TORSION OF ELLIPTIC CURVES OVER THE REAL QUADRATIC FIELD Q(√3)

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:
y2 = x3 + 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:
y2 + axy + ay = x3 + ax2 + ax + awith aK
Around 1900, Henri Poincare showed that the set
E(K) = {(x, y) K:y2 = x3 + Ax + B, A, BK}{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)tors. In this direction, Mazur (1977) proved the following deep theorem:

Theorem 2.1 (Mazur, 1977) Let E be any Elliptic curve over Q. Then E(Q)tors must be one of the following 15 groups:
Zn
1 ≤ n ≤ 12, n≠ 11
Z₂⊕Zn
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/MZ⊕Z/NZ over a quadratic field K, one needs to determine whether the modular curve X1(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) 11G05, 11G18, 11R11, 14H52.

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/MZZ/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/MZZ/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/MZZ/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 ‚ Z2Z10,Z2 Z12,Z3 Z3,Z3 Z6 and Z4 Z4. Since Z3 Z3,Z3 Z6 occur only over Q(√−3) and Z4Z4 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,
y2 + a1xy + a3y = x3 + a2x2 + a4x + a6, aᵢϵ K.  by [a1, a2, a3, a4, a6].

By Baziz (2010), elliptic curves with torsion Z11 over a quadratic field K are induced by solutions over K of the equation, X1(11): y2 y = x3 x2

satisfying
x (x − 1) (x5 − 18x4 + 35x3 − 16x2 − 2x + 1) ≠0
We see that X1(11) is an elliptic curve given by [0, −1, −1,0,0]. The rank of X1(11) (Q (√3)) is 0 and the torsion group is Z5. Since,
X1(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),
X1(13):y2 = x6 − 2x5 + x4 − 2x3 + 6x2 − 4x + 1) satisfying
x (x − 1) (x3 − 4x2 + x + 1) ≠ 0
We see that X1(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(J1(13) (Q (√3))) = 0.

Further we compute that (J1(13) (Q (√3)))tors is isomorphic to Z21. All these points on Z21√are generated by the cusps of X1(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),
X1(14):y2 + xy + y = x3 x satisfying
x(x − 1)(x + 1)(x3 − 9x2 x + 1)(x3 − 2x2 x + 1) ≠ 0
We see that X1(14) is an elliptic curve given by√   [1,0,1,−1,0].

The rank of X1(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 overK of the equation (Baziz, 2010),
X1(15):y2 + xy + y = x3 + x2
satisfying
x(x + 1)(x4 + 3x3 + 4x2 + 2x + 1) (x4 − 7x3 − 6x2 + 2x + 1) ≠ 0
We see that X1(15) is an elliptic curve given by [1,1,1,0,0].
The rank of X1(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,
+ (1916 − 1085√3) xy− (19585 − 11308√3) y= − (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),
X1(16):y2 = x (x2 + 1) (x2 + 2x − 1) satisfying
x (x − 1) (x + 1) (x2 − 2x − 1) (x2 + 2x − 1) ≠ 0
We see that X1(16) is a hyper-elliptic curve. Proceeding as in the case of X1(13), We compute that
rank(J1(16) (Q (√3))) = 0
and                                               
                     (J1(16) (Q (√3)))tors = Z2 ⊕Z10
Since all these torsion points are induced by the cusps of X1(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),
X1(18):y2 = x6 + 2x5 + 5x4 + 10x3 + 10x2 + 4x + 1
Satisfying
x (x + 1)(x2 + x + 1) (x2 − 3x − 1) ≠ 0

We see that X1(18) is a hyper-elliptic curve. Proceeding as in the case of X1(13), We compute that,         
                                  rank(J1(18)(Q(√3))) = 0
and                                                   
                              (J1(18)(Q(√3)))tors = Z21
Since all these torsion points are induced by the cusps of X1(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),
X1(2,10):y2 = x3 + x2 x
satisfying
x (x2 − 1) (x2 − 4x − 1) (x2 + x − 1) ≠ 0
We see that X1(2,10) is an elliptic curve given by [0,1,0, −1,0]. The rank of X1(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),
X1(2,12):y2 = x3 x2 + x
satisfying
x (x − 1) (2x − 1) (2x2 x + 1) (3x2 − 3x − 1) (6x2 − 6x − 1) ≠ 0
We see that X1(2,12) is an elliptic curve given by [0, −1,0,1,0].

The rank of X1(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 X1(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 Jacobians of hyper elliptic curves of genus two,II. 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.



No comments:

Post a Comment

Share Every Sense on Art, Culture & Literature; Travel & Tourism; Commerce & Economy; Science & Technology; etc. Send us your Write-up by Email: dimorianreview@gmail.com

DIMORIAN REVIEW WELCOMES YOUR ASSOCIATION AS CONTRIBUTOR

Share every sense on Art, Culture & Literature; Travel & Tourism; Commerce & Economy; Science & Technology; etc. Publish Article, Story, Poetry, Book Review, Travelogue, Recipe and News. Send us your creative contents written or transcreated in English at E-mail:dimorianreview@gmail.com

Get Free Updates at Your Inbox! Enter Your Email Address:

HAPPY TO SEE YOU HERE. PLEASE VISIT AGAIN. HAVE GOOD TIME!

Twitter Bird Gadget