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.