Edray H Goins
SUMSRI 2008
- Samuel Ivy, Brett Jefferson, Michele Josey, Cheryl Outing, Clifford Taylor, and Staci White
4-Covering Maps on Elliptic Curves with Torsion Subgroup Z2 x Z8
SUMSRI Journal, (2008) - ABSTRACT: In this exposition we consider elliptic curves over Q with the torsion subgroup Z2 x Z8. In particular, we discuss how to determine the rank of the curve E: y2=(1-x2)(1-k2x2), where k=(t4-6t2+1)/(t2+1)2 and t = 9/296. We use a 4-covering map Ĉd2' → Ĉd2 → E in terms of homogeneous spaces for d2 ∈ { -1, 6477590, 2, 7, 37 }. We provide a method to show that the Mordell-Weil group is E(Q) ⋍ Z2 x Z8 x Z3, which would settle a conjecture of Flores-Jones-Rollick-Weigandt and Rathbun.
- Jessica Flores, Kimberly Jones, Anne Rollick, and James Weigandt
A Statistical Analysis of 2-Selmer Groups for Elliptic Curves with Torsion Subgroup Z2 x Z8
SUMSRI Journal, (2007) - ABSTRACT: We consider elliptic curves over Q with torsion subgroup Z2 x Z8. These curves are birationally equivalent to y2 = (1-x2)(1-k2x2) where k= (a4-6a2b2+b4)/(a2+b2)2 for some integers a and b. We perform a computational analysis on the 3148208 curves corresponding to |a|, |b|≤ 5000. The largest rank known in this family is r=3; there are 13 examples in the literature. We exhibit 3 more. In an attempt to find such curves of larger rank, we perform a statistical analysis of the distribution of the ranks of the 2-Selmer groups.
- Terris D. Brooks, Elizabeth A. Fowler, Katherine C. Hastings, Danielle L. Hiance, and Matthew A. Zimmerman
Elliptic Curves with Torsion Subgroup Z2 x Z8: Does a Rank 4 Curve Exist?
SUMSRI Journal, (2006) - ABSTRACT: We consider elliptic curves over Q with torsion subgroup Z2 x Z8. These curves are birationally equivalent to y2 = (1-x2)(1-k2x2) where k = (t4-6t2+1)/(t2+1)2 for some rational number t. The largest known rank for such curves is 3. In this paper we search for a curve of rank at least 4 by computing ranks for t=a/b with |a|, |b| ≤ 2000.
- Kathleen P. Ansaldi, Allison R. Ford, Jennifer L. George, Kevin M. Mugo, and Charles E. Phifer
In search of an 8: rank computations on a family of quartic curves
SUMSRI Journal, (2005) - ABSTRACT: We consider the family of elliptic curves y2 = (1-x2)(1-k2x2) for rational numbers k ≠ -1, 0, 1. Every rational elliptic curve with torsion subgroup either Z2 x Z4 or Z2 x Z8 is birationally equivalent to this quartic curve for some k. We use this canonical form to search for such curves with large rank.
Our algorithm consists of the following steps. We compute a list of rational k by considering those associated to a given list of rational points (x,y). We then eliminate certain k by considering the associated 2-Selmer groups. Finally, we use Cremona's \texttt{mwrank} to find the ranks. Using these steps, we found two elliptic curves with Mordell-Weil group E(Q) ⋍ Z2 x Z4 x Z6.
- Jarrod A. Cunningham, Nancy Ho, Karen Lostritto, Jon A. Middleton, and Nikia T. Thomas
On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell
SUMSRI Journal, (2004) - Jarrod A. Cunningham, Nancy Ho, Karen Lostritto, Jon A. Middleton, and Nikia T. Thomas
On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell
Rose Hulman Institute Undergraduate Mathematics Journal, Vol. 7-2 (2006) - ABSTRACT: In 1659, John Pell and Johann Rahn wrote a text which explained how to find all integer solutions to the quadratic equation u2-d v2 = 1. In 1909, Axel Thue showed that the corresponding cubic equation u3-d v3 = 1 has finitely many integer solutions, so it remains to examine their rational solutions. Our goal was to find "large" rational solutions i.e., a sequence of rational points (un, vn) which increase without bound as n increases without bound. Such cubic equations are birationally equivalent to elliptic curves of the form y2 = x3-D. The rational points on an elliptic curve form an abelian group, so a "large" rational point (u,v) maps to a rational point (x,y) of "approximate" order 3. Following an idea of Zagier, we compute such rational points using continued fractions of elliptic logarithms.
We divide our discussion into two parts. The first concerns Pell's quadratic equation. We give an informal discussion of the history of the equation, illuminate the relation with the theory of groups, and review known results on properties of integer solutions through the use of continued fractions. The second concerns the more general equation uN-d vN = 1. We explain why N=3 is the most interesting exponent, present the relation with elliptic curves, and investigate properties of rational solutions through the use of elliptic integrals.