Math. J. Okayama Univ. 54 (2012), 49–52

ON THE STRUCTURE OF THE MORDELL-WEIL GROUPSOF THE JACOBIANS OF CURVES DEFINED BY y n = f (x)Hyunsuk Moon
Abstract. Let A be an abelian variety defined over a number field K.
It is proved that for the composite field Kn of all Galois extensions over
K of degree dividing n, the torsion subgroup of the Mordell-Weil group
A(Kn ) is finite. This is a variant of Ribet’s result ([7]) on the finitenessof torsion subgroup of A(K(ζ∞ )). It is also proved that for the Jacobiansof superelliptic curves y n = f (x) defined over K the Mordell-Weil groupover the field generated by all nth roots of elements of K is the direct
sum of a finite torsion group and a free Z-module of infinite rank.

Let K be a number field of finite degree. Let A be a nonzero abelianvariety defined over K. For an extension M over K, we denote the group ofM -rational points by A(M ) and its torsion subgroup by A(M )tors . We callA(M ) the Mordell-Weil group of A over M . It is well-known that A(M ) is afinitely generated abelian group for a finite algebraic extension M of K; thenthe Mordell-Weil rank means the number of generators of the torsion-freepart of A(M ).
Frey and Jarden ([1]) have asked whether the Mordell-Weil group of every
nonzero abelian variety A defined over K has infinite Mordell-Weil rank overthe maximal abelian extension K ab of K. Here, the Mordell-Weil rank ofA over an arbitrary M means dimQ (A(M ) ⊗Z Q). They proved that for
elliptic curves E defined over Q, the Mordell-Weil group E(Qab ) has infinite
rank. In [3], [12], [4], this is generalized to the Jacobians of hyperelliptic
curves defined over Q. In fact, they showed the infiniteness of the MordellWeil rank for certain elementary abelian 2-extensions over Q and, in [4], we
studied more precise structures of the Mordell-Weil groups in addition to therank. Murabayashi [6] generalized it to the Jacobians of superelliptic curves
y p = f (x), where p is an arbitrary prime number, by showing the infinitenessof the rank for certain elementary abelian p-extensions over Q(ζp ). Rosenand Wong [9] proved the infiniteness of the rank for the Jacobian of any curvethat can be realized over K as a cyclic geometrically irreducible cover of P1 .
This implies in particular the conjecture of Frey and Jarden is affirmative forMathematics Subject Classification. — 11G10, 11G05.
Key words and phrases. Mordell-Weil group, Jacobian, superelliptic curve.
This work was supported by the Korea Research Foundation(KRF) grant funded by
the Korea government(MEST) (No. 2009-0066564).
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any geometrically irreducible, principally polarized abelian surface definedover K. Recently, Sairaiji and Yamauchi [10] proved the conjecture of Freyand Jarden for the Jacobians of non-singular projective curves defined overK under the assumption that the curves have infinitely many K ab -rational
points. Im and Larsen [2] proved the infiniteness of the Mordell-Weil rank
for abelian varieties over any fields which have topologically cyclic absolute
Galois groups and are not algebraic over finite fields. Our aim in this paper
is to give another proof of Rosen-Wong’s result in the case of the Jacobiansof superelliptic curves together with slightly more precise information on thestructure of the Mordell-Weil group.
Throughout this paper, n is an integer ≥ 2. Our main theorem is thefollowing:
Theorem 1. Let C be a smooth projective curve of genus ≥ 1 which isthe smooth compactification of an affine plane curve defined by the equation
y n = f (x) with coefficients in K, and let J be√its Jacobian variety. Supposethat C has a K-rational point. Let M = K( n m | m ∈ OK ), where OK is
the ring of integers of K. Then the Mordell-Weil group J(M ) is the direct
sum of a finite torsion group and a free Z-module of infinite rank.
Remark. By definition, M contains a primitive nth root ζn of unity. Ingeneral, M is not abelian over K but is abelian over K(ζn ).
The key ingredient in the proof is the following variant of Ribet’s result([7]) which may be of some interest in its own right.
Proposition 2. Let K be a number field and let Kn be the composite fieldof all Galois extensions over K of degree dividing n. Then for any abelianvariety A over K, the torsion group A(Kn )tors is finite.
Proof. It is sufficient to show that there exist integers p0 and N0 such thatfor any prime number p and integer m ≥ 1,(i) if p ≥ p0 , then A(Kn ) does not contain a point of order p, and(ii) if p < p0 and pm ≥ N0 , then A(Kn ) does not contain a point of orderpm .
Consider the following Galois representation
ρ : Gal(K/K) → GL(A[pm ]) ≃ GL2g (Z/pm Z),

