# Birational Equivalence of Diophantine Equations and Elliptic Curves

A while ago I saw this question Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$ which was very relevant to a undergraduate research paper I am currently working on. The answer given for this problem describes a method for determining solutions to diophantine equations by finding a "birational equivalence" to the solution set of an elliptic curve.

The result I am seeking depends on the solution set to a quartic diophantine equation of two variables and I believe that there is only one solution and no more (which was indicated by a program which checked possible values up to a high number).

So my question is, is there a general method for determining such a birational equivalence between solutions sets of diophantine equations and elliptic curves? And where is a good source for an undergraduate to learn to use these tools?

• I'm not quite sure what "a birational equivalence" between diophantine equations and elliptic curves should mean. When we want to find integer points on these "elliptic curves", it is a diophantine problem. Generally speaking it is much easier to identify rational points on the curves (and birational parameterization is one way to answer that), and it can be AFAIK a difficult problem to say whether integer points exist even given some rational solutions. Dec 28 '15 at 20:30
• Perhaps what you have in mind is the subject discussed here, at Ask SageMath. Dec 28 '15 at 20:33
• Certain curves are birationally equivalent to elliptic curves, so in this sense Diophantine equations give rise to elliptic curves. I think this is what the OP meant when he referred to birational equivalence between Diophantine equations and elliptic curves. Jan 2 '16 at 18:27
• Yes that is exactly it :) Jan 2 '16 at 20:32

The following shows how to transform the quartic $$y^2=ax^4+bx^3+cx^2+dx+e ,$$ with $a,b,c,d,e \in \mathbb{Q}$, into an equivalent elliptic curve.

Case 1:

We first consider the case when $a=1$, which is very common. If $a=\alpha^2$ for some rational $\alpha$, we substitute $y=Y/\alpha$ and $x=X/\alpha$, giving \begin{equation*} Y^2=X^4+\frac{b}{\alpha}X^3+cX^2+\alpha d X+\alpha^2 e \end{equation*} Thus, suppose $$y^2=x^4+bx^3+cx^2+dx+e$$

We describe the method given by Mordell on page $77$ of his book Diophantine Equations, with some minor modifications.

We first get rid of the cubic term by making the standard substitution $x=z-b/4$ giving \begin{equation*} y^2=z^4+fz^2+gz+h \end{equation*} where \begin{equation*} f=\frac{8c-3b^2}{8} \hspace{1cm} g=\frac{b^3-4bc+8d}{8} \hspace{1cm} h=\frac{-(3b^4-16b^2c+64bd-256e)}{256} \end{equation*}

We now get rid off the quartic term by defining $y=z^2+u+k$, where $u$ is a new variable and $k$ is a constant to be determined. This gives the quadratic in $z$ \begin{equation*} (f-2(k+u))z^2+gz+h-k^2-u(2k+u)=0 \end{equation*}

For $x$ to be rational, then $z$ should be rational, so the discriminant of this quadratic would have to be a rational square. The discriminant is a cubic in $u$. We do not get a term in $u^2$ if we make $k=f/6$, giving \begin{equation*} D^2=-8u^3+2\frac{f^2+12h}{3}u+\frac{2f^3-72fh+27g^2}{27} \end{equation*} and, if we substitute the formulae for $f,g,h$, and clear denominators we have $$G^2=H^3+27(3bd-c^2-12e)H+27(27b^2e-9bcd+2c^3-72ce+27d^2)$$ with $$x=\frac{2G-3bH+9(bc-6d)}{12H-9(3b^2-8c)}$$ and $$y= \pm \frac{18x^2+9bx+3c-H}{18}$$

Case 2:

If $a \ne \alpha^2$, we need a rational point $(p,q)$ lying on the quartic.

Let $z=1/(x-p)$, so that $x=p+1/z$ giving \begin{equation*} y^2z^4=(ap^4+bp^3+cp^2+dp+e)z^4+(4ap^3+3bp^2+2cp+d)z^3+ \end{equation*} \begin{equation*} (6ap^2+3bp+c)z^2+(4ap+b)z+a \end{equation*} where the coefficient of $z^4$ is, in fact, $q^2$. Define, $y=w/z^2$, and then $z=u/q^2, \, w=v/q^3$ giving $$v^2=u^4+(4ap^3+3bp^2+2cp+d)u^3+q^2(6ap^2+3bp+c)u^2+$$ \begin{equation*} q^4(4ap+b)u+aq^6 \equiv u^4+fu^3+gu^2+hu+k \end{equation*}

We now, essentially, complete the square. We can write \begin{equation*} y^2=u^4+fu^3+gu^2+hu+k=(u^2+mu+n)^2+(su+t) \end{equation*} if we set \begin{equation*} m=\frac{f}{2} \hspace{1cm} n=\frac{4g-f^2}{8} \hspace{1cm} s=\frac{f^3-4fg+8h}{8} \end{equation*} and \begin{equation*} t=\frac{-(f^4-8f^2g+16(g^2-4k))}{64} \end{equation*}

This gives \begin{equation*} (y+u^2+mu+n)(y-u^2-mu-n)=su+t \end{equation*} and if we define $y+u^2+mu+n=Z$ we have $$2(u^2+mu+n)=Z-\frac{su+t}{Z}$$

Multiply both sides by $Z^2$, giving \begin{equation*} 2u^2Z^2+2muZ^2+suZ=Z^3-2nZ^2-tZ \end{equation*} which, on defining $W=uZ$, gives \begin{equation*} 2W^2+2mWZ+sW=Z^3-2nZ^2-tZ \end{equation*}

Define, $Z=X/2$ and $W=Y/4$ giving $$Y^2+2mXY+2sY=X^3-4nX^2-4tX$$ which is an elliptic curve. If we define $Y=G-s-mX$, we transform to the form $$G^2=X^3+(m^2-4n)X^2+(2ms-4t)X+s^2$$

All of the above are easy to program in any symbolic package. Hopefully, I have all the formulae correct.

• Thank you for taking the time to explain this in detail and provide a reference. I really appreciate it. Dec 29 '15 at 14:20
• This will seem to solve one of my longstanding problems. I'm going to try programming this one out and report back. May 14 '16 at 15:39

(This is a long comment re MacLeod's answer.)

We can also combine the two cases together. Assume a quartic polynomial to be made a square,

$$pu^4+qu^3+ru^2+su+t=z_1^2\tag1$$

has a known rational point, call it $$w$$. We substitute $$u=v+w$$ and collect the new variable $$v$$,

$$c_4v^4+c_3v^3+c_2v^2+c_1v+\color{blue}{c_0^2}=z_2^2\tag2$$

where the $$c_i$$ are polynomials in $$w$$ and the coefficients of $$(1)$$. The constant term of $$(2)$$ turns out to be a square, specifically, $$c_0^2:=pw^4+qw^3+rw^2+sw+t$$.

Let $$v=1/x,\,$$ $$z_2=c_0\, z_3/x^2$$ and $$(2)$$ becomes,

$$x^4+d_3x^3+d_2x^2+d_1x+d_0=z_3^2\tag3$$

which is the Case 1 and then be solved as explained by MacLeod.