# Fermat's Last Theorem in finite fields

Consider the finite field $$\mathbb{F}_q$$. Schur (1916) proved that, given $$n$$, when the field is sufficient large, this equation,

$$x^n+y^n= z^n$$

always has a nontrivial solution.

What conditions does the number of solutions satisfy?

• I. Schur, Über die Kongruenz $$x^{m} + y^{m} \equiv z^{m} \pmod{p}$$, Jahresber. Deutschen Math. Verein. 25 (1916), 114–117.
• It's hard for me to make sense of "what the conditions the number of the solutions satisfy", even at the level of syntax.
– YCor
Jul 14, 2017 at 8:25
• Lang-Weil tells you that the number $f(q)$ of solutions in $\mathbf{F}_q$ is $q^2+O(q^{3/2})$. Of course one can write $f(q)=1+(q-1)g(q)$, and equivalently $g(q)=q+O(\sqrt{q})$; here $g(q)$ is the number of points of some smooth irreducible projective curve.
– YCor
Jul 14, 2017 at 8:29

There are some results and references in Lang, Cyclotomic Fields, 1.§6 (p. 22ff. in Cyclotomic Fields I and II, Combined Second Edition).

Let $V(d)$ be the Fermat curve of degree $d$. Theorem 6.1. The number of points of $V(d)$ (in affine space) is $q^2 - (q-1)\sum\chi^{a+b}(-1)J(\chi^a,\chi^b)$, the sum over integers $a,b$ with $0 < a,b < d$ and $a+b \not\equiv 0 \pmod{d}$ and $\chi$ the character such that $\chi(u) = \omega(u)^{(q-1)/d}$ with $\omega: \mathbf{F}_q \to \mu_{q-1}$ the Teichmüller character and the Jacobi sum $J(\chi_1,\chi_2) = -\frac{S(\chi_1)S(\chi_2)}{S(\chi_1\chi_2)}$ and the Gauß sum $S(\chi) = \sum_u\chi(u)\lambda(u)$ and $\lambda: \mathbf{F}_q \to \mu_p, \lambda(x) = \exp(2\pi i/p\mathrm{Tr}(x))$.

It is more or less easy to obtain, via exponential sums, an asymptotic formula for the number $J$ of solutions of the congruence

$$x^{n}+y^{n} \equiv z^{n} \pmod{p}$$

where $$1 \leq x, y, z \leq p-1.$$

Indeed, since

\begin{eqnarray*} J &=& \sum_{x=1}^{p-1} \sum_{y=1}^{p-1} \sum_{z=1}^{p-1} \frac{1}{p}\sum_{a=0}^{p-1}e^{2 \pi i \frac{a(x^{n}+y^{n}-z^{n})}{p}}\\ &=& \frac{(p-1)^{3}}{p}+ \frac{1}{p} \sum_{a=1}^{p-1} \sum_{x=1}^{p-1}\sum_{y=1}^{p-1} \sum_{z=1}^{p-1}e^{2\pi i \frac{a(x^{n}+y^{n}-z^{n})}{p}}\\ &=& \frac{(p-1)^{3}}{p} + \frac{1}{p}\sum_{a=1}^{p-1}\left|\sum_{x=1}^{p-1}e^{2\pi i \frac{ax^{n}}{p}}\right|^{2}\sum_{y=1}^{p-1}e^{2\pi i \frac{ay^{n}}{p}}\\ \end{eqnarray*}

and

$$\sum_{a=0}^{p-1} \left| \sum_{x=1}^{p-1} e^{2\pi i \frac{ax^{n}}{p}} \right|^{2} \leq n\,p\,(p-1)$$

and

$$\left|\sum_{y=1}^{p-1}e^{2\pi i \frac{\alpha y^{n}}{p}}\right| \leq n\sqrt{p}$$

for any integer $\alpha$ which is not divisible by $p$, it follows that

$$J = \frac{(p-1)^{3}}{p}+O\left(n^{2}(p-1)\sqrt{p}\right).$$