**Naba Kanta Sarma**

*Assistant Professor, Department of Mathematics, Assam University, Silchar, Assam, India*

#
Abstract

Filip Najman examined the torsion of elliptic
curves over the number fields Q(i) and Q (√(−3)) in 2010

*.*In this paper, we study the torsion structures of elliptic curves over the real quadratic number field Q(√3)*.*

**Keywords**Torsion Subgroup, Elliptic Curve, Cusp

**1.Introduction**

An
elliptic curve (Knapp, 1992) is a smooth projective variety of genus 1 with a
specified base point O. Over a field

*K*with characteristic not equal to 2 or 3, an elliptic curve can be represented by a Weierstrass equation of the form:
y

^{2 }= x^{3 }+ Ax + B, A, B ∈ K.
Over
a field of characteristic 2 and 3

*,*one needs to work with a more general equation of the form:*y*

^{2 }+

*a*

_{₁}

*xy*+

*a*

_{₃}

*y*=

*x*

^{3 }+

*a*

_{₂}

*x*

^{2 }+

*a*

_{₄}

*x*+

*a*

_{₆}with

*a*

*∈*

_{ᵢ}*K*

Around 1900

*,*Henri Poincare showed that the set*E*(

*K*) = {(

*x, y*) ∈

*K:y*

^{2 }=

*x*

^{3 }+

*Ax*+

*B, A, B*∈

*K}*∪{

*O*}

of all

*K*-rational points on*E,*together with the base point*O*, is an Abelian group. He also conjectured that this group is finitely generated when*K:*=Q*.*In 1922*,*Luis Mordell proved the conjecture and in 1928*,*Andre Weil generalized this result not only for elliptic curves over algebraic number fields but also for Abelian varieties.
We are mainly interested in the torsion part of this
group, which we denote by

*E*(*K*)*In this direction, Mazur (1977) proved the following deep theorem:*_{tors}.**Theorem 2.1**(Mazur, 1977)

*Let E be any Elliptic curve over*Q. Then

*E*(Q)

*15*

_{tors }must be one of the following*groups:*

Z
_{n} |
1 ≤
n
≤ 12, n≠ 11 |

Z₂⊕Z₂
_{n} |
1 ≤
n
≤ 4 |

Then several mathematicians worked on the possible
torsion over quadratic extensions of Q

*,*culminating in the following theorem:**Theorem 2.2.**

*(Kamienny-Kenku-Momose,*1988)

*Let E be an elliptic curve over a quadratic extension of*Q

*. Then E*(

*K*)

_{tors }is isomorphic to one of the following 26 groups:
The
also showed that the groups Z3⊕Z3
and Z3⊕Z6 can occur over the quadratic field Q(√(−3))
only, while the group Z4⊕Z4
can occur over the quadratic field Q(

*i*) only.
In 2010

*,*Filip Najman (2010 & 2011) took a different approach by fixing a quadratic extension*K*of Q and then looking for possible torsions over*K.*They compiled a list of torsion subgroups for the quadratic fields Q(*i*) and Q (√ −3)*.*In 2012*,*Sheldon Kamienny and Filip Najman (2012) described methods that can be used to obtain results of this type i.e. finding all possible torsions over a given quadratic field*K.*In order to find whether there exists an elliptic curve with torsion Z*/M*Z⊕Z*/N*Z over a quadratic field*K,*one needs to determine whether the modular curve*X*_{1}(*M, N*) has a*K*-rational point that is not a cusp.**2.Objective**

In this paper, we use their method to investigate the
possible torsion structures over the real quadratic field Q(√3)

*.***3. Methodology**

Mathematics Subject Classification (2010) 11

*G*05*,*11*G*18*,*11*R*11*,*14*H*52.
The method we follow in this paper is described in
Kamienny (2012). We reproduce the main points here for the sake of
completeness.

i)
If X1(M, N) is an elliptic curve, then a
usual method of computing the rank is to perform 2-descent. If the rank is
positive, then there will be infinitely many elliptic curves with torsion Z/MZ⊕Z/NZ over K. If the rank is zero, then one
has to check whether all the torsion points are cusp. If not, then there will
be finitely many, explicitly computable, elliptic curves with the given torsion
subgroup.

ii)
If X1(M, N) is hyper-elliptic curve, namely
X1(13), X1(16), or X1(18), then by Faltings theorem, there are finitely many
K-rational points, implying that there are finitely many elliptic curves, up to
isomorphism, with torsion Z/MZ⊕Z/NZ
over K.

iii)
In order to find all these points, one can
sometimes proceed to compute the rank of the Jacobian using 2-descent on
Jacobians. This can be done in MAGMA. However, 2-descent is not an algorithm
and so one has no guarantee to obtain the rank using it. It guarantees only an
upper bound not the rank.

iv)
If the rank is zero, after finding the
torsion of the Jacobian, one has to check whether any of the torsion points
arise from a K-rational point that is not a cusp. If the rank is positive, the
problem becomes complicated and one can try to apply the Chabauty method (if
the rank is one) or other similar methods.

