KUMMER THEORY FOR NUMBER FIELDS AND THE REDUCTIONS OF ALGEBRAIC NUMBERS II
ANTONELLA PERUCCA AND PIETRO SGOBBA
ABSTRACT. Let K be a number field, and letG be a finitely generated and torsion-free subgroup ofK×. For almost all primespofK, we consider the order of the cyclic group (Gmodp), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property if`eis a prime power andais a multiple of`(andais a multiple of 4if`= 2), then the density of primespofKsuch that the order of(Gmodp)is congruent to amodulo`eonly depends onathrough its`-adic valuation.
1. INTRODUCTION
If we reduce the number 2 modulo every odd prime number p, then we have the sequence of natural numbers given by the multiplicative order of (2 modp). This sequence is very mysterious, and for example it is not known unconditionally whether the order of(2 modp) equals p−1 for infinitely many primes p, see [3]. Now consider a non-zero integerz: the density of primespfor which the multiplicative order of(zmodp)lies in a given arithmetic progression has been studied in various papers by Chinen and Murata, and by Moree, see, e.g.
[1, 4].
More generally, consider a number fieldK and a multiplicative subgroupGofK×which is finitely generated. For positive integers x, ywithy | xwe denote by Kx := K(ζx) thexth cyclotomic extension ofK, and byKx,y := Kx(√y
G) theyth Kummer extension ofGover Kx. If pis a prime ofK, then we writeordp(G)for the multiplicative order of (Gmodp), which we tacitly assume to be well-defined. As customary, given two integers x, ywe write (x, y)for their greatest common divisor and[x, y]for their least common multiple. Finally, if we assume (GRH) we mean the extended Riemann hypothesis for the Dedekind zeta function of number fields.
In [8] we have generalised results by Ziegler [10] to higher rank and have proven in particular the following statement.
Theorem 1 ([8, Theorem 1.3]). Let K be a number field, and let Gbe a finitely generated and torsion-free subgroup ofK×of positive rank. Fix an integerd>2, fix an integera, and consider the following set of primes ofK:
P :={p: ordp(G)≡amodd}.
2010Mathematics Subject Classification. Primary: 11R44; Secondary: 11R45, 11R18, 11R21.
Key words and phrases. Number field, reduction, multiplicative order, arithmetic progression, density.
1
LetP(x)be the number of primespinPwith norm up tox.
Assuming (GRH), for everyx>1we have
(1) P(x) = x
logx X
n,t>1
µ(n)c(n, a, d, t) [K[d,n]t,nt :K] +O
x log3/2x
,
wherec(n, a, d, t)∈ {0,1}, and wherec(n, a, d, t) = 1if and only if the following conditions hold:
(i) (1 +at, d) = 1;
(ii) (d, n)|a;
(iii) the element ofGal(Q(ζdt)/Q)mappingζdttoζdt1+atis the identity onQ(ζdt)∩Knt,nt. From this result it is not clear whether the natural densitydensK(G, amodd)of the setP is a rational number, if it is strictly positive, or if it is possible to evaluate it. The main results of this paper are the following, whereK,G,a, anddare as in Theorem 1:
Theorem 2. Assume (GRH). Letd=`efor some prime number`and for somee>1. Suppose that K = K` if`is odd, or thatK = K4 if` = 2. Then the densitydensK(G, amod`e) depends onaonly through its`-adic valuation, and it is a computable strictly positive rational number. In particular, it is the same for allacoprime to`.
Although the previous result has an assumption on the base field, we do not need that assump- tion in the following corollary.
Corollary 3(Equidistribution property). Assume (GRH). LetK be any number field, and let d=`efor a prime number`ande>1. Suppose that`|aif`is odd, or that4|aande>2if
`= 2. Then the densitydensK(G, amod`e)depends onaonly through its`-adic valuation, and it is a computable strictly positive rational number.
The following result concerns the case of composite modulus.
Theorem 4. Assume (GRH). Let d > 2and setr := Q
`|d`to be its radical. Suppose that K = Kr ifdis odd, or that K = K2r ifdis even. Then, for a coprime tod, the density densK(G, amodd)is a computable strictly positive rational number which does not depend ona.
The following result generalises the positivity assertion of Corollary 3.
Theorem 5. Assume (GRH). The densitydensK(G, amodd)is strictly positive for any num- ber fieldKifdis a prime power or ifais coprime tod.
