On the Asymptotics of a Prime Spin Relation

For totally real cyclic number fields satisfying certain properties, we give formulas for the density of primes that satisfy a spin relation.


Introduction
In Theorem 7.4, we prove for certain totally real cyclic number fields K that the density of rational primes p that satisfy the spin relation spin(p, σ) = spin(p, σ −1 ) for all σ ∈ Gal(K/Q) where p is a prime of K above p is given by the formula 2 n−1 + (m K n + 1)(n − 1) 2 n n where n := [K : Q] is an odd prime and m K is a computable and bounded invariant of K which we call the Starlight invariant.
Let σ ∈ Gal(K/Q) be a generator. Given an odd principal ideal a, following [FIMR13], we define the spin of a (with respect to σ) to be spin(a, σ) = α a σ where a = (α), α is totally positive, and α b denotes the quadratic residue symbol in K.
M 4 := (O K /4O K ) × / (O K /4O K ) × 2 . The Starlight invariant m K of the number field K is defined to be the number of non-trivial Gal(K/Q)-orbits of M 4 with representative α ∈ O K such that the Hilbert symbol (α, α σ ) 2 = 1 for all non-trivial σ ∈ Gal(K/Q). We expect m K to depend only on [K : Q]. Definition 1.1. Let S be a set of primes and let R ⊆ S. If the limit exists, then the restricted density of R (restricted to S) is defined as where S N and R N denote the set of primes in S and R respectively of absolute norm less than N ∈ Z + . Table 1 gives examples of computed Starlight invariants for cyclic number fields of degree n over Q and conductor ℓ for the given n and ℓ values and it gives the corresponding (restricted) density of primes satisfying the spin relation spin(p, σ) = spin(p, σ −1 ) for all σ ∈ Gal(K/Q) where p is a prime of K above p, and where the restriction is to primes that split completely in K/Q. That is, if we define S to be the set of odd rational primes that split completely in K/Q and R to be the set of p ∈ S such that the given spin relation holds, then the last row of Table 1 gives d(R|S) For example, the second column of Table 1 tells us that if K is defined to be the unique subextension of the 11 th cyclotomic field of degree 5 over Q, then m K = 1 and Theorem 7.2 states that the (restricted) density of R restricted to S is 3 16 . These values of m K were computed using magma [BCP97]; the code can be found in Appendix B of [McM18].
We will prove in Theorem 7.2 that the formula for this restricted density is Furthermore, Theorem 7.6 asserts that this restricted density is bounded between 1 2 n and 1 2 . In Theorem 7.4, we will see that the unrestricted density of primes satisfying the spin relation is given by the formula 2 n−1 + (m K n + 1)(n − 1) 2 n n .

Assumptions on the Number Field
Given an odd prime n and a positive integer ℓ ≡ 1 mod n, we define K(n, ℓ) to be the subfield of the ℓ th cyclotomic field that has degree n over Q. Furthermore, unless n and ℓ are specified, we impose the restrictions that for K := K(n, ℓ), (1) the class number of K is odd, (2) every totally positive unit is a square, (3) 2 and 5 are inert in K/Q.
The last condition is for simplicity. One can easily verify the second property using a computer algebra system such as magma [BCP97]. We will take advantage of the fact that such number fields are automatically totally real and cyclic.
Let O K denote the ring of integers of K, let U := O × K denote the group of units of K and let U T denote the subgroup of totally positive units. Let U 2 denote the group of units squared. Note that in general, and assumption (2) states that U T = U 2 . This condition is a class field theoretic condition as we will see in Section 4. Table 1 gives some examples of such number fields.
If K := K(n, ℓ) for n = 3, then by the following results of Armitage and Fröhlich, U T = U 2 is implied by the condition that the class number of K is odd. Let h(K) denote the class number of K.

The Spin of Prime Ideals
Let K := K(n, ℓ).
Definition 3.1. [FIMR13] Let σ ∈ Gal(K/Q) be a generator. Given an odd principal ideal a, we define the spin of a (with respect to σ) to be where a h(K) = (α), α is totally positive, and α b denotes the quadratic residue symbol in K.
Spin is well-defined; since U T = U 2 , the choice of totally positive generator α does not affect the quadratic residue and Lemma 4.1 asserts the existence of a totally positive generator.
Lemma 11.1 in [FIMR13] states that the product spin(p, σ)spin(p, σ −1 ) is a product of Hilbert symbols at places dividing 2. We restate this more explicitly in Lemma 3.2.
For a place v of K, let K (v) denote the completion of K at v. For a, b ∈ K co-prime to v, the Hilbert Symbol is defined such that (a, b) v := 1 if the equation ax 2 + by 2 = z 2 has a solution x, y, z ∈ K (v) where at least one of x, y, or z is nonzero and (a, b) v := −1 otherwise.