The only other hyper-elliptic curve, X1(16), is usually
easier to deal with, since f(x), where y2 = f(x) is a model of X1(16), is not
irreducible. This allows one to find all the points with more elementary
methods, like covering the hyper-elliptic curve with 2 elliptic curves.

Once a K-rational point on Y1(M, N) is found, one can find
(Rabarison, 2010) how to actually construct an elliptic curve with torsion Z/MZ⊕Z/NZ over K. Models for the curve X1(N) can
be found in [7], while the curves X1(2, N) that we need can be found in 2010.

Note that the torsions appearing in Mazur’s theorem will
appear over any number field. So we are left to examine the remaining 11
torsion structures:Z11 ‚Z13 ‚ Z14 ‚ Z15 ‚ Z16‚ Z18 ‚ Z2⊕Z10,Z2 ⊕Z12,Z3
⊕Z3,Z3
⊕Z6
and Z4 ⊕Z4.
Since Z3 ⊕Z3,Z3
⊕Z6
occur only over Q(√−3) and Z4⊕Z4
occurs only over Q(i) our study is further reduced to the torsions
Z11,Z13,Z14,Z15,Z16,Z18,Z2 ⊕Z10
and Z2 ⊕Z12.

We use MAGMA for our computations (Bosma,2006; Stoll,
2001 and Bosma et al., 1997).

**4. Results and Discussion**

The following are the main results of this paper:

**Theorem 3.1.**

*The torsions of elliptic curves over the real quadratic field*Q(√3)

*are the torsions appearing in Mazur’s theorem together with*Z14

*and*Z15

*.*

**4.1 The torsion Z11**

In
this sequel, we denote the elliptic curve over

*K*defined by the equation,*y*

^{2 }+

*a*

_{1}

*xy*+

*a*

_{3}

*y*=

*x*

^{3 }+

*a*

_{2}

*x*

^{2 }+

*a*

_{4}

*x*+

*a*

_{6,}aᵢϵ K. by [

*a*

_{1}

*, a*

_{2}

*, a*

_{3}

*, a*

_{4}

*, a*

_{6}]

*.*

By
Baziz (2010)

*,*elliptic curves with torsion Z11 over a quadratic field*K*are induced by solutions over*K*of the equation,*X*_{1}(11):*y*^{2 }−*y*=*x*^{3 }−*x*^{2}
satisfying

*x*(

*x*− 1) (

*x*

^{5 }− 18

*x*

^{4 }+ 35

*x*

^{3 }− 16

*x*

^{2 }− 2

*x*+ 1) ≠0

We
see that

*X*_{1}(11) is an elliptic curve given by [0*,*−1*,*−1*,*0*,*0]*.*The rank of*X*_{1}(11) (Q (√3)) is 0 and the torsion group is Z5*.*Since,*X*

_{1}(11) (Q (√3)) = {

*O,*(0

*,*0)

*,*(0

*,*1)

*,*(1

*,*0)

*,*(1

*,*1)}

We see that all torsion points correspond to

*x*= 0*,*1 which does not satisfy the compatibility condition. Hence, Z11 cannot occur as a torsion over the quadratic field Q(√3)*.*

**4.2 The torsion Z13**

Elliptic curves with torsion Z13
over a quadratic field

*K*are induced by solutions over*K*of the equation (Baziz, 2010)*,**X*

_{1}(13):

*y*

^{2 }=

*x*

^{6 }− 2

*x*

^{5 }+

*x*

^{4 }− 2

*x*

^{3 }+ 6

*x*

^{2 }− 4

*x*+ 1) satisfying

*x*(

*x*− 1) (

*x*

^{3 }− 4

*x*

^{2 }+

*x*+ 1) ≠ 0

We see that

*X*_{1}(13) is a hyper-elliptic curve. As the points of a hyper-elliptic curve have no group structure, it becomes convenient to study the Jacobian variety for such curves (Najman, 2010)*.*In MAGMA (Bosman et al., 1997)*,*there exists an implementation of 2-descent on Jacobians, but unfortunately only over Q*.*We find via 2-descent,*rank*(

