# SUMSRI Number Theory Research Seminar

The Summer Undergraduate Mathematical Sciences Research Institute (SUMSRI) is a program hosted by the Department of Mathematics at Miami University. It is a research experience for talented undergraduate students in the mathematical sciences who are interested in pursuing advanced degrees. Because of the shortage of minority and female mathematical scientists, they are especially interested in, but not limited to, African Americans and other underrepresented minorities and women. Here are links to number theory projects during my five years as the leader of the Number Theory Research Seminar:

Related information can be found at Dujella's Rank Records for E(Q)tors ⋍ Z2 x Z8 page. Last updated on October 2, 2016.

SUMSRI 2008:
4-Covering Maps on Elliptic Curves with Torsion Subgroup Z2 x Z8 • Samuel Ivy (Morehouse College), Brett Jefferson (Morgan State University), Michele Josey (North Carolina Central University), Cheryl Outing (Spelman College), Clifford Taylor (Grand Valley State University), and Staci White (Shawnee State University); co-mentored by Elizabeth Fowler (University of Tennessee).

• 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 d2' → 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.

• [Slides from Final Presentation #1] | [Slides from Final Presentation #2] | [Poster] | [Final Report] | [SUMSRI Journal (2008)]

SUMSRI 2007:
A Statistical Analysis of 2-Selmer Groups for Elliptic Curves with Torsion Subgroup Z2 x Z8 • Jessica Flores (University of Puerto Rico at Humacao), Kimberly Jones (Savannah State University), Anne Rollick (John Carroll University), and James Weigandt (Purdue University); co-mentored by Maria Salcedo (Purdue University).

• 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.

• [Slides from Final Presentation #1] | [Slides from Final Presentation #2] | [Poster] | [Final Report] | [SUMSRI Journal (2007)]

SUMSRI 2006:
Elliptic Curves with Torsion Subgroup Z2 x Z8: Does a Rank 4 Curve Exist? • Terris D. Brooks (Central State University), Elizabeth A. Fowler (Maryville College), Katherine C. Hastings (Baldwin Wallace College), Danielle L. Hiance (Campbellsville University), and Matthew A. Zimmerman (Central State University); co-mentored by Holly Attenborough (Miami University/Indiana University).

• 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.

• [Poster] | [Final Report] | [SUMSRI Journal (2006)]

SUMSRI 2005:
In Search of an 8: Rank Computations on a Family of Quartic Curves • Kathleen P. Ansaldi (Loyola College of Maryland), Allison R. Ford (Mary Baldwin College), Jennifer L. George (Miami University), Kevin M. Mugo (Otterbein College), and Charles E. Phifer (Morehouse College); co-mentored with Lakeshia Leggett (Howard University).

• 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.

• [Slides from Final Presentation] | [Poster] | [Final Report] | [SUMSRI Journal (2005)]

SUMSRI 2004:
On Large Rational Solutions of Cubic Thue Equations: What Thue Did to Pell • Jarrod A. Cunningham (University of South Alabama), Nancy Ho (Mills College), Karen Lostritto (Brown University), Jon A. Middleton (SUNY Buffalo), and Nikia T. Thomas (Morgan State University); co-mentored by Lakeshia Leggett (Howard University).

• 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.

• [Slides from Final Presentation] | [Poster] | [Final Report] | [SUMSRI Journal (2004)]

• Rose Hulman Institute Undergraduate Mathematics Journal, Vol. 7-2 (2006)