# elliptic curve

## 1 Basics

An over a field $K$ is a projective nonsingular  algebraic curve  $E$ over $K$ of genus 1 together with a point $O$ of $E$ defined over $K$. The word “genus” is taken here in the algebraic geometry   sense, and has no relation    with the topological notion of genus (defined as $1-\chi/2$, where $\chi$ is the Euler characteristic) except when the field of definition $K$ is the complex numbers   $\mathbb{C}$.

Using the Riemann-Roch theorem for curves, one can show that every elliptic curve $E$ is the zero set  of a Weierstrass equation of the form

 $E:y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6},$

for some $a_{i}\in K$, where the polynomial    on the right hand side has no double roots. When $K$ has characteristic  other than 2 or 3, one can further simpify this Weierstrass equation into the form

 $E:y^{2}=x^{3}-27c_{4}x-54c_{6}.$

The extremely strange numbering of the coefficients is an artifact of the process by which the above equations are derived. Also, note that these equation are for affine curves; to translate  them to projective curves, one has to homogenize the equations (replace $x$ with $X/Z$, and $y$ with $Y/Z$).

## 2 Examples

We present here some pictures of elliptic curves over the field $\mathbb{R}$ of real numbers. These pictures are in some sense not representative of most of the elliptic curves that people work with, since many of the interesting cases tend to be of elliptic curves over algebraically closed fields. However, curves over the complex numbers (or, even worse, over algebraically closed fields in characteristic $p$) are very difficult to graph in three dimensions   , let alone two.

Figure 1 is a graph of the elliptic curve $y^{2}=x^{3}-x$.

Figure 2 shows the graph of $y^{2}=x^{3}-x+1$:

Finally, Figures 3 and 4 are examples of algebraic curves that are not elliptic curves. Both of these curves have singularities at the origin. Figure 3: Graph of y2=x2⁢(x+1). Has two tangents at the origin. Figure 4: Graph of y2=x3. Has a cusp at the origin.

## 4 Elliptic Curves over $\mathbb{C}$

Over the complex numbers, the general correspondence between algebraic  and analytic theory specializes in the elliptic curves case to yield some very useful insights into the structure of elliptic curves over $\mathbb{C}$. The starting point for this investigation is the Weierstrass ${\mathfrak{p}}$function, which we define here.

###### Definition 2.

For any lattice $L$ in $\mathbb{C}$, the Weierstrass ${\mathfrak{p}}_{L}$–function of $L$ is the function ${\mathfrak{p}}_{L}:\mathbb{C}\longrightarrow\mathbb{C}$ given by

 ${\mathfrak{p}}_{L}(z):=\frac{1}{z^{2}}+\sum_{\omega\in L\setminus\{0\}}\left(% \frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right).$

When the lattice $L$ is clear from context, it is customary to suppress it from the notation and simply write ${\mathfrak{p}}$ for the Weierstrass ${\mathfrak{p}}$–function.

Properties of the Weierstrass ${\mathfrak{p}}$–function:

• ${\mathfrak{p}}(z)$ is a meromorphic function with double poles at points in $L$.

• ${\mathfrak{p}}(z)$ is constant on each coset of $\mathbb{C}/L$.

• ${\mathfrak{p}}(z)$ satisfies the differential equation

 ${\mathfrak{p}}^{\prime}(z)^{2}=4{\mathfrak{p}}(z)^{3}-g_{2}{\mathfrak{p}}(z)-g% _{3}$

where the constants $g_{2}$ and $g_{3}$ are given by

 $\displaystyle g_{2}$ $\displaystyle:=$ $\displaystyle 60\sum_{\omega\in L\setminus\{0\}}\frac{1}{\omega^{4}}$ $\displaystyle g_{3}$ $\displaystyle:=$ $\displaystyle 140\sum_{\omega\in L\setminus\{0\}}\frac{1}{\omega^{6}}$

The last property above implies that, for any $z\in\mathbb{C}/L$, the point $({\mathfrak{p}}(z),{\mathfrak{p}}^{\prime}(z))$ lies on the elliptic curve $E:y^{2}=4x^{3}-g_{2}x-g_{3}$. Let $\phi:\mathbb{C}/L\longrightarrow E$ be the map given by

 $\phi(z):=\begin{cases}({\mathfrak{p}}(z),{\mathfrak{p}}^{\prime}(z))&z\notin L% \\ \infty&z\in L\end{cases}$

(where $\infty$ denotes the point at infinity on $E$). Then $\phi$ is actually a bijection (!), and moreover the map $\phi:\mathbb{C}/L\longrightarrow E$ is an isomorphism        of Riemann surfaces as well as a group isomorphism (with the addition operation on $\mathbb{C}/L$ inherited from $\mathbb{C}$, and the elliptic curve group operation on $E$).

We can go even further: it turns out that every elliptic curve $E$ over $\mathbb{C}$ can be obtained in this way from some lattice $L$. More precisely, the following is true:

###### Theorem 3.
1. 1.

For every elliptic curve $E:y^{2}=4x^{3}-bx-c$ over $\mathbb{C}$, there is a unique lattice $L\subset\mathbb{C}$ whose constants $g_{2}$ and $g_{3}$ satisfy $b=g_{2}$ and $c=g_{3}$.

2. 2.

Two elliptic curves $E$ and $E^{\prime}$ over $\mathbb{C}$ are isomorphic if and only if their corresponding lattices $L$ and $L^{\prime}$ satisfy the equation $L^{\prime}=\alpha L$ for some scalar $\alpha\in\mathbb{C}$.

## References

• 1 Dale Husemoller, Elliptic Curves. Springer–Verlag, New York, 1997.
• 2 James Milne, Elliptic Curves, online course notes. http://www.jmilne.org/math/CourseNotes/math679.htmlhttp://www.jmilne.org/math/CourseNotes/math679.html
• 3 Joseph H. Silverman, . Springer–Verlag, New York, 1986.
 Title elliptic curve Canonical name EllipticCurve Date of creation 2013-03-22 12:03:02 Last modified on 2013-03-22 12:03:02 Owner djao (24) Last modified by djao (24) Numerical id 32 Author djao (24) Entry type Definition Classification msc 11G07 Classification msc 11G05 Classification msc 14H52 Related topic Isogeny  Related topic ComplexMultiplication Related topic RankOfAnEllipticCurve Related topic HeightFunction Related topic LSeriesOfAnEllipticCurve Related topic BirchAndSwinnertonDyerConjecture Related topic JInvariant Related topic MordellWeilTheorem Related topic ConductorOfAnEllipticCurve Defines $\mathfrak{p}$-function