The Lumenary · Research pathways

The Langlands programme: an orientation

Name: ______________________

Date: ______________________

You will learn to

Trace reciprocity from quadratic reciprocity through cyclotomic fields to class field theory, prove quadratic reciprocity via Gauss sums and Frobenius elements, and state the Kronecker–Weber theorem
Describe modular forms, Hecke eigenforms, automorphic representations and Galois representations, and explain what it means for them to correspond through matching L-functions
Prove that the coefficients of Ramanujan's Δ are multiplicative, as an example of Hecke theory, and check numerically that an elliptic curve and a weight-one Artin representation are modular
Explain functoriality, the modularity theorem and the geometric Langlands programme as parts of the programme, separating proved cases from conjectures

1.
Use quadratic reciprocity to compute the Legendre symbol $\left(\frac{3}{11}\right)$ . Enter $1$ or $-1$ .
2.
Into how many distinct prime ideals does $p = 7$ split in the cyclotomic field $\Q(\zeta_{12})$ , which has degree $\varphi(12) = 4$ ?
3.
Let $g = \sum_{a=1}^{6} \left(\frac{a}{7}\right)\zeta_7^a$ be the quadratic Gauss sum for $q = 7$ . What is the integer $g^2$ ?
4.
In the Galois-theoretic proof of quadratic reciprocity we used that $\Q(\sqrt{q^*})$ is the unique quadratic subfield of $\Q(\zeta_q)$ . Why is that true?
- A. $\operatorname{Gal}(\Q(\zeta_q)/\Q) \cong (\Z/q)^\times$ is cyclic, so it has exactly one subgroup of index $2$ ; by the Galois correspondence there is exactly one quadratic subfield, and the Gauss sum shows it contains $\sqrt{q^*}$ .
- B.Because $q^* \equiv 1 \pmod 4$ .
- C.By the Kronecker–Weber theorem.
- D.Because every prime $p \neq q$ is unramified in $\Q(\zeta_q)$ .
5.
Which statement about the number of roots of $x^3 - x - 1$ modulo a prime $p \neq 23$ is true?
- A.It depends only on $p \bmod 23$ .
- B.If $\left(\frac{-23}{p}\right) = -1$ , it has exactly one root.
- C.It has three roots for every $p \equiv 1 \pmod{23}$ .
- D.It is always $0$ or $3$ .
6.
By Chebotarev's density theorem, what proportion of primes $p$ have the property that $x^3 - x - 1$ has exactly one root mod $p$ ?
- A. $\tfrac12$
- B. $\tfrac13$
- C. $\tfrac16$
- D. $0$ , because it has $0$ or $3$ roots for almost all $p$
7.
Which of the lesson's Lean theorems is the formal version of the statement "the Frobenius element at $p$ in $\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong (\Z/n)^\times$ is the class of $p$ " used in the proof of quadratic reciprocity?
- A.cyclotomic_frobenius: the image of the stabiliser (decomposition group) of a prime P above p under galEquivZMod is Subgroup.zpowers (ZMod.unitOfCoprime p hn).
- B.cyclotomic_residue_degree: the residue degree of p is orderOf (p : ZMod m).
- C.dirichlet_primes_in_ap: every unit class mod q contains infinitely many primes.
- D.five_square_iff: 5 is a square mod p iff p % 5 = 1 ∨ p % 5 = 4.
8.
Write tau_list(N) returning the list [0, τ(1), …, τ(N-1)] of coefficients of $\Delta = q\prod_{n \ge 1}(1 - q^n)^{24}$ , so that tau_list(N)[n] is $\tau(n)$ . Then test Mordell's theorem $\tau(mn) = \tau(m)\tau(n)$ for coprime $m, n$ with $mn \lt 60$ .
Computational — Practice
9.
Which of the following are proved theorems? Choose all that apply.
- A.Kronecker–Weber: every finite abelian extension of $\Q$ lies in a cyclotomic field.
- B.Functoriality for every $L$ -homomorphism between reductive groups.
- C.Every elliptic curve over $\Q$ is modular.
- D.The symmetric square of a cuspidal automorphic representation of $GL_2$ is automorphic on $GL_3$ .
- E.Artin's conjecture: $L(\rho, s)$ is entire for every non-trivial irreducible Artin representation $\rho$ of $G_\Q$ .
- F.Langlands reciprocity for $GL_n$ over $\Q$ , for all $n$ .

