Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Problem Set 11: More Cyclotomic Fields Due in class Wednesday, November 21

Please submit eight of the following problems. (Note: this version includes known corrections as of class on November 14.)

More on cyclotomic fields.

Let ζ_{n} denote a primitive n-th root of unity and let K = ℚ(ζ_{n} + ζ_{n}^{−1}).

1. Prove that K is the maximal totally real subfield of ℚ(ζ_{n}). (Recall that a number field is totally real if all of its embeddings into ℂ are real, i.e., have their image in ℝ.)

2. Prove that ℤ[ζ_{n} + ζ_{n}^{−1}] is the ring of integers of K.

3. Let E be the group of units of ℤ[ζ_{n}] and let F be the group of units of ℤ[ζ_{n} + ζ_{n}^{−1}]. Prove that F and the roots of unity of ℚ(ζ_{n}) generate a subgroup of E of index at most 2. (Hint: this problem generalizes one of the problems from Problem Set 5.)

4. Let 𝔞 be an ideal of 𝔬_{K} which becomes principal in ℤ[ζ_{n}]. Prove that 𝔞 is principal. That is, the natural map from Cl(𝔬_{K}) to Cl(ℤ[ζ_{n}]) is injective.

Gauss sums and Stickelberger’s theorem.

Reminders: given a character χ : 𝔽_{q} → ℚ(ζ_{m}), the Gauss sum G(χ) is given by

G(χ) = − ∑_{a∈𝔽_{q}^{∗}} χ(a)ζ_{p}^{Trace_{𝔽q/𝔽p}(a)}

(note the minus sign here!); given characters χ_{1}, χ_{2}, the Jacobi sum J(χ_{1}, χ_{2}) is given by

J(χ_{1}, χ_{2}) = − ∑_{a∈𝔽_{q}} χ_{1}(a)χ_{2}(1 − a).

1. Verify the properties of Jacobi sums not proved in class:

J(χ_{1}, χ_{2}) = \begin{cases}χ_{1}(−1) & χ_{1}χ_{2} = 1 \\ G(χ_{1})G(χ_{2})/G(χ_{1}χ_{2}) & χ_{1}χ_{2} ≠ 1. \end{cases}

(Hint: write G(χ_{1}) as a sum over a, G(χ_{2}) as a sum over b, then rewrite G(χ_{1})G(χ_{2}) as a double sum in which a + b is one of the indices.)

2. Stickelberger’s theorem, in full generality, actually says that for any subfield M of ℚ(ζ_{m}) and any fractional ideal 𝔞, 𝔞^{βθ} is principal, where θ = ∑_{a} \{\frac{a}{m}\} σ_{a}^{−1} is the Stickelberger element. Deduce this from the case M = ℚ(ζ_{m}) proved in class. (Again, it suffices to work with 𝔞 being a prime not dividing m.)

3. Let p ≡ 3 (mod 4) be a prime, and let R and N be the number of quadratic residues and nonresidues, respectively, mod p in the interval (0, p/2). Prove that for every fractional ideal 𝔞 of ℚ(\sqrt{−p}), 𝔞^{R−N} is principal. (Hint: use the previous exercise.)

Interlude: the Weil Conjectures.

For q a prime power and d a positive integer, let N(d, q) denote the number of solutions of the equation x^{d} + y^{d} = z^{d} with x, y, z ∈ 𝔽_{q} not all zero, divided by q − 1. (The result is an integer by homogeneity.) In these exercises, we reconstruct Weil’s motivation for proposing the powerful Weil Conjectures about the number of solutions of polynomial equations (i.e., points on algebraic varieties) over finite fields.

1. Suppose d divides q − 1, and let χ : 𝔽_{q}^{∗} → ℂ^{∗} be a character of order d. Show that

N(d, q) = q + 1 − ∑_{a,b} J(χ^{a}, χ^{b}),

where the sum runs over a, b = 1, …, d − 1 but excluding pairs with a + b = d. (Hint: show that the number of d-th roots of u ∈ 𝔽_{q}^{∗} is ∑_{a=1}^{d} χ^{a}(u).)

2. Show that |N(d, q) − (q + 1)| ≤ (d − 1)(d − 2)\sqrt{q}. (Hint: reduce to the case where d divides q − 1.)

3. (Davenport-Hasse) Given a character χ : 𝔽_{q}^{∗} → ℂ^{∗}, define a character χ′ : 𝔽_{q^{r}}^{∗} → ℂ^{∗} by setting χ′(a) = χ(Norm_{𝔽_{q^{r}} /𝔽_{q}}(a)). Show that G(χ′) = G(χ)^{r}; you may assume p > 2 if it helps. (Hint: first show G(χ′)/G(χ)^{r} is a unit and has absolute value 1, so is a root of unity. Then show it is congruent to 1 modulo primes above p. For more details, see Washington, Introduction to Cyclotomic Fields, Chapter 6.)

4. Suppose d divides q − 1, and let χ : 𝔽_{q}^{∗} → ℂ^{∗} be a character of order d. Show that

N(d, q^{r}) = q^{r} + 1 − ∑_{a,b} J(χ^{a}, χ^{b})^{r},

where the sum runs over a, b as above.

5. Show that |J(χ^{a}, χ^{b})| = \sqrt{q} for all a, b as above. This is the “Riemann hypothesis” for the zeta function of the curve x^{d} + y^{d} = z^{d} in the projective plane. (Hint: consider the power series ∑_{i}(1 − α_{i}z)^{−1} in z, where α_{i} runs over the J(χ^{a}, χ^{b}), and use the earlier exercises to bound its coefficients. Or use the fact that |G(χ)| = \sqrt{q} if χ is a character on 𝔽_{q}.)