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«THE RANK AND GENERATORS OF KIHARA’S ELLIPTIC CURVE WITH TORSION Z/4Z OVER Q(t) ´ ´ ANDREJ DUJELLA, IVICA GUSIC AND PETRA TADIC Abstract. For the ...»

THE RANK AND GENERATORS OF KIHARA’S ELLIPTIC

CURVE WITH TORSION Z/4Z OVER Q(t)

´ ´

ANDREJ DUJELLA, IVICA GUSIC AND PETRA TADIC

Abstract. For the elliptic curve E over Q(t) found by Kihara, with torsion

group Z/4Z and rank ≥ 5, which is the current record for the rank of such

curves, by using a suitable injective specialization, we determine exactly

the rank and generators of E(Q(t)).

1. Introduction By Mazur’s theorem, we know that the torsion group of an elliptic curve over Q is one of the following 15 groups: Z/nZ with 1 ≤ n ≤ 10 or n = 12, Z/2Z × Z/2mZ with 1 ≤ m ≤ 4. The same 15 groups appear as possible torsion groups for elliptic curves over the field of rational functions Q(t). The current records for the rank of elliptic curves over Q(t) with prescribed torsion group can be found in the table [3]. Note that in this table for the most of torsion groups only the lower bounds for the rank of record curves are given.

Indeed, it seems that only for the torsion group Z/2Z × Z/4Z the exact rank over Q(t) of the record curve can be found in literature. In fact, in [5], Dujella and Peral proved that the corresponding curve, obtained from the so called Diophantine triples, has rank equal to 4 and they provide the generators for the group. The proof uses the method introduced by Gusi´ and Tadi´ in [6] for c c an efficient search for injective specializations. In this paper, we will prove an analogous result for the curve with record rank over Q(t) with torsion group Z/4Z found by Kihara [9] (see Theorem 2.1). Here we will use results from the recent paper [7], where the authors generalize and extend their method from [6]. In particular, by results of [7], now the method can be applied to curves with only one rational 2-torsion point.

Our main tool is [7, Theorem 1.3]. It deals with elliptic curves E given by y 2 = x3 + A(t)x2 + B(t)x, where A, B ∈ Z[t], with exactly one nontrivial 2-torsion point over Q(t). If t0 ∈ Q satisfies the condition that for every The authors were supported by the Croatian Science Foundation under the project no.

6422.

2010 Mathematics Subject Classification. 11G05, 14H52.

Key words and phrases. Elliptic curve, specialization homomorphism, rank, torsion, generators, mwrank ´ ´ 2 A. DUJELLA, I. GUSIC AND P. TADIC nonconstant square-free divisor h of B(t) or A(t)2 − 4B(t) in Z[t] the rational number h(t0 ) is not a square in Q, then the specialized curve Et0 is elliptic and the specialization homomorphism at t0 is injective. If additionally there exist P1,..., Pr ∈ E(Q(t)) such that P1 (t0 ),..., Pr (t0 ) are the free generators of E(t0 )(Q), then E(Q(t)) and E(t0 )(Q) have the same rank r, and P1,..., Pr are the free generators of E(Q(t)).

We can mention here that by the methods from [7], it is easy to show that the general families of curves with torsion Z/10Z, Z/12Z and Z/2Z × Z/8Z given in Kubert’s paper [10] all have rank 0 over Q(t), as expected from the corresponding entries in the above table (see Remark 3.1).

2. Kihara’s curve with rank ≥ 5 In 2004, Kihara [9] constructed a curve over Q(t) with torsion group Z/4Z and rank ≥ 5. This improved his previous result [8] with rank ≥ 4. We briefly describe Kihara’s construction. The quartic curve H given by the equation

–  –  –

is considered. Forcing five points with coordinates of the form (r, s), (r, u), (s, p), (u, q), (p, m) to satisfy (2.1) leads to a system of certain quadratic Diophantine equations, for which a parametric solution is found. By the transformation X = (a2 − b)y 2 /x2 and Y = (a2 − b)y(b + ax2 + ay 2 )/x3, we get from H the elliptic curve E with equation

