The discussion centers on calculating the genus of the curve defined by the equation $x^2 - x + y - y^5 = 0$ and applying Falting's Theorem to determine the number of rational points. It is established that this curve is hyper-elliptic and has a genus of 2. The participants suggest using Riemann surfaces and the Riemann-Hurwitz formula as methods for computing the genus, emphasizing the importance of understanding these concepts to apply Falting's Theorem effectively.
PREREQUISITESMathematicians, algebraic geometers, and students interested in the properties of curves and their genus, particularly those studying rational points and the applications of Falting's Theorem.
bitsmath said: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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mathbalarka said: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.]
This is no doubt, it is hyper-elliptic curve and has genus 2.
mathbalarka said: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?