Top Student at Their Peak
Chapter 77 - 77 67 I'm Sorry I'm Not Skilled Enough

77: Chapter 67: I’m Sorry, I’m Not Skilled Enough…

77: Chapter 67: I’m Sorry, I’m Not Skilled Enough…

“…

Look, this is an elliptic curve.

But it’s not the ordinary ellipse from a conic curve, it’s a smooth projective curve of genus 1 over a field.

If the characteristic is not equal to 2, the affine equation is y^2 = x^3 + ax^2 + bx + c.

You surely remember the prerequisites for the BSD conjecture, right?

An elliptic curve over the complex field is a Riemann surface of genus 1, and over a global field, it’s a finitely generated abelian group.

The Abelian variety is the higher-dimensional generalization of the elliptic curve.

So at this point, I feel the need to transform the elliptic curve into Weierstrass form.

This is the method I found after studying a lot of related theories.

Such a transformation is quite mechanical, assuming the equation has at least one rational point.

But obviously, this step holds, we’ve already proven it before, so we can derive these two equations…”

Qiao Yu was speaking while writing on the small table with a pen.

Lan Jie listened attentively, craning his neck to look at Qiao Yu’s overall problem-solving process and the figure quickly drawn using the coordinate system.

“…

Clearly, we have now obtained a classic elliptic curve with two real parts.

The line on the right obviously extends continuously to positive and negative infinity, whereas the closed elliptic curve on the left is the key to solving the problem.

Given any solution to this equation, we can restore the numbers we need using the equation.”

“The most critical part of this step is that the triplet (a:b:c) must be a projective curve, so that it can satisfy the equation for any constant multiplication.

Next, the two-way rational equivalence is used; I’ll directly find a rational point on this elliptic curve that’s the most convenient to solve and substitute it back into the original equation to find the solution.

Actually, once you reach this step, it becomes simple.

In elliptic curve theory, the chord-tangent technique is the key tool for generating new rational points.

As long as two known rational points are found on the elliptic curve, P1 and P2, a new rational point can be generated through addition.

The next step is to directly construct the tangent line, and at this moment, an Abelian group naturally forms.

We need to introduce the zero element O from this group, and according to the rule, any point P added to O remains P.

…

Then, by drawing the tangent of point P, find the point of intersection again with the curve and then calculate.

If an integer solution cannot be obtained, continue using the chord between P and 2P to find the third intersection point with the curve, then connect to the point O to find the fourth intersection, and repeat this step to find the fifth intersection…

In short, repeat this step until the corresponding integer solution is found.

However, this step can’t be done by hand, only by a computer, and after finding that value, use a geometry program for iteration.

The final computation of 9P results in an integer, and then use the value obtained from 9P to perform 9 iterations with the geometry program, and finally derive the values of a, b, c in the above equation.

This is the entire problem-solving idea.”

…

Qiao Yu spoke continuously for a whole hour, only feeling a dry mouth and tongue afterwards, he directly took the mineral water inserted in the front seat back and took a few sips before asking: “So, Teacher Lan, do you think my method has universality?”

Lan Jie came to his senses, glanced at Qiao Yu, and didn’t immediately reply.

After all, to determine whether this method has universality, he must fully understand the method.

Asking Qiao Yu to explain was because he originally thought that Qiao Yu wouldn’t use overly complicated content in number theory when solving this equation.

After all, his impression of Qiao Yu had always been talented, but without systematic study in mathematics.

But he was different, having systematically studied abstract algebra and introductory number theory in university, so he shouldn’t have trouble understanding.

But obviously, he was wrong.

Listening to Qiao Yu’s explanation, he even recalled the times in university, the fear dominated by advanced algebraic geometry.

Projective geometry and modular space indeed caused headaches.

He studied hard and barely passed, obtaining the credits.

Of course, there were also many impressive classmates who could easily earn full marks.

This is why in his graduate studies, he chose combinatorics, and after graduation, returned to Star City to become a high school math teacher.

It’s not that he didn’t want to do research, continue with a Ph.D., and then try to teach in universities.

Mainly, his abilities were limited; he just couldn’t carry on.

So he genuinely didn’t fully understand Qiao Yu’s solution to this equation.

As everyone knows, to judge whether a mathematical solution method is universally applicable to a class of equations, the entire solution idea must be completely understood.

This was very awkward.

He initially thought that with his accumulated mathematical knowledge in university, after hearing Qiao Yu’s on-the-spot explanation, he would surely be able to give an answer.

But now he had to choose between embarrassment and finding ways to cover it up.

After pondering for about ten seconds, Lan Jie chose honesty.

Because he really wasn’t good at pretending.

“Qiao Yu, to be honest, my level isn’t sufficient to make a judgment…

So this issue, you can only try it yourself.

Choose several similar equations and solve them using your method.

If you can reach the correct answers in the end, then you can start writing a paper.

I can’t help you resolve the specific problem in the paper, but I can teach you how to write it.

After all, mathematics papers have specific formats and writing requirements, along with some commonly used general standards.”