–  –  –

where A(t) = 4 · (6561t52 − 363636t51 + 9938430t50 − 173377920t49 + 2093381633t48 − 17874840696t47 + 104874633374t46 − 354077727932t45 − 225490507368t44 + 11426935382596t43 − 77979654468618t42 + 278213963503072t41 − 111253886611731t40 − 6018676884563976t39 + 50437302286983990t38 − 272612407880904180t37 + 1164419469986139655t36 − 4197430502686137512t35 + 13172874796371804604t34 − 36632311181492128960t33 + 91270726486568186066t32 − 205234200473064086512t31 + 418565949588822731196t30 − 776853016569799513688 ∗ t29 + 1315193607411569449248t28 − 2034605787348730781688t27 + 2881061692467531957212t26 − 3743290609966430672640t25 + 4481632889126213095506t24 − 4982866114496291088464t23 + 5212858449376876320156t22 − 5228984497845291579880t21 + 5133683278868660716535t20 − 4993730543781934338052t19 + 4784499422236344389062t18 − 4408888972356409443776t17 + 3784174478544654656557t16 − 2930449238068167436056t15 + 1987364904374352466086t14 − 1143489371242947035052t13 + 534571453903095183576t12 − 187222081778006813708t11 + 38190501318649624878t10 + 3640463051927237920t9 − 6612210530622475839t8 + 2873353737226588120t7 − 748108460242930642t6 + 127972510817241756t5 − 14371540294374703t4 + 1013310571582176t3 − 40376902667904t2 + 671143753728t + 2176782336),

–  –  –

Here we give also the factorization of A(t)2 − 4B(t) because it is essential for the proof of Theorem 2.1.

A(t)2 − 4B(t) = 16(6561t52 − 393876t51 + 12044286t50 − 233179992t49 + 3037888017t48 − 27010254024t47 + 156557186174t46 − 431368937388t45 − 1778897440520t44 + 29118669267908t43 − 185554409423562t42 + 692712486737480t41 − 814770507947971t40 − 9637269870939544t39 + 91721999355372182t38 − 511726240316396532t37 + 2209210564256660999t36 − 7976932199520997736t35 + 24934025825524375740t34 − 68735161090418984560t33 + 168954992642195397618t32 − 372775248087820281744t31 + 740995668471372699516t30 − 1328945941188034678776t29 + 2149268803793268828000t28 − 3126651933116879854968t27 + 4072542345072750657372t26 − 4716734217685402094832t25 + 4814383096857805387890t24 − 4296609977077762663152t23 + 3376161521034367049052t22 − 2507937390234783739560t21 + 2175389746176683769207t20 − 2602425451380388744228t19 + 3581059482295238331078t18 − 4560635054158334637368t17 + 4966911843389715455741t16 − 4539431424438631575336t15 + 3452812241784521490182t14 − 2155004986056068938396t13 + 1071003286706676463160t12 − 395973787910697589516t11 + 87894481825369263726t10 + 4136860927499429288t9 − 12949727214730449839t8 + 5873221034815986696t7 − 1551776386124274418t6 + 266291310738984156t5 − 29718680649281967t4 + 2058088618943712t3 − 79201521880704t2 + 1255157987328t + 2176782336) × (81t26 − 2058t25 + 22205t24 − 136914t23 + 569600t22 − 1941994t21 + 7144777t20 − 31865642t19 + 143465455t18 − 557913380t17 + 1796620282t16 − 4792045284t15 + 10672893440t14 − 19973820452t13 + 31471575770t12 − 41625786276t11 + 45790269127t10 − 41147326466t9 + 29240715721t8 − 15417678410t7 + 5182080208t6 − 459229234t5 − 434078947t4 + 182700750t3 − 25979095t2 + 933744t + 46656)2.





Kihara in [9] found five independent points P1,..., P5 on this curve, corresponding to the five points on H mentioned above, showing that the rank of E over Q(t) is ≥ 5. The torsion subgroup is Z/4Z. Indeed, the point T1 = (a2 − b, 2a(a2 − b)) on (2.2) is of order 4 since 2T1 = (0, 0) and 4T1 = O.

