4-COVERING MAPS ON ELLIPTIC CURVES WITH TORSION SUBGROUP Z2 × Z8 SAMUEL IVY, BRETT JEFFERSON, MICHELE JOSEY, CHERYL OUTING, CLIFFORD TAYLOR, AND STACI WHITE Abstract. In this exposition we consider elliptic curves over Q with the tor- sion subgroup Z2 × Z8. In particular, we discuss how to determine the rank of the curve E : y2 = (1 − x2)(1 − k2 x2), where k = (t4 − 6t2 + 1)=(t2 + 1)2 and t = 9=296. We use a 4-covering map C0 ! C ! E in terms of homogeneous bd2 bd2 spaces for d2 2 {−1; 6477590; 2; 7; 37g. We provide a method to show that the 3 Mordell-Weil group is E(Q) ' Z2 × Z8 × Z , which would settle a conjecture of Flores-Jones-Rollick-Weigandt and Rathbun. 1. Introduction Given an elliptic curve E, we denote the set of Q-rational points as E(Q). Since E(Q) is finitely generated, there exists a finite group E(Q)tors and a nonnegative r integer r such that E(Q) ' E(Q)tors × Z , where r is called the (Mordell-Weil) rank. Those elliptic curves with E(Q)tors ' Z2 × Z8 are birationally equivalent to t4 − 6t2 + 1 y2 = (1 − x2)(1 − k2 x2) where k = for some t 2 : (t2 + 1)2 Q The highest known rank of curves with this torsion subgroup is r = 3. It is conjectured in [6] that this bound is obtained when t = 9=296; one knows that 2 ≤ r ≤ 3 for this particular t. In this exposition, we focus on determining the exact rank of this particular elliptic curve. Our main result is as follows: Theorem. Assuming the Birch and Swinnerton-Dyer conjecture, the Mordell-Weil group of the elliptic curve Y 2 + XY = X3 − 71813598680248384341084284771096244120 X E : + 234238430204114181370252185964622864112853337413958990400 r is E(Q) ' Z2 × Z8 × Z , where r = 3. We sketch the idea of the proof. Consider the diagram: φ^0 φ^ E00 / E0 / E ? ? C0 bd2 / Cbd2 2000 Mathematics Subject Classification. Primary 14H52; Secondary 14J27. 1 2 S. IVY, B. JEFFERSON, M. JOSEY, C. OUTING, C. TAYLOR, AND S. WHITE E is 2-isogenous to E0, which itself is 2-isogenous to E00. Using the connecting homomorphisms, × 0 × E(Q) Q 0 E (Q) Q δb : ,! × 2 and δb : ,! × 2 ; φ^(E0(Q)) (Q ) φ^0(E00(Q)) (Q ) we may determine the rank r by finding Q-rational points (z; w) on the homogeneous spaces 2 2 2 2 2 2t Cbd2 : d2w = (1 + d2z )(1 + d2κ z ) where κ = ; t2 − 1 3 0 2 2 02 2 0 4(t − t) Cb : d2w = (1 + d2z )(1 + d2k z ) where k = : d2 (t2 + 1)2 When t = 9=296, we use Cremona's mwrank [4] to find that the images of these connecting homomorphisms are contained in the group generated by −1, 6477590, 2, 7, and 41. We find rational points on the homogeneous spaces for the first four generators; see Tables 2 and 3. Hence 2 ≤ r ≤ 3. The (global) root number of E is wE = −1, so that the rank r must be odd. Hence, assuming the conjecture of Birch and Swinnerton-Dyer, we see that r = 3. Alternatively, it suffices then to find rational points on C and C0 for d = 37. bd2 bd2 2 The various maps in the diagram above involve quadratic polynomials; hence, we consider this diagram to be a \4-cover" of the elliptic curve E. In comparison to finding points on E, it should be half as difficult to find points on Cbd2 and a quarter as difficult to find points on C0 . bd2 The authors would like to thank the Summer Undergraduate Mathematical Sci- ences Research Institute (SUMSRI) at Miami University for the opportunity to conduct this research. They would also like to thank the National Science Foun- dation and the National Security Agency for their funding, as well as Residential Computing (ResComp) at Miami University and the Rosen Center for Advanced Computing (RCAC) at Purdue University for use of their machines. They are grateful to Michael Stoll for helpful conversations, Tom Farmer for careful reading of this document, Elizabeth Fowler for her constant support, and Edray Goins for his guidance. 2. Foundations An elliptic curve E is a set of points which satisifies an equation of the form Y 2 = X3 +AX +B, where A and B are rational numbers such that the discriminant −16(4A3 +27B2) 6= 0. In particular, an elliptic curve is a type of nonsingular cubic curve. Define E(Q) as the set of Q-rational points, where we append a \point at infinity" O. We may use geometry to turn E(Q) into a group. For P; Q 2 E(Q), draw a line through P and Q. (If P = Q, we choose the line tangent to the curve at P .) This line will intersect the curve at a third Q-rational point P ∗ Q { which may possibly be O. By reflecting this point about the x-axis, we obtain another Q-rational point, denoted by P ⊕ Q. For a graphical represention, see Figure 1. Formally define P ⊕ Q = (P ∗ Q) ∗ O. Under this operation, the set E(Q) forms an abelian group with the identity element O and inverse [−1]P = P ∗ O. For more information, see [12]. 