Some Class Field Theory
We now diverge momentarily from the spin of prime ideals to discuss some class field theory in the case when U T = U 2 . We say a modulus is narrow whenever it is divisible by all infinite places. We say a modulus is wide whenever it is not divisible by any infinite place. We say a ray class group or ray class field is narrow or wide whenever its defining modulus is narrow or wide respectively. The following lemma is an exercise in class field theory.
Lemma 4.1. Let K be a totally real number field. The following are equivalent.
(2) The narrow and wide Hilbert class groups of K coincide.
(3) Every principal ideal of K has a totally positive generator.
Proof. To show the equivalence of (1) and (2), apply Theorem V.1.7 in [Mil13] using the modulus given by the product of all infinite places. Statements (3) and (2) are equivalent by the definitions of narrow and wide class groups.
Definition 4.2. Let K := K(n, ℓ). For q a power of 2, we define the group The Galois group Gal(K/Q) acts on M q in the natural way.
We will primarily be interested in M 4 . We will see in Lemma 4.5 that M q is canonically isomorphic to the quotient of the narrow ray class group over K of conductor q modulo squares.
Lemma 4.3. Let K be a cyclic number field of odd degree n over Q such that 2 is inert in K. Then as vector spaces Furthermore, the invariants of the action of Gal(K/Q) are exactly ±1 ∈ M 4 .
Proof. This proof is due to Sam Mundy [Mun]. Consider the exact sequence Viewing F 2 n as an additive group with Galois action by G ∼ = Gal(F 2 n /F 2 ), there is an isomorphism of Galois modules given by This map is easily seen to be a Galois equivariant homomorphism. Injectivity and surjectivity follow from considering 2-adic expansions of elements in O K /4. Since ψ is an isomorphism we can rewrite the exact sequence of Galois modules in equation 1 as The first vertical map is multiplication by 2, which is the zero map. The next two vertical maps are squaring. The third vertical map is an isomorphism because F × 2 n is cyclic of odd order. Recall that Then we apply the snake lemma to the diagram below.
The snake lemma gives us the exact sequence of G-modules Therefore M 4 ∼ = F 2 n as G-modules. The invariants of F 2 n are F 2 . Tracing through the isomorphism we see that this corresponds to the invariants {±1} in M 4 .
Let K := K(n, ℓ). Let M q,G denote the set of Gal(K/Q)-orbits of M q for q a power of 2. Recall that we say a modulus of K is narrow whenever m ∞ divides the modulus where m ∞ is the product of all infinite places of K. Letting m denote a narrow modulus with finite part m 0 , let J m K = J m0 K denote the group of fractional ideals of K prime to m 0 and let P m K = P m0 K denote the subgroup of J m K formed by the principal ideals with generator α ∈ K × such that ord 2 (q) ≤ ord 2 (α) and α ≻ 0. We let P m K = P m0 K denote the set of prime ideals of O K co-prime to m 0 so that J m K is generated by P m K . Definition 4.4. Let K := K(n, ℓ). Let q ≥ 4 be a power of 2.
(1) Define the map (2) Define the map where p is any prime in K above p.
The map r 0 is well-defined out of P 2 K ; recall that by Lemma 4.1, U T = U 2 is equivalent to the coincidence of the narrow and wide Hilbert class groups so U T = U 2 if and only if all principal ideals have a totally positive generator. Since squares are trivial in M q by definition and U T = U 2 , the map r 0 is well-defined.
The map r is well-defined out of P 2 Q because M q,G is the quotient of M q by the Gal(K/Q)-action so different choices of primes p of K above p give the same result; r 0 (p σ ) = r 0 (p) σ for σ ∈ Gal(K/Q) and p an odd prime of K.
Since J q K is generated by P q K = P 2 K , the map r 0 induces a homomorphism We first show the induced homomorphism is well-defined. By Proposition V.1.6 in [Mil13], every element of nCl q is represented by an integral ideal. Let a and b be two integral ideals representing the same element of nCl q . Then by Proposition V.1.6 in [Mil13], there exist nonzero a, b ∈ O K such that ba = ab, a ≡ b ≡ 1 mod q, and ab ≻ 0.
Noting that h(K) is odd and squares are trivial in M q by definition, ϕ 0 maps any principal integral ideal (α) to the class in M q containing the representative α ∈ O K where α is a totally positive generator.
Since U T = U 2 , every principal ideal of O K has a totally positive generator so there exists a unit u ∈ O × K such that ua ≻ 0 and ϕ 0 (a) = ua. Since ab ≻ 0, then u −1 b ≻ 0 so ϕ 0 (b) = u −1 b. We know that a ≡ b ≡ 1 mod q. Since squares are trivial in M q by the definition of M q , this implies =⇒ ϕ 0 (a) = ϕ 0 (b) Therefore the homomorphism ϕ 0 induces a well-defined homomorphism from nCl q K . We now show the homomorphism is a canonical surjective homomorphism. Let m be the narrow modulus with finite part q. Let K m := {a ∈ K × : ord 2 (a) = 0}, Let X ∈ M q . Consider the exact sequence from Theorem V.1.7 in [Mil13]; and the canonical isomorphism Consider only the 2-part of each group. Then since h(K) is odd, we have the short exact sequence Note that since squaring sends all signatures to the trivial signature, the canonical isomorphism in equation 3 induces a canonical isomorphism on the 2-part modulo squares; Consider the squaring map and apply the snake lemma to get the following commutative diagram of exact sequences; Then ψ induces an isomorphism Tracing through the definitions of the maps, ϕ • ψ is surjective (it is essentially the identity). Therefore ϕ is surjective.

