Now, if you’ve ever taken a course in algebraic geometry or number theory, you might have heard of this concept before. But for those who haven’t, let me break it down for you in layman’s terms:

Dual isogeny = fancy math word that means “flipping things around”

Okay, okay I know that doesn’t really help much. Let’s dive a little deeper into what we mean by flipping things around. In algebraic geometry, an isogeny is essentially a map between two varieties (think of them as shapes or curves) that preserves certain properties. For example, if you have two elliptic curves and there’s a way to “glue” them together using some fancy math, then the resulting object is called an abelian variety. And if we can find another map that takes us back from this new object to our original elliptic curves (while preserving all of their properties), then we say that these two maps are isogenous.

Now, what’s so special about flipping things around? Well, it turns out that there’s a whole bunch of cool stuff you can do with dual isogenies! For example:

– They allow us to study the “dual” version of an object (e.g., if we have an elliptic curve, its dual is called a Jacobian variety)
– They help us understand how certain objects are related to each other (e.g., if two varieties are isogenous, then their duals must also be isogenous)
– And they can even give us new insights into the structure of these objects themselves!

So, without further ado, let’s dive headfirst into constructing our very own dual isogeny. First we need to choose a starting point. Let’s say that we have an elliptic curve E defined by the equation y^2 = x^3 + ax + b (where a and b are some arbitrary constants).

Now, in order to construct its dual variety J(E), we need to follow these steps:

1. Define a new set of points on our original elliptic curve E specifically, the “division points” that correspond to each point P on E. These division points are essentially the “dual” versions of P, and they’re defined as follows:

– If P has order n (i.e., it generates a cyclic subgroup of size n), then its dual is given by the sum of all the other points in this subgroup (modulo E). Otherwise, we can use some fancy math to construct the “division point” associated with P.

2. Define a new set of curves on our original elliptic curve E specifically, the “divisor classes” that correspond to each divisor D on E. These divisor classes are essentially the “dual” versions of D, and they’re defined as follows:

– If D is a principal divisor (i.e., it corresponds to a single point P), then its dual is given by the sum of all the other points in this subgroup (modulo E). Otherwise, we can use some fancy math to construct the “divisor class” associated with D.

3. Define a new set of functions on our original elliptic curve E specifically, the “dual isogeny” that takes us from E to J(E) (and vice versa). This dual isogeny is essentially the “flipping around” part we mentioned earlier!

– To construct this dual isogeny, we need to use some fancy math involving elliptic curves and their associated functions. But don’t worry you can just take our word for it that it works!

And there you have it the construction of a dual isogeny in all its glory! Of course, this was just a brief overview of what we covered today. If you want to learn more about dual isogenies (and other cool topics in algebraic geometry and number theory), I highly recommend checking out some of the resources listed below:

– “Algebraic Geometry” by David Mumford This book provides an excellent introduction to the subject, with a focus on modern techniques and applications.
– “Number Theory: An Algorithmic Approach” by Kenneth Rosen While not specifically focused on dual isogenies, this book covers many of the key concepts in number theory that are relevant for understanding them (e.g., modular forms, elliptic curves).
– “The Arithmetic and Geometry of Elliptic Curves” by Joseph H. Silverman This book provides a comprehensive overview of the subject, with an emphasis on applications to cryptography and coding theory.

And that’s all for today!