Theorem 2 is proven in Section 3.1 for`odd, and in Section 3.2 for` = 2, respectively. We prove Corollary 3 in Section 3.3. Theorem 4 is proven in Section 3.4, while Theorem 5 is proven in Section 3.5. Section 4 is devoted to removing from Theorem 1 the assumption that the groupGis torsion-free. Finally, Section 5 contains examples of applications of the above theorems and some numerical data.
Notice that in this paper we rely on Theorem 1 and hence most of our results assume (GRH):
if the density in Theorem 1 is known unconditionally, then our results would also be uncondi- tional.
2. PRELIMINARIES
LetK be a number field, and letGbe a finitely generated and torsion-free subgroup ofK×. In the whole paper we tacitly assume that the primespofKthat we consider are such that the reduction ofGmodulopis a well-defined subgroup of the multiplicative group of the residue field atp. Notice that the results of this section are unconditional.
2.1. Prescribing valuations for the order.
Theorem 6. Let `1, . . . , `n be distinct prime numbers and x1, . . . , xn nonnegative integers.
Then the density of primesp ofK such thatv`i(ordp(G)) = xi for alliis a strictly positive computable rational number.
Proof. The rationality of the density can be seen by neglecting the condition on the Frobenius in [5, Theorem 18]. For the positivity, apply [6, Proposition 12] to a basis g1, . . . , gr ofG consisting ofZ-independent points of the multiplicative groupK×. Corollary 7. Given an integerd>2and a positive divisorgofd, the sum of densities
(2) X
06a<d (a,d)=g
densK(G, amodd)
is a strictly positive computable rational number.
Proof. We will express the sum (2) as a rational combination of densities as in Theorem 6.
Writeg = Qn
i=1`fii, and partition the index set as{1, . . . , n} =I tJ such thatfi < v`i(d) fori∈I, andfi=v`i(d)fori∈J. Then it is easy to check that
(3) X
06a<d (a,d)=g
densK(G, amodd) = densK
p: v`i(ordp(G)) =fi,∀i∈I v`i(ordp(G))>fi,∀i∈J
.
From this expression and Theorem 6 we deduce that (2) is strictly positive. The density on the right-hand side of (3) is given by (applying the inclusion-exclusion principle for the primes up toxand then taking the limit to make the densities)
(4)
|J|
X
s=0
(−1)s X
S⊆J
|S|=s
densK
p: v`i(ordp(G)) =fi,∀i∈I v`i(ordp(G))6fi−1,∀i∈S
,
and each of the densities in (4) exists and equals densK
p: v`i(ordp(Gh)) =fi,∀i∈I v`i(ordp(Gh)) = 0,∀i∈S
, whereh=Q
i∈S`fii−1. Such densities are computable rational numbers by Theorem 6. Hence
the statement is proven.
Remark 8. Corollary 7 implies that the densitydensK(G,0 modd)is known unconditionally to be a strictly positive computable rational number.
2.2. Simplifications by changing the modulus. We keep the notation of Theorem 1. By Remark 8 we may suppose that 0 < a < d. The following lemma allows us to reduce to residue classes coprime todifdis a prime power.
Lemma 9. Letd=`e, where`is a prime number ande>1. Suppose thata=`x·w, where wis coprime to`and0< x < e. Setwj :=w+j`e−xfor06j < `(notice thatwj is also coprime to`). Then the primespofK such thatordp(G) ≡ amoddare exactly those such that
(5) ordp(G`x)≡wmod`e−x
minus those such that
(6) ordp(G`x−1)≡wj mod`e−x+1
for some06j < `. In particular, we have densK(G, amod`e) = densK(G`x, wmod`e−x)−
`−1
X
j=0
densK(G`x−1, wj mod`e−x+1).
Proof. Notice that condition (6) for anyjimplies condition (5) becausewj is coprime to`and hence we must haveordp(G`x) = ordp(G`x−1).
Letpbe a prime ofKsuch thatordp(G)≡amodd. In particular,`xdividesordp(G). Thus we have
ordp(G`x) = ordp(G)
`x and ordp(G`x−1) = ordp(G)
`x−1 . Dividing the congruenceordp(G)≡amod`eby`xand`x−1, respectively, we obtain
ordp(G`x)≡wmod`e−x and ordp(G`x−1)≡w`mod`e−x+1. We have proven one containment becausew`is not congruent to any of thewj modulo`.