Furthermore, from the factorizations of B(t) and A(t)2 − 4B(t) we see that T1 ̸∈ 2E(Q(t) and that E(Q(t) has exactly one point of order 2. An alternative proof of this fact will be given in Section 3.

Our goal is to prove that the rank of E over Q(t) is exactly equal to 5 and to find the generators of E(Q(t)). Computations with several specializations indicate that P1, P2, P3, P4, P5 are not generators of E(Q(t)). Indeed, from our results it will follow that they generate a subgroup of index 64 in E(Q(t)).

In fact, it holds that P1 + Pi ∈ 2E(Q(t)) for i = 2, 3, 4, 5, i.e. there exist points W2, W3, W4, W5 of E(Q(t)) such that P1 +Pi = 2Wi, i = 2, 3, 4, 5. Since the torsion subgroup is Z/4Z, there are two choices for each Wi. We choose KIHARA’S CURVE 5

one of them. The x-coordinates of these points are

x(W2 ) = 4(t − 3)(3t − 1)(t2 + 2)(t4 − 4t3 + 6t2 − 12t + 1)(3t4 − 17t3 + 27t2 − 43t + 6) × (t4 − 9t3 + 15t2 − 19t + 4)(7t2 − 18t + 23)(t4 − 28t3 + 54t2 − 92t + 41)(t − 2)2 × (9t12 − 110t11 + 576t10 − 2333t9 + 7802t8 − 19832t7 + 39488t6 − 57374t5 + 61421t4 − 42914t3 + 16488t2 − 701t − 216)2 (t + 1)4, x(W3 ) = −16(t − 1)3 (t + 1)3 (t − 3)(t2 + 2)(t2 + 6t − 1)(t3 + 4t2 − 5t + 16) × (t4 − 9t3 + 15t2 − 19t + 4)(2t4 − 17t3 + 27t2 − 25t + 1)(3t4 − 17t3 + 27t2 − 43t + 6) × (3t2 − 2t + 7)(t − 2)2 t2 (3t − 1)2 (t − 5)2 (t4 − 28t3 + 54t2 − 92t + 41)2 × (2t4 − 7t3 + 9t2 − 11t − 5)2, x(W4 ) = 16t(t − 1)3 (t + 1)(3t − 1)(t2 + 2)(t2 + 6t − 1)(t3 + 4t2 − 5t + 16) × (3t4 − 17t3 + 27t2 − 43t + 6)(t4 − 9t3 + 15t2 − 19t + 4)(2t4 − 17t3 + 27t2 − 25t + 1) × (3t2 − 2t + 7)(t − 2)2 (t − 5)2 (7t2 − 18t + 23)2 (t4 − 4t3 + 6t2 − 12t + 1)2 × (2t4 − 7t3 + 9t2 − 11t − 5)2 (3t7 − 29t6 + 51t5 − 136t4 + 175t3 − 267t2 + 179t − 24)2 / (8 − 55t − 29t2 + 103t3 − 120t4 + 69t5 − 27t6 + 3t7 )2, x(W5 ) = 16(t − 1)3 (t − 2)3 (t − 3)2 (t2 + 2)2 (3t2 − 2t + 7)2 (t3 + 4t2 − 5t + 16)2 × (t4 − 28t3 + 54t2 − 92t + 41)(t4 − 4t3 + 6t2 − 12t + 1)(3t4 − 17t3 + 27t2 − 43t + 6) × (t4 − 9t3 + 15t2 − 19t + 4)(5t4 − 17t3 + 27t2 − 79t + 16)(2t4 − 17t3 + 27t2 − 25t + 1) × (t2 + 6t − 1)(t − 5)(3t − 1)(t + 1)t.