An Equidistribution Lemma
Let K := K(n, ℓ). Recall that P m Q denotes the set of rational primes not dividing m and P m K denotes the set of primes of K not dividing m. Letting p be a prime of K above a rational prime p, denote the corresponding inertia degree f K/Q (p) = f K/Q (p) (well-defined because K is Galois over Q). That is, where D is the decomposition group of p for the extension K/Q and E is the inertia group.
Definition 5.1. Let K = K(n, ℓ). Define the following sets of rational primes.
Q is the set of odd rational primes that split completely in K/Q and I ⊆ P 2ℓ Q is the set of odd rational primes that are inert in K/Q. Furthermore, S ′ is the set of primes of K laying above the primes in S and I ′ is the set of primes of K laying above the primes in I.
Since K/Q is cyclic of prime degree n, then f K/Q (p) = 1 or n for all p ∈ P 2ℓ Q so in this case, P 2ℓ Q is the disjoint union of S and I. The next Lemma asserts that for K := K(n, ℓ), the primes are equidistributed in M 4 via the map r 0 .
Although the equidistribution generalizes to M q , note that the number of elements of M 8 for example is different than the number of elements of M 4 so the generalized statement would need to be adjusted accordingly.
(1) For any α ∈ M 4 , the density of p ∈ P 2ℓ K such that ϕ(p) = α is 1 2 n . That is, (2) Furthermore, the density does not change when we restrict to primes of K that split completely in K/Q. That is, Proof. Recall that nR 4 = nR 4 K denotes the narrow ray class field over K of conductor 4m ∞ . Let G := Gal(nR 4 /K). Define H ≤ G to be where Art denotes the Artin isomorphism. In other words, we define H by the following commutative diagram of exact sequences where surjectivity of ϕ is proven in Lemma 4.5. Let L be the fixed field of H so that Gal(L/K) ∼ = G/H.

This induces a canonical isomorphism
For α ∈ M 4 , define P (α) to be the set of odd unramified prime ideals of K which map to α via ϕ. Let σ ∈ G/H corresponding to α. Then where P 2ℓ K is the set of odd unramified prime ideals of K and Art L|K denotes the Artin map for the extension L|K.
Then by Dirichlet/Chebotarev's Density Theorem (which is true for natural density by Theorem 4 in [Ser81]), P (α) has a density and it is given by The first asserted equality of part (1) is proved. The second equality of part (1) is true by Lemma 4.3.
To prove part (2), observe that

Property Star and the Starlight Invariant
Let K := K(n, ℓ). Recall that M 4,G denotes the set of Gal(K/Q)-orbits of M 4 and recall the statement of Lemma 3.2 which motivates the following.
Then ⋆ is a well-defined map.
Proof. We will show that ⋆ is well-defined out of M 4 . Then because ⋆ is a property of the full Galois orbit, ⋆ is well-defined out of M 4,G . Let α, β ∈ O K be two representatives of the same class in M 4 so α ≡ βγ 2 mod 4O K for some γ ∈ O K .
Recall that by Lemma 4.3, the elements of M 4 that are invariant under the Gal(K/Q)-action are exactly ±1. The following lemma fully describes ⋆ on these invariants.
Remark 6.4. By Lemma 6.2, it is equivalent to define the starlight invariant of K, as m K := # ker(⋆) − 1.
We now define ⋆ : P 2 K → {±1} and ⋆ : P 2 Q → {±1} as the composition of ⋆ as defined in Theorem 6.1 and r 0 and r respectively as defined in Definition 4.4.
Definition 6.5. Let p ∈ P 2 Q and let p ∈ P 2 K . Define ⋆(p) := ⋆•r 0 and ⋆(p) := ⋆•r, the composition of the maps r 0 and r respectively with the map ⋆ from Definition 4.4.
We say that a prime p ∈ P 2 K (respectively p ∈ P 2 Q ) has property ⋆ or that ⋆ is true for p (respectively p) whenever ⋆(p) = 1 (respectively ⋆(p) = 1).
The main results of this paper, Theorem 7.2 and Theorem 7.4 give formulas in terms of n and m K for the density of rational primes (assumed to split completely in Theorem 7.2) that satisfy property ⋆.