Now suppose that (5) holds, and that (6) does not hold for anyj. In particular we must have ordp(G`x−1)6= ordp(G`x). We deduceordp(G`x−1) =`·ordp(G`x), and thereforeordp(G) =
`x·ordp(G`x). We may conclude because multiplying (5) by`xgives
`x·ordp(G`x)≡amodd .
Remark 10. Consider the condition ordp(G) ≡ amodd. Decompose (a, d) = sh where s= Q
`|(a,d)`is its radical, and writea0 = ah,d0 = dh. Notice that(a0, d0) = sis squarefree.
We claim that the following equivalence holds:
ordp(G)≡amodd ⇐⇒ ordp(Gh)≡a0 modd0.
If the first congruence is satisfied, then(a, d)dividesordp(G), so in particular we have ordp(G)
h ≡a0 modd0.
Since h divides ordp(G), we have ordph(G) = ordp(Gh) and the second congruence holds.
Conversely, if the second congruence is satisfied, thens= (a0, d0)dividesordp(Gh). Sinceh introduces no new prime factors, we have
ordp(Gh)·h= ordp(G) and hence the congruenceordp(G)≡amoddholds.
2.3. A general result. We keep the notation from Theorem 1, and we denote byDensK(G, d) the density of primespofKsuch thatordp(G)is coprime tod.
Remark 11. From the results in[2]and[7], under the assumptions of Theorems 2 and 4, the density DensK(G, d) depends onGonly through thed-parameters for the`-divisibility ofG for each ` | d. As a consequence of the results of this paper, the same holds for the density densK(G, amodd)considered in Theorems 2 and 4 and in Corollary 3.
Theorem 12. Let`be a prime number. Suppose that for everyGand for everye>1we have densK(G, wmod`e) = densK(G, w0 mod`e)
as long asw, w0 are coprime to`. Then for everyGand for everye>1the density densK(G, amod`e)
depends onaonly through its`-adic valuation, and it is a computable rational number.
Proof. We know from [2, Theorem 3] that the quantity
DensK(G, `) = 1−densK(G,0 mod`)
is a computable rational number. Then for every acoprime to `, by the assumption on the equidistribution, we have
densK(G, amod`e) = 1
ϕ(`e) ·DensK(G, `),
so thatdensK(G, amod`n)is a computable rational number which does not depend ona.
For0 < a < `enot coprime to`we apply Lemma 9, which allows us to compute the density densK(G, amod`e)as the difference of densities which we know to be computable rational numbers. More precisely, by the equidistribution condition the formula given in Lemma 9 becomes
densK(G, amod`e) =
densK(G`x, wmod`e−x)−`·densK(G`x−1, wmod`e−x+1),
wherea=w`xandx =v`(a). In particular, this formula shows that what matters aboutais only its`-adic valuation.
Finally the density fora= 0is given as the complementary density of all the considered cases,
and hence it is also a computable rational number.
Remark 13. Notice that for0< a < `ewith some fixed valuationv`(a) =xwhere06x < e, the previous theorem says that we have the following density:
(7) densK(G, amod`e) = 1
ϕ(`e−x) ·densK({p:v`(ordp(G)) =x}) .
Proposition 14. With the assumptions of Theorem 12, we have that the densitydensK(G, amod
`e)is strictly positive for everya.
Proof. Fora= 0we know this unconditionally by Remark 8. For0 < a < `e, by Theorem 6
the densities (7) in Remark 13 are strictly positive.
We say that a primepofK is of degree1if both its ramification index and its residue class degree overQare equal to1.
Lemma 15. LetKbe a number field, and letGbe a finitely generated and torsion-free sub- group ofK×. Leta, dbe integers withd>2and letr:=Q
`|d`be the radical ofd. Letm=r ifdis odd, andm= 2rotherwise. Consider the following set of primespofK:
S :={p: ordp(G)≡amodd, Np≡1 modm}. Then the density of the setSexists and it is equal to
(8) 1
[Km :K]·densKm(G, amodd).
Remark 16. Notice that, assuming (GRH), a formula for the density of the setS is given in [8, Corollary 5.2]. By Theorems 2 and 4 it follows that the density(8)is a computable strictly positive rational number ifdis a prime power or ifais coprime tod. Moreover, ifd=`efor a prime `, then the density ofS depends onaonly through its`-adic valuation, while if dis composite and(a, d) = 1, then it does not depend ona.
