4-Covering Maps on Elliptic Curves with Torsion Subgroup Z2 × Z8

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' $ 4-Covering Maps on Elliptic Curves with Torsion Subgroup Z2 × Z8 Samuel Ivy (Morehouse College), Brett Jeﬀerson (Morgan State University), Michele Josey (North Carolina Central) Cheryl Outing (Spelman College), Cliﬀord Taylor (Grand Valley State), Staci White (Shawnee State University)

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' $ Abstract Poincar´e’s Conjecture m-Isogenies and Quotient Groups 2-Covering Maps Proof of Main Result

We consider elliptic curves over Q with the torsion subgroup In 1901, Henri Poincaréconjectured (7) that this abelian group is Mordell’s proof (6) of Poincaré’s conjecture began by considering Theorem 5. Using φ : E → E′ and φˆ : E′ → E, define the By Theorem 4, the Mordell-Weil group of the elliptic curve Z2 × Z8. In particular, we discuss how to determine the rank of the finitely generated. This was proved by Louis Mordell in 1922. 2 3 2 2 2 2 4 2 2 2 the quotient group, E(Q)/2E(Q). Although E(Q) in general is an connecting homomorphisms: Y + XY = X − 71813598680248384341084284771096244120 X curve E : y = (1 − x )(1 − k x ) where k =(t − 6t +1)/(t +1) E : and t =9/296. Theorem 1 (Mordell, (6)). If E is an elliptic curve over Q, then infinite group, the quotient group is always finite. In fact, E′(Q) Q× + 234238430204114181370252185964622864112853337413958990400 −→ , the abelian group E(Q) is finitely generated. Furthermore, there × 2 r ′ E(Q) r+2 δ : φ E(Q) (Q ) is Z2 × Z8 × Z for some r. It suffices to show that the rank r = 3. We use a 4-covering map Cd2 → Cd2 → E in terms of homogeneous E(Q)tors ≃ Z2 × Z2m =⇒ ≃ Z2 , exists a finite group E(Q)tors and a nonnegative integer r such that 2 E(Q) (X : Y : 1) 7−→ 4X + 39141876845580405121; Using Cremona’s mwrank (1) we find that spaces for d2 ∈ {−1, 6477590, 2, 7, 37}. We show that the Mordell- b b × Weil group is r where r is the Mordell-Weil rank. Hence, the basic idea to compute E(Q) Q ′ E(Q) ≃ E(Q)tors × Z . −→ , E (Q) E(Q) 3 r is to calculate |E(Q)/2E(Q)|. ˆ ′ × 2 δ = {1}, δ ⊆ h−1, 6477590, 2, 7, 37i; E(Q) ≃ Z × Z × Z δ : φ E (Q) (Q ) ′ 2 8 φ E(Q) ! φˆ E (Q) ! X Y 7−→ X − . thereby proving a conjecture of Flores-Jones-Rollick-Weigandt and Mordell-Weil Group To this end, we will compute two smaller, yet related, quotient ( : : 1) 4892734605697550640 b so by Proposition 3 and Theoremb 5, Rathbun. groups. One constructs quotient groups for elliptic curves by con- Both δ and δ are injective group homomorphisms with finite images. The set E(Q)tors is the collection of points with finite order; it is ′ sidering a special type of group homomorphism. For instance, let E r+2 E(Q) E (Q) E(Q) 5 called the torsion subgroup of E. This implies that the quotient To be more precise, let Σ(k) = {82207, 87697, 92863} and Σ(κ) = 2 = = δ δ ≤ 2 . ′ 2 E(Q) φ(E(Q)) ˆ ′ Q Introduction r and E be two elliptic curves over Q. We say that a group homo- b φ(E ( ))! group E(Q)/E(Q)tors ≃ Z is a torsion-free group. {2, 3, 5, 7, 37, 41, 61} be the set of primes dividing k and κ, respec- morphism φ : E → E′ is an m-isogeny if b tively, as in Theorem 4. Then The authors in (3) found the following two Q-rational points: Given an elliptic curve E, we denote the set of Q-rational points The nonnegative integer r is called the Mordell-Weil rank of E; it signifies the number of independent generators having infinite (i) the coordinates of φ involve rational functions with Q-rational ′ 1256911215674901177485830929368441344290 as E(Q). Since E(Q) is finitely generated, there exists a finite group E (Q) P = 2 order. 3 16027875241 r coefficients, and δ E(Q)tors and a nonnegative integer r such that E(Q) ≃ E(Q)tors×Z , φ E(Q) 786698395807855729596742215337306893212760400533639780 ! : 160278752413 : 1 ; where r is called the (Mordell-Weil) rank. Those elliptic curves with (ii) there are exactly m points in the kernel E(Q)[φ] of the map. 2 2 2 2 Cd1 : d1 w = 1 − d1 z 1 − d1 k z 419146355190134411415222739650581610769161840 Mazur’s Theorem e(ℓ) P4 = 92556465261312 E(Q)tors ≃ Z2 × Z8 are birationally equivalent to = d1 ≡± ℓ ;   105211386192778469849488967231903854157073005646773612879981880 We will be interested in the case where m = 2. ℓ∈Σ(k) has a Q-rational point (z,w)   : 92556465261313 : 1 . t4 − 6t2 +1 While the rank r is mysterious, the torsion subgroup is well- Y 2 2 2 2 Q y = (1−x )(1−k x ) where k = 2 2 for some t ∈ . understood. In 1977, Barry Mazur completely classified the struc- (t + 1) E(Q)  The torsion subgroup is generated by P1 and P2, and the points P3 ture of torsion subgroups of elliptic curves. Counting Cosets to Determine the Rank δ φˆ E′(Q) ! and P4 have infinite order. Hence 2 ≤ r ≤ 3.

