Abelian variety

An abelian variety is a special type of mathematical object that has both addition and multiplication.

Let's start with addition. Imagine you have a box with some objects inside. When you put two objects together, you get a new object that represents the sum of the two original objects. This is similar to how you can add two numbers together to get a new number. An abelian variety behaves in a similar way, but instead of objects, we have points on a curve.

Now, let's talk about multiplication. When you have a number and you multiply it by another number, you get a new number that represents the product of the two original numbers. In an abelian variety, instead of numbers, we have points on a curve. When you multiply two points together, you get a new point that represents the product of the two original points.

The really cool thing about abelian varieties is that they have some special properties. For example, if you start with a point on the curve and you add it to itself multiple times, you will get a new point that is somehow related to the original point. This is called a group structure. It means that you can keep adding and multiplying points on the curve, and you will always get another point on the curve.

Another important idea in abelian varieties is symmetry. Think of symmetry like a mirror. When you look in a mirror, you see a reflection of yourself. In an abelian variety, you can reflect a point on the curve and get another point that is somehow related to the original point. This symmetry is really useful for solving problems and studying the properties of abelian varieties.

So, in summary, an abelian variety is a special mathematical object that has both addition and multiplication. It has a group structure, which means you can keep adding and multiplying points on the curve and always get another point on the curve. It also has symmetry, which helps us study and solve problems related to abelian varieties.