeometric equation:



 1 1 1 D(D − ) − x(D + λ1 − )(D + λ2 − ) u(x) = 0. d d d

(4.2)

Y. Aikawa / Journal of Number Theory 202 (2019) 332–346

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Recall that the vector space Vα is the two dimensional vector  space over C spanned  1 1 λ1 − d ,λ2 − d by 2 F1 (α) and 2 F1 (1 − α) where 2 F1 (x) = 2 F1 ; x is the Gaussian hyper1 1− d geometric function. Then Vα is the space of local solutions of the differential equation (4.2). Put   Hλd1 ,λ2 := Im π 1 (S, α) → Aut(Vα ) . Lemma 4.1 says that   Hλd1 ,λ2 ∼ = Im π1 (S, α) → Aut(Ldα,n (χ)) . Therefore, the following proposition finishes the proof of Proposition 3.5. Proposition 4.1. For d > dmin , Hλd1 ,λ2 is an infinite group. Proof. For d > dmin , from the Table 4.1, we have 0 < λ1 −

1 1 1 < λ2 − < 1 − < 1. d d d

According to Theorem 4.8 in [3], this inequality implies the infiniteness of the group Hλd1 ,λ2 . 2 Acknowledgments The author is deeply grateful to Masanori Asakura for useful discussion and his constant encouragement. Without his support, this work has not been accomplished. References [1] M. Asakura, N. Otsubo, Regulators on K1 of hypergeometric fibrations, in: Proceedings of Conference “Arithmetic L-functions and Differential Geometric Methods (Regulators IV)”, in press, arXiv:1709.04144. [2] W. Barth, K. Hulek, C. Peters, A. Van de Ven, Compact Complex Surfaces, second edition, SpringerVerlag, Berlin, 2004. [3] F. Beukers, G. Heckman, Monodromy for the hypergeometric function n Fn−1 , Invent. Math. 95 (2) (1989) 325–354. [4] J.S. Ellenberg, Selmer groups and Mordell-Weil groups of elliptic curves over towers of function fields, Compos. Math. 142 (5) (2006) 1215–1230. [5] L.A. Fastenberg, Mordell-Weil groups in procyclic extensions of a function field, Duke Math. J. 89 (2) (1997) 217–224. [6] L.A. Fastenberg, Computing Mordell-Weil ranks of cyclic covers of elliptic surfaces, Proc. Amer. Math. Soc. 129 (7) (2001) 1877–1883. [7] K. Kodaira, On compact complex analytic surface I, Ann. of Math. 71 (1960) 111–152, vol. II, Ann. of Math. 77 (1963) 563–626, vol. III, Ann. of Math. 78 (1963) 1–40. [8] B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972) 183–266. [9] A. Pál, Hodge theory and the Mordell-Weil rank of elliptic curves over extensions of function fields, J. Number Theory 137 (2014) 166–178.

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