Answer key — The Langlands programme: an orientation

1.
$\left(\frac{3}{11}\right) = -\left(\frac{11}{3}\right) = -\left(\frac{2}{3}\right) = -(-1) = 1$ . Check directly: $5^2 = 25 \equiv 3 \pmod{11}$ .
2.
$7^2 = 49 \equiv 1 \pmod{12}$ , so $f = 2$ and $7$ splits into $4/2 = 2$ primes of residue degree $2$ . In Galois terms, $\operatorname{Frob}_7 = \sigma_7$ has order $2$ in $(\Z/12)^\times \cong (\Z/2)^2$ .
3.
$g^2 = (-1)^3 \cdot 7 = -7$ , so $\sqrt{-7} = g \in \Q(\zeta_7)$ : the unique quadratic subfield of $\Q(\zeta_7)$ is $\Q(\sqrt{-7})$ , not $\Q(\sqrt 7)$ . (Indeed $\Q(\sqrt 7)$ has discriminant $28$ , so it lies in $\Q(\zeta_{28})$ but not in $\Q(\zeta_7)$ .)
4.
A cyclic group of even order $q - 1$ has a unique subgroup of index $2$ (the squares). By the Galois correspondence $\Q(\zeta_q)$ has a unique quadratic subfield, and since $\sqrt{q^*} = \pm g \in \Q(\zeta_q)$ by Lemma 1, that subfield is $\Q(\sqrt{q^*})$ .
5.
The number of roots is the number of fixed points of $\operatorname{Frob}_p \in S_3$ : $3$ for the identity, $1$ for a transposition, $0$ for a $3$ -cycle. The sign of $\operatorname{Frob}_p$ is its action on $\Q(\sqrt{-23})$ , i.e. $\left(\frac{-23}{p}\right)$ ; when this is $-1$ , $\operatorname{Frob}_p$ is a transposition and there is exactly one root. When it is $+1$ , the cases $3$ and $0$ are distinguished not by a congruence but by which quadratic form of discriminant $-23$ represents $p$ — equivalently, by the coefficient of $q^p$ in $\eta(z)\eta(23z)$ .
6.
Exactly one root $\iff$ $\operatorname{Frob}_p$ is a transposition. The transpositions form a conjugacy class of size $3$ in $S_3$ , so the density is $3/6 = \tfrac12$ . The lesson's experiment finds $0.520$ for primes below $2000$ — statistics consistent with the theorem, not a proof of it.
7.
cyclotomic_frobenius says the stabiliser of P — the decomposition group, generated by $\operatorname{Frob}_{\mathfrak{P}}$ — maps onto the subgroup of $(\Z/n)^\times$ generated by $p$ . Since $\operatorname{Frob}$ is the canonical generator ( $x \mapsto x^p$ on the residue field), this is the precise sense in which " $\operatorname{Frob}_p = p \bmod n$ ". The other theorems are true and related, but they say less (b), something different (c), or something derived from reciprocity (d).
8.
```
def tau_list(N):
    c = [0] * N
    c[0] = 1
    for n in range(1, N):
        for _ in range(24):
            for k in range(N - 1, n - 1, -1):
                c[k] -= c[k - n]
    return [0] + c[:N - 1]
```
The checks confirm Mordell's theorem for $mn \lt 60$ and the prime-power relation $\tau(4) = \tau(2)^2 - 2^{11}$ . This is evidence for small $n$ ; Theorem 5 proves it for all $n$ , from the fact that $S_{12}(SL_2(\Z))$ is one-dimensional.
9.
Theorems: (a) Kronecker–Weber, (c) modularity of elliptic curves over $\Q$ , (d) symmetric-square functoriality. Open: (b) functoriality in general, (e) Artin's conjecture in general, (f) reciprocity for $GL_n$ , $n \ge 3$ .