# Understanding a result of Serre about zeros of $x^3 - x - 1$ in $\mathbb{F}_p$

I'm trying to understand a result of Serre which relates the number of zeros in the finite field $\mathbb{F}_p$ of $f(x) = x^3 - x - 1$ to a modular form. The result can be found in the section 5.2 of the paper "On a Theorem of Jordan".

Let $E = \mathbb{Q}[x]/(f(x))$ be the field obtained by adjoining a single root of $f$, and let $L$ be the splitting field of $f$ over $\mathbb{Q}$. The Galois group $\textrm{Gal}(L/\mathbb{Q})$ is isomorphic to $S_3$. In the paper, Serre lets $\rho$ be the "natural" embedding of $S_3$ into $\textrm{GL}_2(\mathbb{C})$ (which I assume is the representation of $S_3$ as a dihedral group of rotation and reflection matrices) and considers the associated Artin-L function $$L(\rho,s) = \sum_{n=1}^{\infty}\frac{a_n}{n^s}.$$ He goes on to say that we can characterize the above function by $$L(\rho,s) = \zeta_E(s)/\zeta(s),$$ where $\zeta$ is the Riemann-zeta function and $\zeta_E$ is the Dedekind zeta function of the number field $E$. He says that this is equivalent to saying that $\rho \oplus 1$ (presumably $1$ denotes the trivial representation of $S_3$) is isomorphic to the 3-dimensional permutation representation of $S_3$.

I understand that if $\nu$ is the permutation representation of $S_3$ (i.e. the regular representation), then the Artin $L$-function $L(\nu,s)$ is equal to the Dedekind zeta function $\zeta_L(s)$. In particular, the $L$-function associated to the trivial representation should just be the usual Riemann zeta function. I also know that the Artin $L$-function of a sum of representations is given by the product of the $L$-functions. So this should say that $$\zeta_L(s) = L(\rho \oplus 1,s) = L(\rho,s)\zeta(s),$$ which would give us $L(\rho,s) = \zeta_L(s)/\zeta(s)$ instead.

The other part I am having trouble with is the following: Serre claims that since $S_3$ is a dihedral group, Hecke's theory applies and shows that the power series $F(\tau) = \sum_{n=1}^{\infty}a_nq^n, q = e^{2\pi i\tau}$ is a cusp form of weight $1$ and level $23$. I can't seem to find the specific result of Hecke being used here and would like a reference.

Thank you.

• For what it is worth, $x^3 - x - 1$ factors as three linear factors $\pmod p$ when $p = u^2 + uv + 6 v^2,$ is irreducible when $p = 2 u^2 + uv + 3 v^2,$ and has one root when $(-23|p) = -1.$ May be necessary to check primes $2,3,23$ separately, can't recall. – Will Jagy Mar 13 '18 at 4:29
• zakuski.utsa.edu/~jagy/Hudson_Williams_1991.pdf – Will Jagy Mar 13 '18 at 4:31
• @will you certainly do have to check $p=23$ separately as this modulus gives a double root, $x^3-x-1\equiv (x-3)(x-10)^2$. – Oscar Lanzi Mar 13 '18 at 10:02
• For your second question, dihedral Galois representations correspond to CM modular forms. Here $\rho$ is induced from a character $\chi$ of $\mathrm{Gal}(L/K)$, where $K$ is the Galois quadratic subfield of $L$. By class field theory, $\chi$ corresponds to a Hecke character, which can be used to build a modular form; this modular form will be your $F$. A good reference for the Hecke character -> modular form side is Miyake, chapter 4.8. – Mathmo123 Mar 13 '18 at 22:50
• Essentially what's going on is a basic case of Langlands functoriality. Hecke chars (i.e. automorphic representations for $\mathrm{GL}(1)$ correspond to 1-dimensional Galois representations (this is exactly class field theory). Modular forms are examples of automorphic representations for $\mathrm{GL}(2)$ and have 2-dimensional Galois representations. The representation theoretic induction on the Galois side corresponds to an automorphic induction (i.e. a theta lift) on the automorphic side. The fact that the whole picture commutes is Langlands functoriality. – Mathmo123 Mar 14 '18 at 18:24