Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

On the number of ramified primes in
specializations
François LegrandSchool of Mathematical Sciences, Tel Aviv University
Department of Mathematics and Computer Science, The Open University of Israel

Konstanz - July 23, 2015

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Let E /Q(T ) be a (non-trivial) regular finite Galois extension.
EEn
T = n ∈ N \ {0}

Q(T )

/

Q

Given a positive integer n, letRam(n)
be the number of ramified prime numbers in the specialization
En /Q of E /Q(T ) at n.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Three kinds of results:(1) results for suitable positive integers n,(2) results for a given positive integer n,(3) statistical properties of the function Ram (joint work with
Bary-Soroker).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Grunwald Problem. Let- G be a (non-trivial) finite group,- S a finite set of prime numbers,- for each prime number p ∈ S, Fp /Qp a finite Galois extension
with Galois group contained in G .
Can we find some finite Galois extension F /Q with group G suchthat the completion at each prime number p ∈ S is the extension
Fp /Qp ?
The Grunwald Problem- holds if G has odd order (Grunwald in the cyclic case, Neukirchin the general case),- does not hold if G = Z/8Z (Wang).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Proposition
Let G be a (non-trivial) finite group. Assume that the Grunwald
Problem holds for the finite group G . Then the following holds:(∗) given a positive integer m, there exists at least one Galois
extension F /Q with group G and at least m ramified primes.
It is not clear that any finite group G which occurs as a Galois
group over Q satisfies condition (∗). For example, given a“general” prime number p, the group PSL2 (Fp ) is a Galois groupover Q but all known realizations of this group over Q ramify onlyat 2 and p (Zywina).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Theorem (L.)
Let E /Q(T ) be a (non-trivial) regular finite Galois extension withgroup G . Then, given a finite set S of large enough “suitable”
primes (depending on the extension E /Q(T )), there exist infinitely
many positive integers n such that
(1) Gal(En /Q) = G ,(2) the extension En /Q ramifies at each prime of S.
Moreover, for at least one such n, we can require the discriminantdEn of En /Q to satisfyYYp ≤ |dEn | ≤ α ·
pβ
p∈S

p∈S

for some positive constants α and β (depending only on E /Q(T )).
François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Remark(1) A prime p is “suitable” if p satisfies some necessary condition
to ramify in at least one specialization of E /Q(T ) at a positive
integer. This necessary condition is related to the arithmetic of the
branch points of E /Q(T ).
(2) At least infinitely many primes are “suitable”. Hence, given a
positive integer m, there exist positive integers n such that
Gal(En /Q) = G and Ram(n) ≥ m (in particular, condition (∗)
holds for any non-trivial regular Galois group over Q).
(3) If E /Q(T ) has at least one branch point in Q, then any prime
is “suitable”. Examples: abelian groups of even order, Sn (n ≥ 2),
An (n ≥ 4), many non abelian simple groups...

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Let N ≥ 3 and E /Q(T ) be the splitting extension of the trinomial
Y N − Y N−1 − T . The extension E /Q(T ) has Galois group SN , isregular and has branch points 0, ∞ and −(N − 1)N−1 /N N .
Corollary
Let S be a finite set of primes p > N. Then there exist infinitely
many positive integers n such that
(1) Gal(En /Q) = SN ,(2) the extension En /Q ramifies at each prime of S.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Let N ≥ 3 and E /Q(T ) be the splitting extension of the trinomial
Y N − Y N−1 − T . The extension E /Q(T ) has Galois group SN , isregular and has branch points 0, ∞ and −(N − 1)N−1 /N N .
Corollary
Let S be a finite set of primes p > N. Then there exist infinitely
many positive integers n such that
(1) Gal(En /Q) = SN ,(2) the extension En /Q ramifies at each prime of S.
Theorem (Bary-Soroker and Schlank)
There exist positive integers n such that
(1) Gal(En /Q) = SN ,(2) Ram(n) ≤ 3.
François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Natural question. What can we expect for a given positive
integer n (such that the specialization En /Q has Galois group G )?