Proof of Lemma 15. We may assume that the primespofSare of degree1and unramified in Km. Hence for a primepinSwe haveNp ≡ 1 modmif and only ifpsplits completely in Km. Therefore, the set of primes ofKm lying above the primes ofS is the set
{P⊆Kmof degree1 : ordP(G)≡amodd},
which has density densKm(G, amodd). Thus we obtain that the density of the setS exists and it is equal to 1/[Km : K] timesdensKm(G, amodd)(see for instance [7, Proposition
1]).
3. PROOF OF THE RESULTS IN THEINTRODUCTION
We keep the notation of Theorem 1.
3.1. Proof of Theorem 2 for`odd.
Lemma 17. Let`be an odd prime number. Suppose thatK =K`. For everyGand for every e>1we have
c(n, x, `e, t) =c(n, x0, `e, t) as long asx, x0are coprime to`.
Proof. Letd= `e. Letavary among the integers strictly between0anddand coprime to`.
Sinceais coprime to`andd=`e, the condition(d, n)|ameans`-nand it is independent ofa. Ifc(n, a, d, t)is non-zero, then the integertmust be divisible by`becauseζ` ∈ Kand hence it must be fixed if raised to the power1 +at(recall thatais coprime to`). In particular, the condition(1 +at, d) = 1holds independently ofa.
We are left to check that Condition (iii) of Theorem 1 does not depend on a, provided that Conditions (i) and (ii) hold. WriteF := Knt,ntand defineτ := v`(t). We thus have to show that the following is independent ofa: the Galois group ofF`e+τ/Fcontains the automorphism σ1+tasatisfyingζ`e+τ 7→ζ`1+ate+τ. SinceK =K`, we have some largest integerx>τ >1such thatFcontainsQ`x, and this integer determines the Galois group ofF`e+τ/F, which is a finite cyclic`-group.
Ifx > e+τ, then the field extensionF`e+τ/F is trivial and the coefficientc(n, a, `e, t)is0 independently ofa. Now suppose thatτ 6x < e+τ. The exponents for the action onζ`e+τ are those corresponding to the automorphisms of order dividing`e+τ−x. Sincev`(at)does not depend ona, we have that
v`((1 +at)`n−1) =τ+n
independently ofaand we conclude.
Proof of Theorem 2 for`odd. Lemma 17 implies that the conditions of Theorem 12 are satis- fied ifK=K`(compare with formula (1)). Thus the densitydensK(G, amodd)depends on aonly through its`-adic valuation, and it is a computable rational number. By Proposition 14
this rational number must be strictly positive.
3.2. Proof of Theorem 2 for`= 2.
Lemma 18. SupposeK=K4. For everyGand for everye>1we have c(n, x,2e, t) =c(n, x0,2e, t)
as long asx, x0are odd.
Proof. Letd= 2e. Notice that the claim is clear fore= 1, so supposee> 2. Letavary in the odd integers strictly between 0andd. Similarly to the proof of Lemma 17, the condition (n, d) |ameans that2 -nand is independent ofa. Moreover,tmust be an even integer and hence the condition(1 +at, d) = 1is satisfied independently ofa. Now suppose that the above conditions are satisfied, and let us focus on Condition (iii) of Theorem 1.
Set τ := v2(t), and callF the fieldKnt,nt. Similarly to the proof of Lemma 17, we check that the following condition is independent of a: the Galois group ofF2e+τ/F contains the automorphismσ1+tasatisfyingζ2e+τ 7→ζ21+ate+τ.
Recall thatK =K4, and callx >2the largest integer such thatF containsQ2x (we clearly havex>τ). We then need to investigate the cyclic groupGal(Q2e+τ/Q2x).
Ifx>e+τ, then this field extension is trivial and we havec(n, a,2e, t) = 0independently ofa (whereais odd). Ifx=τ thenGal(Q2e+τ/Q2x)contains2eautomorphisms acting distinctly onζ2e+τ and fixingζ2τ: we deduce thatc(n, a,2e, t) = 1independently ofa(whereais odd).