Theorem 2 (Mazur, (5)). If E is an elliptic curve over Q, then ′ b C : d w2 = 1 + d z2 1 + d κ2 z2 The highest known rank of curves with this torsion subgroup is Proposition 3. Let E and E be elliptic curves over Q. Let e(ℓ) d2 2 2 2 E(Q) is isomorphic to one of the following 15 groups: = d2 ≡± ℓ . tors ′ ˆ ′ ˆ   P on E d2 = δ(P ) Point (z, w) on Cd2 r = 3. It is conjectured in (3) that this bound is obtained when φ : E → E and φ : E → E be 2-isogenies such that φ◦φ = [2] is the ℓ∈Σ(κ) has a Q-rational point (z,w)  Y b  ′ Q ˆ Q b b 9 (i) Zn, with 1 ≤ n ≤ 10 or n = 12; map which sends P 7→ P ⊕ P . Assume that E ( )[φ] = φ(E( )[2]). P1 −1 (1, 0) t = . Here,e(ℓ) is either 0 or 1.  296 Then, (ii) Z × Z , with 1 ≤ m ≤ 4. 305 8143806511 2 2m E(Q) E′(Q) E(Q) The various curves introduced in this theorem fit together in the P2 6477590 7992 , 200779379640 One knows that r ≥ 2 for this particular t. We focus on determining = . 2 E(Q) φ(E(Q)) φˆ(E′(Q)) diagrams: 116263507795895 119018475593848690746927861139 the exact rank of this particular elliptic curve. Here we denote Z as the cyclic group of order n. We will focus P3 2 683172154272384 , 163644731958920474581067710080 n φ φˆ more specifically on those curves with torsion subgroup Z2 × Z8. ′ ′ 9477908247062185 2802930777448484302006837377371105071 E o E E / E P4 7 147942254073677904 , 7311568666378397912349147334466220240 ˆ _?? ? Proof: Using the isogeny φ, we have the exact sequence: ?? Acknowledgements ?? ?? ′ ′ ?? Example E [φˆ] E (Q) φˆ E(Q) E(Q) The authors would like to thank the Summer Undergraduate {O} −→ −→ −−→ −→ −→ {O}. Cd1 ′ ′ ′ ′ ′ Cd2 P E d2 δ P z, w C φ(E[2]) φ(E(Q)) 2 E(Q) ˆ ′ on = ( ) Point ( ) on d2 Mathematical Sciences Research Institute (SUMSRI) at Miami Uni- φ(E (Q)) We consider these diagrams to be “2-covers” because the diagonal versity for the opportunity to conduct this research. They would 2 3 b b 7690763809 b Y + XY = X − 71813598680248384341084284771096244120 X Using the assumption above, Lagrange’s Theorem and the First maps involve quadratic polynomials. φ(P1) −1 932772960 , 0 also like to thank the National Science Foundation and the National E : + 234238430204114181370252185964622864112853337413958990400 Isomorphism Theorem imply the equalities: 87697 8623536769 Security Agency for their funding, as well as Residential Computing φ(P2) 6477590 77731080 , 6816782522760 has torsion subgroup E(Q)tors ≃ Z2 × Z8 generated by ′ ′ ′ (ResComp) at Miami University and the Rosen Center for Advanced E(Q) E (Q) E(Q) E (Q) E (Q) E(Q) 4-Covering Maps 402721445539793209371967 Computing (RCAC) at Purdue University for use of their machines. = = . 16689898742884224439568 , P = (4892533141966211376 : −2446266570983105688 : 1); 2E(Q) φ(E(Q)) 2E(Q) φ(E(Q)) φ(E(Q)) φˆ(E′(Q)) φ(P3) 2 546790296971729700371447998389073244144667329763 1 − 5319738216468998004248404711869988353017875184 ′ ′ ′′ ˆ′ ′′ ′ They are grateful to Michael Stoll for helpful conversations, Tom Theorem 6. Using φ : E → E and φ : E → E , define the P = (6793371071343566640 : 7739207808589340925333304680 : 1). 1434298275377041049916550061461 Farmer for careful reading of this document, Elizabeth Fowler for 2 78309867527655769246487587761 , ′ connecting homomorphisms: φ(P4) 7 If E(Q)tors ≃ Z2 × Z2m, then E (Q)[φˆ] = φ(E(Q)[2]) holds. 