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Proposition
Let E /Q(T ) be a regular finite Galois extension. Then there exists
some positive real number C (depending only on the extension
E /Q(T )) such thatRam(n) ≤ C · log(n)for any positive integer n ≥ 2 (not a branch point).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Proof.
Let P(T , Y ) ∈ Z[T ][Y ] be a monic separable polynomial withsplitting field E over Q(T ) and ∆(T ) ∈ Z[T ] its discriminant. If n
is large enough, the specialization En /Q of E /Q(T ) at n is the
splitting extension over Q of P(n, Y ). We then obtain that any
prime p ramifying in En /Q divides ∆(n). Hence
Ram(n) ≤ ω(∆(n)) := |{p : p | ∆(n)}|As any positive integer m satisfies trivially m ≥ 2ω(m) , we haveRam(n) ≤log(|∆(n)|)
log 2It then remains to use that |∆(n)| ≤ α · nβ for some positive real
numbers α and β (not depending on n) to finish the proof.
François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Next step: study lim Ram(n), give an asymptotic as n → ∞...
n→∞François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Next step: study lim Ram(n), give an asymptotic as n → ∞...
n→∞

Example√Take E /Q(T ) = Q( T )/Q(T ). For any positive integer n, one√
has En /Q = Q( n)/Q.
(1) If n = , then Ram(n) = 0.
(2) If n is a prime, then Ram(n) = 1 or 2.
(3) If n = p1 . . . ps with n -free, then Ram(n) = s or s + 1.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Next step: study lim Ram(n), give an asymptotic as n → ∞...
n→∞

Example√Take E /Q(T ) = Q( T )/Q(T ). For any positive integer n, one√
has En /Q = Q( n)/Q.
(1) If n = , then Ram(n) = 0.
(2) If n is a prime, then Ram(n) = 1 or 2.
(3) If n = p1 . . . ps with n -free, then Ram(n) = s or s + 1.
Conclusion: it seems to be difficult to say more about the number
Ram(n) for a given positive integer n.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

Let E /Q(T ) be a (non-trivial) regular finite Galois extension.
Theorem (Bary-Soroker and L.)(1) One has
1 X
Ram(n) ∼
x→∞
x
0<n≤x

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

Let E /Q(T ) be a (non-trivial) regular finite Galois extension.
Theorem (Bary-Soroker and L.)(1) One has
1 X
Ram(n) ∼ r log log(x)
x→∞
x
0<n≤x

with r the number of branch points in Q modulo the action of GQ .

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

Let E /Q(T ) be a (non-trivial) regular finite Galois extension.
Theorem (Bary-Soroker and L.)(1) One has
1 X
Ram(n) ∼ r log log(x)
x→∞
x
0<n≤x

with r the number of branch points in Q modulo the action of GQ .
(2) One has
1 X
xRam(n) − r log log(x)0<n≤xFrançois Legrand2∼ r log log(x)x→∞On the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

Theorem (Bary-Soroker and L.)
For any real number a, one has()
Ram(n) − r log log(x)1p0<n≤x :lim
≤ a = I (a)x→∞ x
r log log(x)with1
I (a) = √2πFrançois Legrand

Z a

−t 2

e 2 dt−∞On the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

The same results hold if we consider the set of all positive integers
n such that 0 < n ≤ x and Gal(En /Q) = G (withG = Gal(E /Q(T ))).
This follows from the main result and the following two facts:(1) Ram(n) = O(log(n)),n→∞√(2) N(x) := |{n : 0 < n ≤ x ∧ Gal(En /Q) < G }| = O( x).
x→∞François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)In particular, from1x

X

Ram(n) ∼ r log log(x)0<n≤xGal(En /Q)=Gx→∞

we reobtain the following:
Given a positive integer m, there exist positive integers n such that
Gal(En /Q) = G and Ram(n) ≥ m.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)Let {t1 , . . . , tr } be a set of representatives of the branch points ofthe extension E /Q(T ) lying in Q under the action of the absolute
Galois group of Q.
For each index i ∈ {1, . . . , r }, denote the ramification index ofhT − ti i in E Q/Q(T ) by ei and let Pi (T ) ∈ Z[T ] be an
Q irreducible
polynomial such that Pi (ti ) = 0. Finally set PE (T ) = ri=1 Pi (T ).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

Proposition (based on Beckmann)
There exists some positive real number p0 (depending only on theextension E /Q(T )) satisfying the following. For any prime p > p0
and any positive integer n, not a branch point, the following two
conditions are equivalent:(1) p ramifies in the specialization En /Q of E /Q(T ) at n,(2) there exists a unique index i ∈ {1, . . . , r } such that p divides
Pi (n) and vp (Pi (n)) is not a multiple of ei .