From now on, supposeτ < x < e+τ. We seeGal(Q2e+τ/Q2x)as a subgroup of the cyclic Galois groupGal(Q2e+τ/Q4). That subgroup contains the elements of order dividing2e+τ−x. The Galois automorphisms are determined by the image ofζ2e+τ, and they are determined by the exponent to which they raise this element.
Ifτ = 1, then we do not have the automorphismσ1+ta inGal(Q2e+τ/Q4) (independently of a) because ais odd and hence ζ41+ta 6= ζ4. This means that in this casec(n, a,2e, t) = 0 independently ofa(foraodd).
Finally suppose1< τ < x < e+τ. Sinceτ >1, the automorphismσ1+ta ∈Gal(Q2e+τ/Q4) is well-defined. We have to check whether σ1+ta also belongs to Gal(Q2e+τ/Q2x) or not independently ofa. It is then sufficient to show that the order ofσ1+ta does not depend ona.
This order is a power of2, namely the smallest power2nsuch thatv((1 +at)2n−1)>e+τ. Sincev2(at)>2, then for everyn>1we havev2((1 +at)2n−1) =τ+nindependently of aand hence the order of the automorphismσ1+tadoes not depend ona.
Proof of Theorem 2 for`= 2. Analogously to the proof for the odd case, it suffices to combine
Lemma 18 with Theorem 12 and Proposition 14.
3.3. Proof of Corollary 3.
Proof of Corollary 3. Letm = `if`is odd, and m = 4if` = 2. Letpbe a prime ofK of degree1, and which does not ramify inKm. In view of our hypothesis ona, we have that ifp is such thatordp(G)≡amod`e, thenNp≡1 modm. We deduce from Lemma 15 that
densK(G, amod`e) = 1
[Km :K]·densKm(G, amod`e).
By Theorem 2 we conclude that densK(G, amod`e) depends onaonly through its`-adic valuation and that it is a computable strictly positive rational number.
3.4. Proof of Theorem 4.
Lemma 19. Letd>2be an integer and writed=Q
`efor its prime decomposition. For the coefficients of Theorem 1, with respect to any fixed groupG, we have
c(n, a, d, t) =Y
`|d
c(n, a, `e, t).
Proof. We prove thatc(n, a, d, t) = 1if and only ifc(n, a, `e, t) = 1for every prime divisor
` of d. It is clear that (1 +at, d) = 1and (d, n) | aif and only if (1 +at, `e) = 1and (`e, n) | afor every `. Now suppose that these conditions hold. Let σ be the element of Gal(Q(ζdt)/Q)such thatσ(ζdt) =ζdt1+at, and letσ`be the element ofGal(Q(ζ`et)/Q)such
thatσ`(ζ`et) =ζ`1+atet . We are left to show thatσis the identity onQ(ζdt)∩Knt,ntif and only ifσ` is the identity onQ(ζ`et)∩Knt,nt for every`. This follows from the fact thatQ(ζdt)is the compositum of the fieldsQ(ζ`et), andσ`is the restriction ofσtoQ(ζ`et)for each`.
Lemma 20. Letd>2be an integer and letr:=Q
`|d`be its radical. Suppose thatK =Kr ifdis odd, or thatK =K2rifdis even. For the coefficients of Theorem 1, with respect to any fixed groupG, we then have
c(n, x, d, t) =c(n, x0, d, t) as long asx, x0are coprime tod.
Proof. We have to show that, wheneverais coprime tod, the coefficientc(n, a, d, t)is inde- pendent ofa. By Lemma 19 we may reduce to the case in whichdis a prime power, and then we may conclude by Lemma 17 ifdis odd, and Lemma 18 ifdis even.
Proof of Theorem 4. By [7, Corollary 12] and [2, Theorem 3] the density DensK(G, d) of primespofKsuch thatordp(G)is coprime todis an explicitly computable rational number.
This density can be decomposed as the sum over a, with a coprime to d, of the densities densK(G, amodd). SinceKr =Kifdis odd, andK2r =K ifdis even, by Lemma 20 the above densities have equal value, so that for everyacoprime todwe have
densK(G, amodd) = 1
ϕ(d) ·DensK(G, d),
which is then a computable rational number. Moreover, this density is also strictly positive because by Theorem 6 the densityDensK(G, d)is strictly positive.
3.5. Proof of Theorem 5.
Proof of Theorem 5. Letrbe the radical ofd, and letm=rifdis odd, andm= 2rotherwise.
