For some time now, I've had the pleasure of trying to describe my theorems to complete laypersons. It's fun, but I don't really know why, perhaps because it's something math types don't do very often. Maybe I just like to hear myself talk. In any case, here's my well-rehearsed explanation. It's not short enough to utter in one breath, but I feel pretty good about its brevity.

My result

There are mod 4 Galois representations that do not come from elliptic curves*.
*-if you want to get picky, this needs to be qualified a bit.

Elliptic Curves

These are fairly easy to describe in terms of high-school algebra. y^2=x^3+ax+b. That's an equation for an elliptic curve. The x and the y are variables. The a and the b are numbers (I only care when a and b are whole numbers). The curve consists of all the x and y that make the equation true for any particular choice of a and b. Every elliptic curve can be made to have an equation like that, but there are other ways to represent them too. Most important: an elliptic curve is not an ellipse. That's just a fluke of history that they have similar names.

Galois Representations

These are much more abstract. That is, in order for the name to make sense, you have to have a whole lot of school behind you, and only a graduate education in math seems to do it. But the idea of these objects is that they are good stand ins for a lot of other things Number Theorists care about, a way to talk about all of them in the same breath. They are a very general way of thinking about a large class of objects.

mod n Galois representations

Again, a tough one to zero in on very quickly, but you can think of them as a certain kind of Galois representation. More importantly, there is always a very easy way (for mathematicians) to get a mod n Galois representation from any elliptic curve. It's like shooting fish in a barrel. It's one of the first things I remember learning about elliptic curves.

mod 4 Galois representations

Well, if you remember anything from math, n means any number I want. And in this case I want 4. Why? Well, here it's the perfect number for what's not yet known and what I might be able to figure out. Also what happens when n=2 is very easy to see, so I have something to work with.
Mathematicians don't ask the kind of questions good teachers are supposed to ask, they are usually closed ended questions that have very short answers. Of course, they've been at it a while so what's short and easy to ask may be long and hard to answer.

The million dollar (I wish) question

If it's so easy to get a mod 4 representation from any elliptic curve, can you always do the reverse: get an elliptic curve from a mod 4 Galois representation?

My answer

No. In fact I found more than a million mod 4 Galois representations that don't come from any elliptic curve. This is important because we now know that mod 4 Galois representations are truly a more generic kind of object than elliptic curves.

How did I figure this out?

that's a topic for another day.