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)This proposition is a natural motivation to introduce the following
definition.
Definition
Given two positive integers a and n, letma (n)
be the number of primes p such that vp (n) is a non-zero multipleof a.
Remark: one has m1 (n) = ω(n) for any positive integer n.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)Conjoining the proposition and the definition provides the following.
Proposition
There exists some real number C ≥ 1 (depending only on theextension E /Q(T )) such thatRam(n) − ω(PE (n)) +rXmei (Pi (n)) ≤ C

i=1

for any positive integer n (not a branch point).

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

By the previous proposition, one has
1 X
1 X
Ram(n) =ω(PE (n))
x
x
0<n≤x

0<n≤x

−

r
X
1 X
i=1

x

mei (Pi (n))

0<n≤x

+ O(1)

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

By the previous proposition, one has
1 X
1 X
Ram(n) =ω(PE (n))
x
x
0<n≤x

0<n≤x

−

r
X
1 X
i=1

x

mei (Pi (n))

0<n≤x

+ O(1)
By some classical results, one has
1 X
ω(PE (n)) ∼ r log log(x)
x→∞
x
0<n≤x

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

It then remains to prove the following result.
Proposition
Let a be an integer ≥ 2 and P(T ) ∈ Z[T ] a non-constantpolynomial. Then there exists some positive real number C (P)(depending only on the polynomial P(T )) such thatX
ma (P(n)) ≤ C (P) · x
0<n≤x

for any positive integer x.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

It then remains to prove the following result.
Proposition
Let a be an integer ≥ 2 and P(T ) ∈ Z[T ] a non-constantpolynomial. Then there exists some positive real number C (P)(depending only on the polynomial P(T )) such thatX
ma (P(n)) ≤ C (P) · x
0<n≤x

for any positive integer x.
Remark(1) The proposition does not hold if a = 1.
(2) Key-point in the proof: a ≥ 2 =⇒ ma (P(n)) ≤ {p : p 2 |P(n)}.
François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)1

Results for suitable positive integers n

2

Result(s) for a given positive integer n

3

Statistical properties
Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)First we need to generalize the previous proposition.
Proposition
Let a be an integer ≥ 2, k a positive integer and P(T ) ∈ Z[T ] a
non-constant polynomial. Then there exists some positive real
number C (P, k) (depending only on the polynomial P(T ) and theinteger k) such thatX
mak (P(n)) ≤ C (P, k) · x
0<n≤x

for any positive integer x.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)Conjoining this proposition and the last proposition from the firstpart of the proof.
Proposition
Given a positive integer k, there exists some positive real number
C (k) (depending only on the integer k and the extension E /Q(T ))such thatkX Ram(n) − ω(PE (n))≤ C (k) · x
0<n≤x

for any positive integer x.

François LegrandOn the number of ramified primes in specializations

Results for suitable positive integers n
Result(s) for a given positive integer nStatistical properties

Statement of the main result
First part of the proof
Second part of the proof (for the mean value)
Second part of the proof (general case)

By a result of Halberstam (1956), one has


Z +∞
−t 2
1 X
ω(PE (n)) − r log log(x) k
1
p
lim
=√
t k e 2 dt
x→∞ x
2π −∞
r log log(x)0<n≤xfor any positive integer k.
Conjoining this and the previous proposition provides


Z +∞
−t 2
1 X
Ram(n) − r log log(x) k
1
p
√
lim
=
t k e 2 dt
x→∞ x
2π −∞
r log log(x)0<n≤xfor any positive integer k.
Apply this result with k = 1 and k = 2 to get the results about themean value and the variance respectively. It then remains to use
the method of moments to get the result about the probability
distribution.
François LegrandOn the number of ramified primes in specializations