A Density ofRamified
PrimesChristine
McMeekin
Introduction

A Density of Ramified Primes

Construction
Density
FormulaChristine McMeekin

Conditions
Proof Ideas

Max Planck Institute for Mathematics
PhD, Cornell University

CIRM, 6th International Conference on Uniform
Distribution Theory, October 2018

Motivation
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
FormulaConditions
Proof Ideas

Let K (p) be a number field depending on a rational prime p(fixing the isomorphism class of Gal(K (p)/Q)).
What is the density of rational primes p that exhibit aprescribed factorization in K (p)/Q?
(Non-Interesting) Examples:√1 K (p) := Q( p).
2Let K (p) ⊆ Q(ζp ) such that [K (p) : Q] = 3 where we
restrict to p ≡ 1 mod 3.

In both of these examples, p is totally ramified in K (p) withdensity 1.

Summary
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
Formula[McMng] For cyclic totally real number fields K of odd prime
degree n satisfying certain assumptions, a spin dependence
relation occurs with density given by a formula depending on nand mK , a computable and bounded invariant of the number
field K .

Conditions
Proof Ideas[McM18] A consequence of this result is a (conditional)
theorem giving a formula for the density of rational primes
exhibiting a prescribed (ramified) factorization in a number
field depending on the prime in question. This density isstrictly between 0 and 1.

Fix a Number Field K .
A Density ofRamified
PrimesChristine
McMeekin

We now construct our family of number fields K (p). First, we
fix a number field K .

Introduction
Construction
Density
FormulaConditions
Proof Ideas

Let n be an odd prime and let ` ∈ Z+ such that ` ≡ 1 mod n.
Let K ⊆ Q(ζ` ) such that [K : Q] = n. Note that such a cyclic
number field is totally real. Assume the following.
1K has odd class number.

2

Every totally positive unit of K is a square.

3(for simplicity) 2 and 5 are inert in K /Q.

(1) =⇒ (2) for n = 3 by [AF67].

A Family of Number Fields
A Density ofRamifiedPrimes
Christine
McMeekin
Introduction
Construction
Density
FormulaConditions
Proof Ideas

Definition
Let p - 2` be a rational prime. Define the number field K (p)depending on p (and K ) to be the composite of the fields{R(p)}p as p varies over all primes of K laying above p whereR(p) is the unique quadratic subextension of nRp /K , thenarrow ray class field over K of conductor p.

How can p factor in K (p)/Q?
A Density ofRamifiedPrimes
Christine
McMeekin
Introduction

p is always ramified in K (p)/Q with ramification index 2.
If p splits completely in K /Q then there are 2 ways that p
can factor in K (p)/Q. Let p denote a prime of K above p.

Construction
Density
Formula

1Conditions

If fR(pσ )/K (p) = 1 for all σ ∈ Gal(K /Q) thenfK (p)/K (p) = 1.

Proof Ideas

In other words, p “splits as completely as possible” inK (p)/Q (given the ramification).
2

If there exists σ ∈ Gal(K /Q) such that fR(pσ )/K (p) = 2
then fK (p)/K (p) = 2.

With what density does p split as completely aspossible?
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
FormulaConditions
Proof Ideas

S : = {rational primes that split completely in K /Q}
F : = {p ∈ S : fK (p)/K (p) = 1}
The density of F ⊆ S restricted to S is defined as{p ∈ F : p < N}
N→∞ {p ∈ S : p < N}d(F |S) := lim

Conditional Theorem (5.5.3 in [McM18])
d(F |S) =mK n + 1n−1 .
2n+ 2
where mK is a computable and bounded invariant of K .

Examples and Bounds
A Density ofRamifiedPrimes
Christine
McMeekin

Recall n = [K : Q] and ` is the conductor.

Introduction
Construction
Density
FormulaConditions
Proof Ideas  3n−1  n+1211 2≤ d(F |S) ≤22

A Conjecture on the Independence of Spin
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
FormulaPK := {prime ideals of OK }. For p ∈ PK , let α ∈ OK be a
totally positive generator of ph(K ) . Define αspin(p, σ) :=pσΛσ := {p ∈ PK : spin(p, σ) = 1}.

Conditions
Proof Ideas

Conjecture (An [McM18])
Let σ, τ ∈ Gal(K /Q) non-trivial such that σ 6= τ and σ 6= τ −1 .
Thend(Λσ ∩ Λτ |PK ) = d(Λσ |PK )d(Λτ |PK ).
This conjecture is within reach due to recent (not yetpublished) results of Djordjo Milovic and Peter Koymans.

A Conjecture on Character Sums
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
FormulaConditions

The following is a conjectural improvement to Burgess’s resulton short character sums coming from [FIMR13].
Conjecture (Cn [FIMR13])
Let χ be a non-principal real character mod q. Let n ≥ 3,1Q ≥ 3, N ≤ Q n , and q ≤ Q. ThenX1−δχ(n) << Q n + ,Proof Ideas

M<n<M+Nwith some δ = δ(n) > 0 and any  > 0, the implied constantdepending only on  and δ.
RemarkConjecture Cn is true for n = 3 by Burgess’s bound.

Computational Aspect
A Density ofRamified
PrimesChristine
McMeekinM4 := (OK /4)× / (OK /4)×

2∼
= (Z/2)nIntroduction
Construction
Density
FormulaConditions
Proof Ideas

mK is defined as the number of Gal(K /Q)-orbits of M4 ofsize-n such that (α, ασ )2 = 1 for all σ 6= 1 ∈ Gal(K /Q)
where α is a representative in OK .
Currently, finding mK involves a computation dependingon the number field K , though I am working towards aformula for mK depending only on n = [K : Q].
In particular, with an added assumption on K , I am close
to a proof that mK = 1 whenever n = 3.

Ideas Behind the ProofA Density ofRamified
PrimesChristine
McMeekin

R := {p ∈ S : spin(p, σ) = spin(p, σ −1 )}F ⊆R⊆S

Introduction
Construction
Density
FormulaConditions
Proof Ideasd(F |S) = d(F |R)d(R|S)
Using some class field theory and Dirichlet/Chebotarev’s
Density Theorem, we arrive at Theorem 7.2 of [McMng], whichgives
mK n + 1.
d(R|S) =2n
If Conjectures Cn and An are true, then by the main results of[FIMR13],
  n−1
1 2.
d(F |R) =2

Bibliography
A Density ofRamified
PrimesChristine
McMeekin
Introduction
Construction
Density
FormulaConditions
Proof Ideas

J. V. Armitage and A. Fröhlich, Classnumbers and unit
signatures, Mathematika 14 (1967), 94–98. MR 0214566
J. B. Friedlander, H. Iwaniec, B. Mazur, and K. Rubin,The spin of prime ideals, Invent. Math. 193 (2013), no. 3,697–749. MR 3091978
Christine McMeekin, A density of ramified primes,
ProQuest LLC, Ann Arbor, MI, 2018, Thesis(Ph.D.)–Cornell University, (pending publication).
Christine McMeekin, On the asymptotics of prime spin
relations, (publication pending), arXiv:1807.00892.