We also give the x-coordinate of P1 :

x(P1 ) = 16t(t − 2)2 (t − 3)(t − 5)(3t − 1)(t2 + 2)(3t2 − 2t + 7)(7t2 − 18t + 23) × (t2 + 6t − 1)(t3 + 4t2 − 5t + 16)(2t4 − 17t3 + 27t2 − 25t + 1) × (2t4 − 7t3 + 9t2 − 11t − 5)(3t4 − 17t3 + 27t2 − 43t + 6)(t4 − 4t3 + 6t2 − 12t + 1) × (t2 − 2t + 3)(5t4 − 17t3 + 27t2 − 79t + 16)(t4 − 9t3 + 15t2 − 19t + 4) × (t4 − 28t3 + 54t2 − 92t + 41)(3t13 − 128t12 + 1185t11 − 5018t10 + 13628t9 − 27704t8 + 44162t7 − 63956t6 + 84827t5 − 100976t4 + 92061t3 − 52802t2 + 10662t − 552)2 / (12t11 − 219t10 + 1699t9 − 7248t8 + 21004t7 − 45434t6 + 72862t5 − 90128t4 + 77496t3 − 46283t2 + 10095t − 768)2.

The points P1, W2,..., W5 are a natural guess for the generators, and we will show that this is indeed true by proving the following theorem in the next section.

–  –  –

3. An injective specialization As described in the introduction, we use [7, Theorem 1.3] to find rational numbers t0 for which the specialization map at t0 is injective. The condition is that for each nonconstant square-free divisor h of B(t) or A(t)2 − 4B(t) in Z[t] the rational number h(t0 ) is not a square in Q. The condition is easy to check, and we can find many rationals t0 satisfying it. However, the coefficients of the curve E are polynomials with large degrees and coefficients. Thus, for the success of our approach, it is crucial to find suitable specialization t0 of reasonably small height. Furthermore, we need a specialization for which the rank of Et0 over Q is equal to 5, so it is reasonable to consider only specializations for which the the root number of Et0 is −1 (conjecturally implying that the rank is odd).

We find that the specialization at t0 = − 11 satisfies all requirements, see preceding section for the factoriozation of B(t) and A(t)2 − 4B(t). It remains to compute the rank and generators of E−11/4. For that purpose, we use the excellent program [2] of Cremona, which is included in the program package Sage [13]. By extending significantly the default precision (we use options -p 800 -b 11), we get the elliptic curve E−11/4 over Q, given by the equation y 2 = x3 + 484371205173916954475505177386303655600428018856419825361x2 + 1991079455035325445414429226070637115564958481993681648002 x, which is of rank 5 with five free generators G1,..., G5 and the generator of the torsion group T0, given by their x-coordinates

–  –  –

References

[1] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.

[2] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge Univ. Press, 1997.

[3] A. Dujella, Infinite families of elliptic curves with high rank and prescribed torsion, http://web.math.pmf.unizg.hr/ duje/tors/generic.html [4] A. Dujella and J. C. Peral, Elliptic curves and triangles with three rational medians, J. Number Theory 133 (2013), 2083–2091.

[5] A. Dujella and J. C. Peral, High rank elliptic curves with torsion Z/2Z × Z/4Z induced by Diophantine triples, LMS J. Comput. Math. 17 (2014), 282–288.

[6] I. Gusi´ and P. Tadi´, A remark on the injectivity of the specialization homoc c morphism, Glas. Mat. Ser. III 47 (2012), 265–275.

[7] I. Gusi´ and P. Tadi´, Injectivity of the specialization homomorphism of elliptic c c curves, J. Number Theory 148 (2015), 137–152.

[8] S. Kihara, On the rank of elliptic curves with a rational point of order 4, Proc.

Japan Acad. Ser A Math. Sci. 80 (2004), 26–27.

[9] S. Kihara, On the rank of elliptic curves with a rational point of order 4, II, Proc. Japan Acad. Ser A Math. Sci. 80 (2004), 158–159.

[10] D. S. Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. 33 (1976), 193–237.

[11] M. Kuwata and J. Top, An elliptic surface related to sums of consecutive squares, Expos. Math. 12 (1994), 181–192.

[12] T. Shioda, On the Mordell-Weil lattices, Comment. Math. Univ. St. Pauli 39 (1990), 211–240.

[13] W. A. Stein et al., Sage Mathematics Software (Version 6.0), The Sage Development Team, 2013, http://www.sagemath.org.

[14] J. Top and F. de Zeeuw, Explicit elliptic K3 surfaces with rank 15, Rocky Mountain J. Math. 39 (2009), 1689–1698.



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