Density Theorems
We first state and prove Theorem 7.2 which gives a formula describing d(R|S), the restricted density of rational primes p ∈ S that lay in R. Handling the inert case separately, we then apply Theorem 7.4 to obtain Theorem 7.4 which gives a formula for the (unrestricted) density of rational primes that satisfy the spin relation spin(p, σ) = spin(p, σ −1 ) for all σ ∈ Gal(K/Q) for K := K(n, ℓ).
Recall the definitions of S, S ′ , I, and I ′ from Definition 5.1.
Recall from Definition 1.1 that d(R|S) denotes the restricted density of primes p ∈ R restricted to S.
Theorem 7.2. Let K := K(n, ℓ) such that n = 2 is prime. Then Proof. Let N ∈ Z + . Let R N and S N denote the sets of primes in R and S respectively of norm less than N . We will show that Let R ′ N ⊆ P 2ℓ K denote the set of primes of K that lay above rational primes in R N ⊆ P 2ℓ Q and define S ′ N similarly with respect to S N ⊆ P 2ℓ Q . Let r 0,N denote the restriction of r 0 to S ′ N ⊆ P 2ℓ K . Since we have restricted to primes that split completely in K/Q, where the above is a disjoint union over elements α ∈ M 4 such that ⋆(α) = 1. Therefore By Lemma 5.2, this implies This proves the first equality in equation 4. Let σ be a generator of Gal(K/Q). By Lemma 4.3, the elements of α ∈ M 4 such that α σ = α are α = ±1 and we know that ⋆(1) = 1 and ⋆(−1) = −1 by Lemma 6.2. Recalling that m K = #{[α] ∈ M 4,G : α σ = α, ⋆(α) = 1}, this implies #{α ∈ M 4 : ⋆(α) = 1} = m K n + 1.
since n = [K : Q] is prime so Galois orbits X ∈ M 4,G such that X σ = X each contain n elements.
We now state an extended version of Lemma 5.2 which handles the inert case allowing us to give a formula in Theorem 7.4 for d(B|P 2ℓ Q ), the overall density of rational primes that satisfy ⋆. Lemma 7.3. Let K := K(n, ℓ).
(1) For any α ∈ M 4 , the density of p ∈ P 2ℓ K such that ϕ(p) = α is 1 2 n . That is, (2) Restricting to primes of K that split completely in K/Q, (3) Restricting to inert primes of K, Proof. Part (a) and part (b) were proven in Lemma 5.2.
If α = ±1 (for α ∈ M 4 ) then r −1 0 (α) ∩ I ′ = ∅ since ±1 are the only invariants of the Gal(K/Q)-action on M 4 by Lemma 4.3. Therefore d(r −1 Then where α 4 K denotes the quadratic residue symbol in O K for α ∈ O K a totally positive generator of p h(K) . This is a congruence condition so it is routine to show that by Dirichlet/Chebotarev's Density Theorem. Theorem 4 in [Ser81] asserts Dirichlet/Chebotarev's Density Theorem for natural density, or see [Neu99] Theorem VII.13.4 for an easier proof using Dirichlet density.
Proof. Let N ∈ Z + . Let I N and S N denote the sets of (rational) primes in I and S respectively with positive generator less than N . Let I ′ N ⊆ P 2ℓ K denote the set of primes of K which lay above rational primes in I N ⊆ P 2ℓ Q and define S ′ N similarly with respect to S N ⊆ P 2ℓ Q . Note that while S ′ N = {p ∈ S ′ : Norm K/Q (p) < N }, I ′ N = {p ∈ I ′ : Norm K/Q (p) < N n }. Since we have restricted to primes that are inert in K/Q, where B ′ := {p ∈ P 2ℓ K : ⋆(p) = 1} = {p ∈ P 2ℓ K : p lays above some p ∈ B}. Let r 0,N denote the restriction of r 0 to I ′ N ⊆ P 2ℓ K . Observe that p ∈ I ′ implies p σ = p so r 0 (p) = ±1 for all p ∈ I ′ by Lemma 4.3. Lemma 6.2 states that ⋆(1) = 1 and ⋆(−1) = −1. Therefore Note that since K/Q is cyclic, P 2ℓ Q is the disjoint union of S and I. Proof. By Lemma 6.2, (−1, −1) 2 = −1.
Recall the Definitions 5.1 and 7.1 defining S and R.
Lemma 7.5 implies the upper bound; α = −α in M 4 because −1 is not a square modulo 4O K .