How do I show this /sci/?

[Anonymous No.8334947]

Let [math]{X_5}[/math] be the variety over [math]{\mathbb{F}_5}[/math] defined by taking the elliptic curve [math]X = V\left( {{y^2} - {x^3} + 1} \right)[/math] mod 5.

Show the number of [math]{\mathbb{F}_{{5^n}}}[/math] valued points of [math]{X_5}[/math] is [math]1 + {5^n}[/math]/

[Anonymous No.8335051]

>>8334947
do you know this is true? what's the context?

[Anonymous No.8335060]

>>8335051
>do you know this is true
yes

>what's the context?
just an example building towards some weil stuff

[Anonymous No.8335119]

>>8335060
it's not quite true, you might want to use a different textbook then whereever you're getting this info from. you can find it on page 7 here (https://eprint.iacr.org/2005/374.pdf)) but basically you just:
1) write a=#E(F_5)-(5+1) and x^2-ax+5=(x-b)(x-c)
2) by theorem 4.12 in reference 15 in the above link we have #E(F_(5^n))=(5^n+1)-(b^n+c^n)
3) manually check that there's 6 points over F_5
4) so 6=#E(F_5)=5+1-a implies a=0
5) so x^2+5=(x-b)(x-c) implies b = -c = sqrt(-5)
6) so #E(F_5^n)=5^n+1 if n is odd, and #E(F_5^n)=5^n +- 2*5^(n/2)+1 if n is even

[Anonymous No.8335120]

Try induction, although if I'm not mistaken the amount of valued points is always 1 in that field for all n out of Natural Numbers.
Not sure what valued points are though.

[Anonymous No.8335149]

>>8335119
btw these a,b,c are all natural coefficients that show up in the hasse weil zeta function in case they look unmotivated to you

[Anonymous No.8335188]

>>8335119
>https://eprint.iacr.org/2005/374.pdf
Thank you for the reference. And I should have specified, the example restricted to n odd.

[Anonymous No.8335271]

>>8335120
>Not sure what valued points are though.
Best thought via the functor of points.

Let [math]X[/math] be a scheme. Then we think of it via a functor [math]{h_X}:{\operatorname{Sch} ^{op}} \to \operatorname{Set} [/math] defined by [math]{h_X}\left( Y \right) = \operatorname{Hom} \left( {Y,X} \right)[/math].

So in general, The "Y-valued points of X", is [math]\X\left( Y \right) \equiv {h_X}\left( Y \right) = \operatorname{Hom} \left( {Y,X} \right)[/math].

[Anonymous No.8335311]

>>8334947
Also Weil's Zeta function. Also Silverman's book for proof of Hasse's theorem. Also why you use X notations for elliptic curve. Also scheme are the natural way of doing geometry. Also bitches don't know 'bout muh big amble coherent and invertible canonic sheaf.
>tfw they rly don't know about it
>tfw ihnf

[Anonymous No.8335314]

>>8335271
>Best thought via the functor of points.
Or the set of solutions of the equation in extension field if you're not used to the the really natural and immediate way to do geometry with representable functors and shit

[Anonymous No.8335374]

>>8335120
It's all a fancy way of saying that you're looking for the solutions of the equation mentioned in the OP with coefficients in each [math]\mathbb F_{5^n}[/math]

[Anonymous No.8336205]

>>8335311
>Also Weil's Zeta function.
What about it?
>Also Silverman's book for proof of Hasse's theorem.
I don't have that book
> Also why you use X notations for elliptic curve.
Same thing I use for any scheme?
>Also scheme are the natural way of doing geometry
Point being?
> Also bitches don't know 'bout muh big amble coherent and invertible canonic sheaf.
what?

[Anonymous No.8336383]

>>8334947
Wah, there are some knowledgeable people here.
I'm going to ask a question, then. Does anyone know where I can find a proof of the Lang-weil bound, as stated here
>https://terrytao.wordpress.com/2012/08/31/the-lang-weil-bound/
Tao gives an outline of proof but I don't have enough machinery to make it rigorous. Specially does anyone know were I can learn about the stuff he uses to prove the bertini type lemma (lemma 6) or in general any other proof of the induction on dimension step?