What Is the P-adic Volume of the P-adic Circle x² + y² = 1?

[LordCalculus]

Find the p-adic volume of the p-adic circle x^2 + y^2 = 1.

This isn't hw.

 

[Goongyae]

>Find the p-adic volume of the p-adic circle x^2 + y^2 = 1.

p-adic pi?

 

[ebola1717]

I would guess you'd need measure and integration. I think Koblitz's P-adic numbers, p-adic analysis, and zeta-functions covers this. That said, are you sure this curve encloses a finite area?

 

[LordCalculus]

I'm not even sure what a p-adic number is. The professor handed out this sheet for us to "enjoy." It's not collected or anything. It's just for us - for fun. This was one of the questions on the sheet, and it doesn't elaborate on it at all. So I have no idea if it's a finite area.

 

[blue_raver22]

I can provide an explanation of the p-adic volume of the p-adic circle. The p-adic numbers are a type of number system that extends the rational numbers and are used in number theory and algebraic geometry. The p-adic volume is a measure of the size or magnitude of a geometric object in the p-adic numbers.

To find the p-adic volume of the p-adic circle x^2 + y^2 = 1, we first need to define what we mean by "volume" in the p-adic numbers. In the real numbers, volume is measured in terms of length, width, and height. However, in the p-adic numbers, we use a different concept of "volume" called the Haar measure.

The Haar measure is a way of assigning a measure or size to a geometric object in the p-adic numbers. It is defined in such a way that it is invariant under certain transformations, such as translations and rotations. In other words, the Haar measure of an object remains the same even if we move or rotate it.

In the case of the p-adic circle x^2 + y^2 = 1, we can use the Haar measure to find its p-adic volume. Since the circle is invariant under rotations, we can choose any point on the circle as the center and calculate the Haar measure of the circle around that point. This Haar measure will be the same for all points on the circle, so we can simply choose the center to be (0,0).

Using the Haar measure, we can calculate the p-adic volume of the circle as the integral of 1 over the circle. This integral can be evaluated using techniques from p-adic analysis and algebraic geometry. The result will be a p-adic number, representing the p-adic volume of the circle.

In summary, the p-adic volume of the p-adic circle x^2 + y^2 = 1 can be found using the Haar measure, which assigns a measure to geometric objects in the p-adic numbers. This measure is invariant under certain transformations and can be calculated using techniques from p-adic analysis and algebraic geometry.