Calculating Genus of a Curve Using Falting's Theorem

[bitsmath]

Hello!
I would like to learn finding genus of my curve $x^2-x+y-y^5$. Also, I want to know how to apply Falting's Theorem to conclude finitely many rational points of my curve.
Thanks in advance!

 

[mathbalarka]

Where did you find this problem?

For the question, Falting's theorem states that if the genus of a given curve is $> 1$ then it has finitely many $\Bbb Q$-points. So having the genus automatically answers your second question about Falting, so that part is superfluous.

Not really sure how to compute genus of this particular curve. Have you looked at the corresponding Riemannsurface over $\Bbb P^1$? Perhaps Riemann-Hurwitz or some such?

 

[bitsmath]

Hello!
I would like to learn finding genus of my curve x^2-x+y-y^5 = 0. This is no doubt, it is hyper-elliptic curve and has genus 2. But, I do not know the technique of finding genus. Awaiting suitable reply.

Thanks in advance! Please explain

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Where did you find this problem?

For the question, Falting's theorem states that if the genus of a given curve is $> 1$ then it has finitely many $\Bbb Q$-points. So having the genus automatically answers your second question about Falting, so that part is superfluous.

Not really sure how to compute genus of this particular curve. Have you looked at the corresponding Riemannsurface over $\Bbb P^1$? Perhaps Riemann-Hurwitz or some such?[Sir, I got your message and I re-edited my question by equating to 0. Please explain how to find genus of my curve. This problem is self made.]

 

[mathbalarka]

Well, I have already given you a plausible way to approach this. There are pretty well-known formulas for calculating genus of Riemann surfaces, try them out! If you don't know how to compute genus of Riemann surfaces, try getting a decent book.

This is no doubt, it is hyper-elliptic curve and has genus 2.

If you already know that it has genus 2, what is your question?

 

[bitsmath]

Well, I have already given you a plausible way to approach this. There are pretty well-known formulas for calculating genus of Riemann surfaces, try them out! If you don't know how to compute genus of Riemann surfaces, try getting a decent book.
If you already know that it has genus 2, what is your question?

I Don't know how to find genus of my curve. I understand by your quote, it has 2. But I want to learn how you came to know that, genus of my curve is 2?[/QUOTE]