Consider the following set of primespofKof degree1, and unramified inKm: S :={p: ordp(G)≡amodd, Np≡1 modm}. By Lemma 15 the setS has density equal to
1
[Km :K]·densKm(G, amodd).
By Theorems 2 and 4, the density densKm(G, amodd) is strictly positive if dis a prime power or ifais coprime tod, so the same holds for the density ofS. Consequently, the density
densK(G, amodd)is also strictly positive.
4. MULTIPLICATIVE GROUPS WITH TORSION
Stating Theorem 1 for a finite group is trivial (the given density is either0or1). However it is not trivial to remove the assumption that the multiplicative group is torsion-free: this is what we achieve in this section. As a side remark, notice that our strategy also applies to the density considered in [8, Theorem 1.4], i.e. if we introduce a condition on the Frobenius conjugacy class with respect to a fixed finite Galois extension of the base field.
LetK be a number field, and letG0 be a finitely generated (and not necessarily torsion-free) multiplicative subgroup of K× of positive rank. Then we can write G0 as G0 = hζi ×G, whereζ is a root of unity ofK generating the torsion part ofG0andGis torsion-free. Let us exclude finitely many primes pofK so that the reduction of G0 is well-defined and we have ordp(ζ) = ord(ζ). The order ofG0 modulopis then the least common multiple between the order ofGmodulopand a fixed integer:
ordp(G0) = [ordp(G),ord(ζ)]. We may then reformulate the given problem.
Remark 21. Let Gbe a finitely generated and torsion-free subgroup of K×, and fix some integern>2. Given two integersaandd>2, we investigate the density of primespofKfor which
(9) [ordp(G), n]≡amodd .
Assuming (GRH), the case n = 1 is known, and our aim is reducing to this case. Notice that our method also shows that the considered density exists. We denote this density by dens0K(G, n;amodd).
Let`be a prime divisor of n. The aim is finding a way to replacenwith n` (or to conclude directly). We distinguish various cases.
Case (i): If ` | dand ` - a, then we have dens0K(G, n;amodd) = 0 because ` divides [ordp(G), n]and (9) cannot hold.
Case (ii): If`|dand`|a, then the congruence[ordp(G), n]≡amoddis equivalent to ordp(G`),n
` ≡ a
` mod d
`, so we have
dens0K(G, n;amodd) = dens0K G`,n
`;a
` mod d
`
.
Case (iii): Suppose that ` - d. Let `˜be a multiplicative inverse for ` modulo d, and set v:=v`(n). If`v |ordp(G), then we have
(10) [ordp(G), n]≡amodd ⇐⇒
ordp(G),n
`
≡amodd . If`v -ordp(G), then we have
[ordp(G), n]≡amodd ⇐⇒
ordp(G),n
`
≡a`˜modd . The condition`v |ordp(G)amounts to
ordp(G),n
`
≡0 mod`v
and hence (recalling that`anddare coprime) we can easily combine this congruence and the congruence in (10) with the Chinese Remainder Theorem. The first subcase thus amounts to
ordp(G),n
`
≡a`˜v`vmodd`v.
Similarly, the second subcase amounts to letting
ordp(G),n`
be in the difference of congru- ence classes
(a`˜modd)\(a`˜v+1`vmodd`v).
Notice that the congruence classes for the first and second subcase are distinct. Thus if` -d we can explicitly write
dens0K(G, n;amodd) = dens0K
G,n
`;a0 modd`v
+ dens0K
G,n
`;a`˜modd
−dens0K G,n
`;a0`˜modd`v , where we have seta0 :=a`˜v`v modd`v.
We have thus proven the following result.
Theorem 22. Assume (GRH). Let K be a number field, and letG0 be a finitely generated subgroup ofK×of positive rank. Letn>1be the order of the torsion ofG0, and letGbe a torsion-free subgroup ofG0 such thatG0 =G× hζni. Letaandd>2be fixed integers. The density of the set of primespofK
{p: ordp(G0)≡amodd}
exists and can be expressed as a finite sum of terms of the type (−1)k·densK(Gm, a0modd0) wherek,m,a0,d0are integers andm|n.
5. EXAMPLES
In this last section we work out some examples and collect some numerical data to illustrate our results.