45852135158706046452821064687371285272034083270789515130064924 her constant support, and Edray Goins for his guidance. − 419825279533017414897518986779479787597378016792616598177777 E′′(Q) Q× Rank Records with Prescribed Torsion −→ , ′ ′ ′ × 2 δ : φ E (Q) (Q ) 2-Isogenies for Quartic Models The Group Law for Elliptic Curves In contrast to the torsion subgroup, less is known about the rank X Y 7−→ X Using our “2-cover”, we seek a point P5 such that d2 = δ(P5) = 37. ( : : 1) 4 + 40011942240487566721; r of an elliptic curve. As of 2006, the largest known rank satisfies To be more precise, we seek a Q-rational point (z,w) on the curve Theorem 4 (Goins, (4)). Say that E is an elliptic curve over Q E′(Q) Q× An elliptic curve E is a set of points which satisifies an equation of r ≥ 28. Andrej Dujella (2) has a listing of the largest known ranks −→ , b the form Y 2 = X3 + AX + B, where A and B are rational numbers ′ ˆ′ ′′ Q× 2 2 2 2 2 28387584 among families of elliptic curves with prescribed torsion. The table with torsion subgroup Z2 × Z8. δ : φ E (Q) ( ) C37 : 37 w = 1+37 z 1+37 κ z where κ = . 3 2 7662376225 such that the discriminant −16(4A + 27B ) =6 0. In particular, an below contains this information. (X : Y : 1) 7−→ X − 5001492780060945840. elliptic curve is a type of nonsingular cubic curve. (i) There exists a rational number t such that E is birationally ′ b Alternatively,b using our “4-cover”, we seek a point P5 = φ(P5) such Note that the highest known rank for elliptic curves E with ′ ′ Define E(Q) as the set of Q-rational points, where we append a equivalent to the curve Both δ and δ are injective group homomorphisms with finite im- that d = δ′(P ′) = δ(P ) = 37. To be more precise, we seek a E(Q)tors ≃ Z2 × Z8 is r = 3. 2 5 5 ′ ′ “point at infinity” O. 4 2 ages. To be precise, let Σ(κ ) = {82207, 92863} and Σ(k ) = Q 2 2 2 2 t − 6t +1 -rational point (z,w) on the curve y = (1 − x )(1 − k x ) where k = . b ′ 2 2 {2, 3, 5, 7, 37, 41, 61, 87697} be the set of primes dividing κ = (1 − b b (t + 1) 2 4 Torsion Highest ′ 2 2 ′ 2 ′ 932772960 Author(s) ′ ′ ′ C37 : 37 w = 1+37 z 1+37 k z where k = 7690763809. Subgroup Known Rank k )/(1 + k ) and k , respectively, as in Theorem 4. Then (ii) There exists a 2-isogeny φ : E → E′ in terms of the curve 3 P ∗ Q = (2,3) Z1 28 Elkies (2006) ′′ ′ E (Q) b Z2 18 Elkies (2006) 2 δ ′ 2 2 2 2 2t ′ ′ 2 φ E (Q) We used John Cremona’s algorithm QuarticMinimise() to find Z3 13 Eroshkin (2007, 2008) E : y = (1+ x )(1 + κ x ) where κ = 2 . ! t − 1 ′ Z4 12 Elkies (2006) ′ 2 2 ′2 2 “minimal” integral models for C37 and C then Michael Stoll’s pro- 1 37 ′ e ℓ Cd1 : d1 w = 1 − d1 z 1 − d1 κ z Q 0 1 Z5 6 Dujella - Lecacheux (2001) ( ) P −1 0 = ( , ) Moreover, E has torsion subgroup Z2 × Z4. = d1 ≡± ℓ ; ratpoints = ( , )   gram (9) to find rational points. We are running these on -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Eroshkin (2008) ℓ∈Σ(κ′) has a Q-rational point (z,w) b b ′ ′ ′′  Y  both Miami University’s 128-node high-powered computing cluster, Dujella - Eroshkin (2008) (iii) There exists a 2-isogeny φ : E → E , and hence a 4-isogeny Z6 8 ′ -1 Elkies (2008) ′ ′′ E (Q)  RedHawk, and Purdue University’s Euler. φ ◦ φ : E → E , in terms of the curve δ′ Dujella (2008) ˆ′ ′′ Q 3 φ E ( ) ! -2 Dujella - Kulesz (2001) ′′ 2 2 ′2 2 ′ 4(t − t) REFERENCES Z7 5 E : y = (1+ x )(1 + k x ) where k = . 