MATH 470 Lecture 22

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Algebraic Curves

Out of 52 fields medals , 25 have gone to work in or around algebraic geometry

Newton Polygons

If $K$ is a field (e.g. $\mathbb{\mathbb{Q}}$, $\mathbb{ℝ}$, $\mathbb{ℂ}$, $\mathbb{\mathbb{Z}}/19\mathbb{\mathbb{Z}}$)

$K$ adjoint $x$ and $y$ (written $K[x,y]$ is the set of polynomials in the two variables $x$ and $y$ with coefficients in $K$.

For any polynomial $f\in K[x,y]$, written $f(x,y)=\sum _{(a_1,a_2)\in {\mathbb{\mathbb{Z}}}^2}{c}_a{x}^{a_1}{y}^{a_2}$

We define the support of $f$ to be $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)=\{(a_1,a_2)\mid c_a\ne 0\}$

For example, $x+y-3=x^1{y}^0+x^0{y}^1-3x^0{y}^0$ has support $\{(1,0),(0,1),(0,0)\}$

If we plot these as points on a graph, this forms the Newton polygon, a convex hull of $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)$

Algebraic Curves

An algebraic curve over $K$ ($/K$) is just the zero set in $K^2$ of a non-constant polynomial $f\in K[x,y]$.

For example, a circle represents the rational roots of $f(x,y)=x^2+y^2-1$.

what about $y^2-(x+1)$ over $\mathbb{\mathbb{Z}}/5\mathbb{\mathbb{Z}}$? The squares mod 5 are {0, 1, 4}:

$x$	$x+1$	$y$?
0	1	±1
1	2	no
2	3	no
3	4	±2
4	0	0

Singularities

$\zeta =({\zeta }_1,{\zeta }_2)$ is a singularity (or degeneracy, singular point, or degenerate point) for the curve defined by $f$ iff

$f({\zeta }_1,{\zeta }_2)={\frac{\partial f}{\partial x}|}_{({\zeta }_1,{\zeta }_2)}={\frac{\partial f}{\partial y}|}_{({\zeta }_1,{\zeta }_2)}=0$

For example, the curve $f(x,y)=x^2+y^2-1$ represents a circle of radius 1 centered at the origin.

Then $\frac{\partial f}{\partial x}=2x$ and $\frac{\partial f}{\partial y}=2y$, so the only singularity would be at $(0,0)$, which is not on the curve. Therefore, the curve is smooth, (i.e. no singularities)

Another example: $y^2=x^3$ (rewritten as $y^2-x^3$) has $\frac{\partial f}{\partial x}=-3x^2$ and $\frac{\partial f}{\partial y}=2y$, which are both 0 at $(0,0)$, and that is a point on the curve!

Elliptic Curve

We call the zero set $C$ of $f\in K[x,y]$ an elliptic curve iff

1. the Newton polygon of $f$ has exactly one lattice point in its interior.
2. $C$ is smooth.

For example, $y^2+a_{1}xy-(a_2{x}^3+a_3{x}^2+a_{4}x+a_6)$ (Weierstrass Normal Form) has newton polygon

Now for most choices of $a_i$, you get an elliptic curve!

For example, $\overrightarrow{a}=(0,1,-6,11,\mathrm{N}\mathrm{U}\mathrm{L}\mathrm{L},-6)$ gives a smooth curve:

Recall that $y=a{x}^2+bx+c$ has 0, 1, or 2 real roots, and the discriminant ($b^2-4ac$) tells you this.

For cubics, we also have a determinant. A simplified version for $y=x^3+bx+c$ is $4b^3+27c^2$ (note this is not an elliptic curve since its Newton polygon has no interior points)

FACT: $y^2=x^3+bx+c$ defines a smooth curve (i.e. an elliptic curve) iff the descriminant is nonzero. (this is what "most" meant)

Alternate Notation: Edwards Normal Form

$x^2+y^2-a^2+b^2{x}^2{y}^2$

Newton polygon is a square { (0,0), (2,0), (2,2), (0,2) }, and has exactly one lattice point on its interior, namely (1,1)

FACTs:

1. Any elliptical curve, after a change of variables, can be put in Weierstrass Normal Form (or Edwards Normal Form)
2. Elliptic curve addition is much faster for curves in Edwards Normal Form.