4-COVERING MAPS ON ELLIPTIC CURVES 3 4 3 P Q = (2,3) ∗ 2 1 Q = (0,1) P = ( 1,0) -2 -1.5 − -1 -0.5 0 0.5 1 1.5 2 2.5 3 -1 -2 P Q = ( 2,3) -3 ⊕ − -4 Figure 1. The Group Law on an Elliptic Curve In 1901, Henri Poincar´econjectured [10] that this abelian group is finitely gen- erated. This was proved by Louis Mordell in 1922. Theorem 1 (Mordell, [9]). If E is an elliptic curve over Q, then the abelian group E(Q) is finitely generated. Furthermore, there exists a finite group E(Q)tors and a nonnegative integer r such that r E(Q) ' E(Q)tors × Z : This motivates a few definitions. The set E(Q)tors is the collection of points with finite order; it is called the torsion subgroup of E. This implies that the quotient r group E(Q)=E(Q)tors ' Z is a torsion-free group. The nonnegative integer r is called the Mordell-Weil rank of E; it signifies the number of independent generators having infinite order. While the rank r is mysterious, the torsion subgroup is well-understood. In 1977, Barry Mazur completely classified the structure of torsion subgroups of elliptic curves. Theorem 2 (Mazur, [8]). If E is an elliptic curve over Q, then E(Q)tors is iso- morphic to one of the following 15 groups: (i) Zn, with 1 ≤ n ≤ 10 or n = 12; (ii) Z2 × Z2m, with 1 ≤ m ≤ 4. Here we denote Zn as the cyclic group of order n. We will focus more specifically on those elliptic curves with torsion subgroup Z2 × Z8. In contrast to the torsion subgroup, less is known about the rank r of an elliptic curve. As of 2006, the largest known rank satisfies r ≥ 28. Andrej Dujella [5] has a listing of the largest known ranks among families of elliptic curves with prescribed torsion. Table 1 contains this information. Note that the highest known rank for curves E with E(Q)tors ' Z2 × Z8 is r = 3. 4 S. IVY, B. JEFFERSON, M. JOSEY, C. OUTING, C. TAYLOR, AND S. WHITE Table 1. Rank Records Torsion Subgroup Lower Bound Author(s) T for B(T ) Z1 28 Elkies (2006) Z2 18 Elkies (2006) Z3 13 Eroshkin (2007, 2008) Z4 12 Elkies (2006) Z5 6 Dujella - Lecacheux (2001) Eroshkin (2008) Dujella - Eroshkin (2008) Z 8 6 Elkies (2008) Dujella (2008) Dujella - Kulesz (2001) Z 5 7 Elkies (2006) Z8 6 Elkies (2006) Dujella (2001) MacLeod (2004) Z 3 9 Eroshkin (2006) Eroshkin - Dujella (2007) Dujella (2005) Z 4 10 Elkies (2006) Dujella (2001, 2005, 2006) Z 3 12 Rathbun (2003, 2006) Z2 × Z2 14 Elkies (2005) Elkies (2005) Z2 × Z4 8 Eroshkin (2008) Dujella - Eroshkin (2008) Z2 × Z6 6 Elkies (2006) Connell (2000) Dujella (2000, 2001, 2006) Z2 × Z8 3 Campbell - Goins (2003) Rathbun (2003, 2006) Flores - Jones - Rollick - Weigandt (2007) 3. Computing The Mordell-Weil Rank Mordell's proof [9] of Poincar´e'sconjecture began by considering the quotient group, E(Q)=2E(Q). Although E(Q) in general is an infinite group, the quotient r+2 group is always finite. In fact, if E(Q)tors ' Z2 × Z2m, then E(Q)=2E(Q) ' Z2 , where r is the Mordell-Weil rank. Hence, the basic idea to compute r is to calculate jE(Q)=2E(Q)j. To this end, we will compute two smaller, yet related, quotient groups. One constructs quotient groups for elliptic curves by considering a special type of group homomorphism. For instance, let E and E0 be two elliptic curves over Q. We say that a group homomorphism φ : E ! E0 is an m-isogeny if (1) the coordinates of φ involve rational functions with Q-rational coefficients and (2) there are exactly 4-COVERING MAPS ON ELLIPTIC CURVES 5 m points in the kernel E(Q)[φ] of the map. We will be interested in the case where m = 2. Proposition 3. Let E and E0 be elliptic curves over Q. Let φ : E ! E0 and φ^ : E0 ! E be 2-isogenies such that φ^◦ φ = [2] is the map which sends P 7! P ⊕ P . Assume that E0(Q)[φ^] = φ(E(Q)[2]). Then, E( ) E0( ) E( ) Q Q Q = : 2E(Q) φ(E(Q)) φ^(E0(Q)) It is easy to show that if E is an elliptic curve over Q with torsion subgroup 0 ^ E(Q)tors ' Z2 × Z2m, then the criterion E (Q)[φ] = φ(E(Q)[2]) of Proposition 3 is always satisfied.