2 Elkies (2006) 2 2 b C′ : d w2 = 1 + d z2 1 + d k′ z2 P ⊕ Q = (−2,3) (t + 1) e(ℓ) d2 2 2 2 John Cremona. mwrank and related programs for elliptic curves over Q. -3 = d2 ≡± ℓ . Z8 6 Elkies (2006)   http://www.maths.nott.ac.uk/personal/jec/mwrank/, 2006. Moreover, E′′ has torsion subgroup Z × Z . ℓ∈Σ(k′) has a Q-rational point (z,w) Dujella (2001) 2 2  Y b  Andrej Dujella. High rank elliptic curves with prescribed torsion. -4 MacLeod (2004) http://www.math.hr/ duje/tors/tors.html, 2007. The Group Law on an Elliptic Curve Z9 3 Here,e(ℓ) is either 0 or 1.  Eroshkin (2006) Jessica Flores, Kimberly Jones, Anne Rollick, James Weigandt. A Statistical Analysis of 2-Selmer Groups for Elliptic Curves with Torsion Subgroup Z2 × Z8. SUMSRI We may use geometry to turn E(Q) into a group. For P,Q ∈ Eroshkin - Dujella (2007) Example The various curves introduced in this theorem fit together in the Journal, 2007. Dujella (2005) Z10 4 Edray Herber Goins. SUMSRI Number Theory Research Seminar Lecture Notes. In E(Q), draw a line through P and Q. (If P = Q, we choose the diagrams: preparation, 2008. Elkies (2006) We focus on the case where t =9/296, which was first considered ′ ′ line tangent to the curve at P .) This line will intersect the curve Dujella (2001, 2005, 2006) φ φ φˆ φˆ ´ Z12 3 ′′ o ′ o ′′ / ′ / B. Mazur. Modular curves and the Eisenstein ideal. Inst. Hautes Etudes Sci. Publ. Rathbun (2003, 2006) E _? E _? E E ?E ? E Math., (47):33–186 (1978), 1977. at a third Q-rational point P ∗ Q – which may possibly be O. By in (3). Weierstrass models are as follows: ?? ?? ? ?? Z2 × Z2 14 Elkies (2005) ?? ? 2 3 ? ?? L. J. Mordell. Diophantine equations. Academic Press, London, 1969. reflecting this point about the x-axis, we obtain another Q-rational Y + XY = X − 71813598680248384341084284771096244120 X ?? ?? Elkies (2005) E : ′ ′ Henri Poincaré. Sur les propriétés arithmétiques des courbes algébriques. Journal de C o Cd1 C / point, denoted by P ⊕Q. For a graphical represention, see the figure Z2 × Z4 8 Eroshkin (2008) + 234238430204114181370252185964622864112853337413958990400; d1 d2 Cd2 mathématiques pures et appliquées, 7(5):161–233, 1901. above. Formally define Dujella - Eroshkin (2008) 2 3 We consider these diagrams to be “4-covers” because they both con- Joseph H. Silverman. The arithmetic of elliptic curves. Springer-Verlag, New York- ′ Y + XY = X − 71828384105861957682230266860325044120 X Berlin, 1986. Z2 × Z6 6 Elkies (2006) E : b b + 234137152575130885252407456517423577517272419831108430400; tain pairs of diagonal maps, each involving quadratic polynomials. P ⊕ Q =(P ∗ Q) ∗ O. Connell (2000) Michael Stoll. ratpoints-2.0.1. In the diagram on the right, a Q-rational point on the elliptic curve http://www.faculty.iu-bremen.de/stoll/programs/index.html/, 2008. Dujella (2000, 2001, 2006) 2 3 ′′ Y + XY = X − 87910414011578700645569436440051772120 X E (E′, respectively) will correspond to a Q-rational point on the Under this operation, the set E(Q) forms an abelian group with Z2 × Z8 3 Campbell - Goins (2003) E : + 121529333528097780380319085871651105820656760543386956800. homogeneous space C ,(C′ , respectively) having half as many the identity element O and inverse [−1]P = P ∗ O. For more Rathbun (2003, 2006) d2 d2 digits. information, see (8). Flores - Jones - Rollick - Weigandt (2007) This poster was prepared with Brian Wolven’s Poster LATEX b b macros v2.1.

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