Basics of Arithmetic Geometry

Here is a mathematics problem well-known since ancient times: Find integers a, b, and c that solve the famous equation in the Pythagorean theorem,

\displaystyle a^{2}+b^{2}=c^{2}

Examples are a=3, b=4, c=5, and a=5, b=12, c=13 (a and b are of course interchangeable).

The general solution was already known to the ancient Greek mathematician Euclid. Let m and n be integers; then a, b, and c are given by

\displaystyle a=m^{2}-n^{2}
\displaystyle b=2m^{2}n^{2}
\displaystyle c=m^{2}+n^{2}

Direct substitution and a little algebra completes the proof.

Now for some geometry. If we divide both sides of the equation by c^{2}, and let x=\frac{a}{c}, and y=\frac{b}{c}, then the equation becomes

x^{2}+y^{2}=1

which is the equation of a circle of radius 1 centered at the origin. The problem of finding integer solutions to the equation of the Pythagorean theorem now becomes the problem of finding points in the unit circle whose coordinates are rational numbers.

There are analogous problems of finding “rational points” in “shapes” other than circles (the technical term for shapes described by polynomial equations is “variety”). For the other quadratic equations like the conic sections (parabola, hyperbola, and ellipse) this problem has already been solved.

However for cubic equations (like the so-called “elliptic curves”) and equations with an even higher degree this is still a very fruitful area of research, part of a field of mathematics called arithmetic geometry (also called Diophantine geometry).

One famous theorem in this field is Faltings’ theorem (formerly the Mordell conjecture): The number of rational points on a curve (a curve is a one-dimensional variety – take note that over the complex numbers this is actually a surface) with rational coefficients and genus greater than one (the genus is a number related to the degree) is finite.

References:

Diophantine Geometry on Wikipedia

Diophantine Equation on Wikipedia

Elliptic Curve on Wikipedia

Faltings’s Theorem on Wikipedia

Rational Points on Elliptic Curves by Joseph H. Silverman

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