*J*

_{1}(13) (Q (√3))) = 0

*.*

Further we compute
that (

*J*_{1}(13) (Q (√3)))*is isomorphic to Z21*_{tors }*.*All these points on Z21√are generated by the cusps of*X*_{1}(13)*.*Hence Z13 cannot occur as a torsion over Q(√3)*.***4.3 The torsion Z14**

Elliptic
curves with torsion Z14 over a quadratic field

*K*are induced by solutions over*K*of the equation (Baziz, 2010)*,**X*

_{1}(14):

*y*

^{2 }+

*xy*+

*y*=

*x*

^{3 }–

*x*satisfying

*x*(

*x*− 1)(

*x*+ 1)(

*x*

^{3 }− 9

*x*

^{2 }−

*x*+ 1)(

*x*

^{3 }− 2

*x*

^{2 }−

*x*+ 1) ≠ 0

We see that

*X*_{1}(14) is an elliptic curve given by√ [1*,*0*,*1*,*−1*,*0]*.**X*1(14)(Q(√3) is 1 and the torsion subgroup is Z6

*.*Taking the non-torsion point (1 +√3

*,*−3 − 2√3) on this curve, we obtain the elliptic curve, in which (0

*,*0) is a point of order 14

*.*

#
4.4
The
torsion Z15

Elliptic curves with torsion Z15
over a quadratic field

*K*are induced by solutions over*K*of the equation (Baziz, 2010)*,**X*

_{1}(15):

*y*

^{2 }+

*xy*+

*y*=

*x*

^{3 }+

*x*

^{2}

satisfying

*x*(

*x*+ 1)(

*x*

^{4 }+ 3

*x*

^{3 }+ 4

*x*

^{2 }+ 2

*x*+ 1) (

*x*

^{4 }− 7

*x*

^{3 }− 6

*x*

^{2 }+ 2

*x*+ 1) ≠ 0

We see that

*X*_{1}(15) is an elliptic curve given by [1*,*1*,*1*,*0*,*0]*.*
The
rank of

*X*_{1}(15) over Q(√3)is 1 and the torsion subgroup is Z4*.*Taking the non-torsion point (1 +√3*,*2 +√3) on this curve, we obtain the elliptic curve,*y²*+ (1916 − 1085√3)

*xy*− (19585 − 11308√3)

*y*=

*x³*− (19585 − 1130√3)

*x*

in
which (0

*,*0) is a point of order 15*.***4.5 The torsion Z16**

Elliptic curves with torsion Z16
over a quadratic field

*K*are induced by solutions over*K*of the equation (Baziz, 2010)*,**X*

_{1}(16):

*y*

^{2 }=

*x*(

*x*

^{2 }+ 1) (

*x*

^{2 }+ 2

*x*− 1) satisfying

*x*(

*x*− 1) (

*x*+ 1) (

*x*

^{2 }− 2

*x*− 1) (

*x*

^{2 }+ 2

*x*− 1) ≠ 0

We
see that

*X*_{1}(16) is a hyper-elliptic curve. Proceeding as in the case of*X*_{1}(13)*,*We compute that*rank*(

*J*

_{1}(16) (Q (√3))) = 0

and

(

*J*_{1}(16) (Q (√3)))*= Z2 ⊕Z10*_{tors }
Since
all these torsion points are induced by the cusps of

*X*_{1}(16)*,*so Z16 does not appear as torsion over Q(√3)*.*#
4.6
The
torsion Z18

Elliptic curves with torsion Z18
over a quadratic field

*K*are induced by solutions over*K*of the equation (Baziz, 2010)*,**X*

_{1}(18):

*y*

^{2 }=

*x*

^{6 }+ 2

*x*

^{5 }+ 5

*x*

^{4 }+ 10

*x*

^{3 }+ 10

*x*

^{2 }+ 4

*x*+ 1

Satisfying

*x*(

*x*+ 1)(

*x*

^{2 }+

*x*+ 1) (

*x*

^{2 }− 3

*x*− 1) ≠ 0

We
see that

*X*_{1}(18) is a hyper-elliptic curve. Proceeding as in the case of*X*_{1}(13)*,*We compute that,*rank*(

