y^2=x^3-t^n*x-t^n

Our research considered elliptic surfaces En defined by the equation:

y^2=x^3-t^n*x-t^n (*)

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

r^2-2=0

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.

Blog Post #3: Solved for some coefficients

The past couple of weeks have been interesting. At this point in time, I understand the Sage programming language completely (for the domain of my research), which has allowed for faster progress. I tested the roots that Professor Fastenberg had done earlier, such as the sections when r=1, 2, 3, 7, 10, and 12. All of her roots worked, but since my Sage code differed from Professor Fastenberg’s previous code in Maple, I also discovered other roots. Additionally, by using the solve(…) function in Sage, it allowed me to solve each of the coefficients for the values necessary. The main issue that we encountered at this moment in time is that the roots that Sage puts out in the solve function are approximate. Our goal is to get the exact value, such as in terms of radicals. I tried searching the Sage information database for increased functionality regarding exact roots, but the engine has limits in this regard. Therefore, I need to devise another method to solve for exact roots, probably through a more complex algorithm.

In the upcoming weeks, I will be developing that new algorithm and working on solving for the coefficients of the higher powers of the functions. Another issue that I encountered was that at higher powers, the solve function becomes computationally inefficient. This means that the solver times out, due to the fact that it deals with an insanely large amount of roots, leading to an exponential growth in time.

Blog #2

During this period of time, I explored obtaining the coefficients of the group, y^2=x^3-t^n x-t^n. I specifically explored the groups when n = 3 and 1. I attempted to obtain the coefficients of the group y^2=x^3-t^3 x-t algebraically, by replacing at+b for x and ct^2+dt+e for y.

This resulted in (ct^2+dt+e)^2 = (at+b)^3-(at+b)t^3-t, obtaining a^3t^3 + 3a^2bt^2 + 3ab^2t + b^3 – at^4 – bt^3 – t

By doing this, we know that c^2 = -a, e^2 = b^3, 2cd = a^3-b, 2ce+d^2=3a^2b, de=3ab^2-1

I attempted to solve for these values in maple by using the collect(simplify(equation, variable to solve for)) function to go further, since the variables cannot be simplified any further using basic algebraic methods. Maple has many useful functions that include the simplify, collect, and solve functions that allow one variable to be isolated. Then, we can set all instances of a variable, such as “c,” into what it is equal to. We know that a = -c^2, so all instances of a can be replaced with (-c^2).

I hope to continue doing this over Winter Break and apply this methodology to more complex functions. I also hope that I will be able to develop a more efficient method, possibly not using Maple, so that I can compute the larger coefficients.

Blog #1: The Sections of y^2=x^3-t^n x-t^n

This research is a continuation on Professor Lisa Fastenberg’s research, “Mordell-Weil groups in procyclic extensions of a function field.” In her research, she discovered that the Mordell-Weil group of the sections of  have a maximum rank of 56 when n is a multiple of 2520. These sections are pullbacks of sections for   when n = 1, 2, 3, 7, 8, 10, 12, 15, 18, 20, 42. However, we currently lack an explicit list of such sections, and computations using conventional methods are difficult for larger values of n. Therefore, the goal of this research is to write a computer program on Maple to find all sections of , and show that these sections are generators of the Mordell Weil Group.

Currently, I am learning the Maple programming language and doing preliminary testing on these elliptic curves:

R

k, k’ Minimal Model at 0 Sections: (???)
1 1  y^2=x^3-xt-t (-1,i)
2 1  y^2=x^3-xt^2-t^2  (t, it) or (0, it)
3 2  y^2=x^3-xt^3-t^3  (t, it^2)

As I become more experienced in Maple, I will hopefully be able to replicate Professor Fastenberg’s previously computed sections, and continue to make further progress past what she did.