Calculating Genus of a Curve Using Falting's Theorem

  • Context: MHB 
  • Thread starter Thread starter bitsmath
  • Start date Start date
  • Tags Tags
    Curve
Click For Summary

Discussion Overview

The discussion revolves around calculating the genus of the curve defined by the equation $x^2 - x + y - y^5 = 0$ and the application of Falting's Theorem to determine the finiteness of rational points on the curve. The conversation includes theoretical aspects of algebraic geometry, specifically focusing on the genus of curves and the implications of Falting's Theorem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant seeks to learn how to find the genus of their curve and its relation to Falting's Theorem.
  • Another participant notes that Falting's Theorem indicates that if the genus is greater than 1, the curve has finitely many rational points, suggesting that knowing the genus addresses the second question about Falting.
  • A different participant claims the curve is hyper-elliptic and asserts it has genus 2 but does not provide a method for calculating the genus.
  • There are suggestions to explore Riemann surfaces and the Riemann-Hurwitz formula as potential methods for computing the genus.
  • One participant expresses confusion about how to determine the genus despite being told it is 2, indicating a desire for a detailed explanation of the process.

Areas of Agreement / Disagreement

There is no consensus on the method for calculating the genus of the curve. While some participants agree on the genus being 2, the discussion remains unresolved regarding the specific techniques to arrive at that conclusion.

Contextual Notes

Participants reference well-known formulas for calculating the genus of Riemann surfaces but do not specify which formulas or methods to use. There is an indication that the problem is self-made, which may imply a lack of established context or assumptions.

bitsmath
Messages
3
Reaction score
0
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!
 
Physics news on Phys.org
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 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

- - - Updated - - -

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.]
 
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?
 
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?

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]
 

Similar threads

Undergrad What Is the Genus of One-Dimensional Curves?
  • · Replies 1 ·
Replies
1
Views
2K
High School How to draw a plane intersecting a cylinder at a "compound angle"
  • · Replies 8 ·
Replies
8
Views
3K
High School Can Curvature and Geodesics be Generalized in Higher Dimensions?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Finding Perpendicular Spirals in a Family of Curves
  • · Replies 3 ·
Replies
3
Views
2K
Graduate How Can I Differentiate Curves Where the Real Part of \( Y(t) \) Vanishes?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Differential movement along a curved surface
  • · Replies 3 ·
Replies
3
Views
3K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
High School Question about units for "area under curve"
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Finding the radius of a curved surface
  • · Replies 5 ·
Replies
5
Views
4K
Graduate Lie derivative of vector field defined through integral curv
  • · Replies 4 ·
Replies
4
Views
2K