where g is the dimension of A. Then by Serre (cf. [11], p. 6), there existsan integer c = c(A) ≥ 1, independent of p and m, such that the image of ρcontains Z c , where Z is the center of GL2g (Z/pm Z) and Z c is the group ofcth power elements of Z. If we let C := Im(ρ) ∩ Z, then C ⊃ Z c and hence
(p−1
pm−1
max{ (c,p−1)
if p 6= 2,, (c,pm−1 ) }(exponent of C) ≥m−22
if p = 2.
(c,2m−2 )MORDELL-WEIL GROUPS OF JACOBIANS OF y n = f (x)51Take integers p0 and N0 which satisfy
p0 − 1N0> n.
> n andccp0
Then we have(exponent of C) > n
if p and m are the same as in (i), (ii).
Let p and m be the same as in (i), (ii) and let P be an element of A(K) oforder pm (m = 1 in case (i)). We denote the field generated by the coordinates of P over K by K(P ) and its Galois closure over K by K(P )Gal . We
also denote by K(A[pm ]) the field obtained by adjoining all coordinates ofpm -torsion points of A. Now, we show that K(P )Gal is not contained in Kn .
Let G := Im(ρ) = Gal(K(A[pm ]/K)), HP := Gal(K(A[pm ])/K(P )Gal ) ⊂ G,
GP := Gal(K(P )Gal /K) = G/HP . Since the group HP acts on P triviallyand the action of a scalar matrix ( 6= 1) is not trivial, we have HP ∩ C = 1,
and hence C embeds into GP by the canonical projection G ։ GP . Therefore the exponent of GP is greater than n. On the other hand, the exponent of Gal(Kn /K) is n. We conclude that there cannot exist a surjection
Gal(Kn /K) ։ GP , and hence P 6∈ A(Kn )tors .
Remarks. (1) The above proposition remains true if we replace the definitionof Kn by the composite field K(n) of all extensions over K of degree ≤ n,because we have K(n) ⊂ KN with N = n!.
(2) A proof of the Proposition which uses a different method is given in [5].
In [4], we showed that the Mordell-Weil group with finite torsion grouphas free Z-module structure modulo torsion:
Proposition 3. Let A be an abelian variety over a number field K. Let
M be a Galois extension of K such that A(M )tors is finite. Then the group
A(M )/A(M )tors is a free Z-module of at most countable rank.
Proof of Theorem 1. We may assume that C is a smooth compactificationof the affine plane curve C0 : y n = f (x), where f (x) is a polynomial withcoefficients in OK . Let P0 be a K-rational point of C. Let j : C → J be
the embedding defined over K such that j(P0 ) = O, where O is the identity
point of J. By Propositions 2 and 3, it only remains to show that J(M ) is
not finitely generated. If J(M ) is finitely generated, then it is equal to J(L)for some finite extension L/K. Indeed, such L is constructed by adjoining
to K all coordinates of a finite set of generators of J(M ). Then we have the52

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following commutative diagram:
C0 (M )

֒→J(M )

kC0 (L)

k
֒→

J(L)

Here, the left hand equality follows because C0 (M ) = C0 (K) ∩ J(M ) =C0 (K) ∩ J(L) = C0 (L). By Siegel’s theorem ([8], Chap. 8, Thm. 2.4), C0 (L)contain only finitely many integral points. Thisp contradicts the fact thatthe set C0 (M ) contains the infinite set {(x, n f (x)) | x ∈ OK } of integralpoints.
References[1] G. Frey and M. Jarden, Approximation theory and the rank of abelian varieties overlarge algebraic fields, Proc. London Math. Soc. 28 (1974), 112–128.
[2] B. Im and M. Larsen, Abelian varieties over cyclic fields, Amer. J. Math. 130 (2008),1195–1210.
[3] H. Imai, On the rational points of some Jacobian varieties over large algebraic numberfields, Kodai Math. J. 3 (1980), 56–58.
[4] H. Moon, On the Mordell-Weil groups of Jacobians of hyperelliptic curves over certain
elementary abelian 2-extensions, Kyungpook Math. J. 49 (2009), 419–424.
[5], On the structure of Mordell-Weil groups over infinite number fields,
Surikaisekikenkyusho Kokyuroku 1639 (2009), 56–60.
[6] N. Murabayashi, Mordell-Weil rank of the Jacobians of curves defined by y p = f (x),Acta Arith. 64 (1993), 297–302.
[7] K. Ribet, Torsion points of abelian varieties in cyclotomic extensions, appendix to N.
Katz and S. Lang, Finiteness theorems in geometric classfield theory, Enseign. Math.
(2) 27(1981), no. 3-4, 285–319.
[8] S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, 1983.
[9] M. Rosen and S. Wong, The rank of abelian varieties over infinite Galois extensions,J. Number Theory 92 (2002), 182–196.
[10] F. Sairaiji and T. Yamauchi, The ranks of Jacobian varieties over the maximal abelian
extensions of algebraic number fields: Toward Frey-Jarden’s conjecture, preprint.
[11] J-P. Wintenberger, Démonstration d’une conjecture de Lang dans des cas particuliers,
J. reine angew. Math. 553 (2002), 1–16
[12] J. Top, A remark on the rank of Jacobians of hyperelliptic curves over Q over certain
elementary Abelian 2-extension, Tohoku Math. J. 40 (1988), 613–616.
Hyunsuk Moon: Department of Mathematics, College of Natural Sciences
Kyungpook National University, Daegu 702-701, KOREA
e-mail address: hsmoon@knu.ac.kr(Received January 20, 2011 )