Our research considered elliptic surfaces En defined by the equation:
A section of En is a pair of rational expressions (x(t), y(t) that satisfy the above equation. Because the elliptic surface can be viewed as an elliptic curve over the function field C(t), the Mordell‐Weil Theorem says that the group of sections is finitely generated. The theorem means that while there are infinitely many sections for each n, these sections can be constructed from a finite set of sections, the generators of the Mordell‐Weil group, using a process that has the same properties as regular addition. Our goal was to find these generators (or perhaps multiples of these generators). In particular, finding generators for En where n= 1,2,3,7,8,10,12,15,18,20,42, can be used to construct generators for all values of n.
I spent the summer and much of the fall semester learning about elliptic curves to prepare myself for the project. I also needed to familiarize myself with the Sage programming language which I thought we could use. To do this, I wrote various programs in Sage to tackle the problem of finding the generators of En. While my programs worked for smaller values of n, such as when n was 1, 2, or 3, they didn’t work as well for larger values. They didn’t work because we were interested in exact solutions rather than decimal approximations. Exact solutions take much more computing power than approximations. For example, solving
gives us the exact solutions r = ±√2. However, the decimal approximations are r = ±1.4142. The decimal approximations can only be as long as the number of digits presented. However, when we looked at the generators of En, for some values of n, the exact solution of the coefficient could potentially be an entire page or longer, with multiple roots within roots. We then knew that Sage was not appropriate for this purposes, and so we looked for other programming options. After exploring other options, I decided that Mathematica would be more efficient due to its more efficient algorithm that could solve polynomial equations. I then learned Mathematica and then wrote a program that would work for all values of n.
The general algorithm was that I let x(t) = at^2+bt+c, and y(t) = dt^3+et^2+ft+g, and plug it into the equation (*), and then solve for the coefficients a, b, c, d, e, f, and g. My program was able to generate solutions for n=1, 2, 3, 7, 8, by using this parametric substitution. For the remaining values, I needed to use polynomials of higher degree. I was able to find a large number of candidates for generators for all values of n.
The next steps in this research will be to compute intersection numbers for certain choices to determine which are generators of the Mordell-Weil group, or at least a subgroup of finite index. Once we can discover the generators for the groups, we can look at the degree of the extension field that these groups are defined over.