*J*

_{1}(18)(Q(√3))) = 0

and

(

*J*_{1}(18)(Q(√3)))*= Z21*_{tors }
Since
all these torsion points are induced by the cusps of

*X*_{1}(18)*,*so Z18 does not appear as torsion over Q(√3)*.*#
4.7
The
torsion Z2
⊕Z10

Elliptic
curves with torsion Z2 ⊕Z10
over a quadratic field

*K*are induced by solutions over*K*of the equation (Rabarison, 2010)*,**X*

_{1}(2

*,*10):

*y*

^{2 }=

*x*

^{3 }+

*x*

^{2 }−

*x*

satisfying

*x*(

*x*

^{2 }− 1) (

*x*

^{2 }− 4

*x*− 1) (

*x*

^{2 }+

*x*− 1) ≠ 0

We
see that

*X*_{1}(2*,*10) is an elliptic curve given by [0*,*1*,*0*,*−1*,*0]*.*The rank of*X*_{1}(2*,*10) over Q(√3) is 0 and the torsion subgroup is Z6*.*Hence all torsion point corresponds to cusps and hence Z2 ⊕Z10 cannot appear as torsion over Q(√3)*.*#
4.8
The
torsion Z2
⊕Z12

Elliptic
curves with torsion Z2 ⊕Z12
over a quadratic field

*K*are induced by solutions over*K*of the equation (Rabarison, 2010),*X*

_{1}(2

*,*12):

*y*

^{2 }=

*x*

^{3 }−

*x*

^{2 }+

*x*

satisfying

*x*(

*x*− 1) (2

*x*− 1) (2

*x*

^{2 }−

*x*+ 1) (3

*x*

^{2 }− 3

*x*− 1) (6

*x*

^{2 }− 6

*x*− 1) ≠ 0

We
see that

*X*_{1}(2*,*12) is an elliptic curve given by [0*,*−1*,*0*,*1*,*0]*.*
The
rank of

*X*_{1}(2*,*12) over Q(√3) is 0 and the torsion subgroup is Z8*.*Hence all torsion point corresponds to cusps and hence Z2 ⊕Z12 cannot appear as torsion over this quadratic field.**5.Conclusion**

We
have seen that all the torsion structures that appear over an arbitrary
quadratic field need not necessarily occur over a particular quadratic field.
In this article, we have compiled a list of all possible torsions that can
occur over the quadratic field Q(√3).

**Acknowledgement**

The
author is extremely grateful to Dr. Filip Najman for his valuable comments and
suggestions during the preparation of this paper.

**References**- A. Knapp.
*Elliptic curves*. New Jersey: Princeton University Press, 1992.

- B. Mazur (1977). Modular curves and
Eisenstein Ideal.
*IHES publications Mathematics*, 47, p.186*.*

- Bosma,Cannon.
*Handbook of Magma Functions*, Sydney, 2006 ,http://www.msri.org/about/computing/docs/magma/handbook.pdf

- F. Najman (2010). Complete classification of
torsion of elliptic curves over quadratic cyclotomic field.
*Journal of Number Theory*,130, pp.1964− 1968*.*

- F. Najman (2011). Torsion of elliptic curves
over quadratic cyclotomic field.
*Journal of Okayama University*, 53*,*pp.75− 82*.*

- F. Najman and S. Kamienny (2012). Torsion
groups of elliptic curves over Quadratic fields.
*Acta Arithmetica,*152*,*pp.29− 305*.*

- H. Baziz (2010). Equations for the modular
curve
*X*_{1}(*N*) and models of elliptic curves with torsion points.*Mathematical Computation*, 79*,*pp.237− 2386*.*

- M. Kenku (1988). Torsion points on elliptic
curves defined over quadratic fields.
*Nagoya Mathematical Journal*, 109*, pp.*125− 149*.*

- M. Stoll (2001). Implementing 2-descent on
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*Acta Arithmatica*, 98*,*pp.245− 277*.*

- Rabarison (2010). Structure de torsion des
courbes elliptiques d´efinies sur les corps de nombres quadratiques.
*Acta Arithmetica*, 144(1)*,*pp.17− 52*.*

- The MAGMA computer algebra system is
described in W. Bosma, J. Cannon and Catherine Playoust (1997). The Magma
algebra system I: The user language.
*Journal of Symbolic Comput*, 24*,*pp.235− 265*.*

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