Example 23. LetK = Q(ζ3) and consider the groupG = h5,7i 6 Q(ζ3)×. We compute the densitydensK(G, amod 9)for0 6a <9. Sinceζ3 ∈K, we can use [2, Theorem 2] to compute the density of primespofKfor which the order ofGmodpis coprime to3, and we have
DensK(G,3) = 1 13. Then by Theorem 2 we have:
densK(G, amod 9) = 1
78 fora∈ {1,2,4,5,7,8}. Fora= 3ora= 6, by [2, Theorem 3] we have
DensK(G3,3) = 9 13
and applying Lemma 9 we obtain by the equidistribution property
densK(G, amod 9) = densK(G3,1 mod 3)−3 densK(G,1 mod 9)
= 9
2·13 −3· 1 78 = 4
13.
Fora= 0we get the complementary density ofDensK(G3,3)and hence densK(G,0 mod 9) = 4
13.
Example 24. LetK=Q(ζ4)and consider the groupG=h3,5i6Q(ζ4)×. We compute the density of primesdensK(G, amod 8)for0 6 a < 8. Sinceζ4 ∈ K, by [2, Theorem 2] the density of primespofKfor which the order ofGmodpis odd is given by
DensK(G,2) = 1 28. Then by Theorem 2 we have:
densK(G, amod 8) = 1
112 fora∈ {1,3,5,7}. Fora= 2ora= 6, by [2, Theorem 3] we have
DensK(G2,2) = 1 7,
and applying Lemma 9 we obtain by the equidistribution property
densK(G, amod 8) = densK(G2,1 mod 4)−2 densK(G,1 mod 8)
= 1
14 −2· 1 112 = 3
56. Fora= 4we proceed similarly. By [2, Theorem 3] we have
DensK(G4,2) = 4 7,
and then, by Lemma 9, we obtain by the equidistribution property
densK(G,4 mod 8) = densK(G4,1 mod 2)−2 densK(G2,1 mod 4)
= 4 7 −1
7 = 3 7. Finally fora= 0we obtain the complementary density
densK(G,0 mod 8) = 3 7.
Example 25. LetK =Q(ζ12)and consider the groupG=h7,11i6Q(ζ12)×. We compute the density of primesdensK(G, amod 12)fora∈ {1,5,7,11}, which are all equal by The- orem 4 asζ12∈ K. By [7, Corollary 12] the density of primespofK for which the order of Gmodpis coprime to12can be computed as in the previous examples:
DensK(G,12) = DensK(G,4)·DensK(G,3) = 1 364. Hence we obtain by the equidistribution
densK(G, amod 12) = 1 1456.
In the following two examples we also compute with SageMath [9] approximated densities to support the validity of the equidistribution property of Corollary 3.
Example 26. Consider the grouph2i6Q×. Focusing on the set of primes up to106, we find with SageMath the following approximated values for the densitydensQ(2, amodd):
amodd densQ(2, amodd) primes up to106 4 mod 16 1/6≈0.1667 0.1676 12 mod 16 1/6≈0.1667 0.1652 3 mod 9 1/8 = 0.125 0.1236 6 mod 9 1/8 = 0.125 0.1266 9 mod 27 1/24≈0.0417 0.0422 18 mod 27 1/24≈0.0417 0.0411 3 mod 27 1/24≈0.0417 0.0416 6 mod 27 1/24≈0.0417 0.0421 15 mod 27 1/24≈0.0417 0.0420 21 mod 27 1/24≈0.0417 0.0405
For instance, by Corollary 3, for3|aandd= 9ord= 27we have densQ(2, amodd) = 1
[Q(ζ3) :Q]·densQ(ζ3)(2, amodd),
and similarly for4 | aandd = 16. Thus we can compute these densities by following the same procedure as in the previous examples.
Example 27. We consider the group G = h2,3i 6 Q× and we compute all the densities densQ(G, amodd) using the methods of the previous examples. Again we study the set of primes up to106and find with SageMath the following approximated values for the considered densities:
amodd densQ(G, amodd) primes up to106 4 mod 16 17/112≈0.1518 0.1522 12 mod 16 17/112≈0.1518 0.1508 3 mod 9 2/13≈0.1538 0.1538 6 mod 9 2/13≈0.1538 0.1540 9 mod 27 2/39≈0.0513 0.0513 18 mod 27 2/39≈0.0513 0.0513 3 mod 27 2/39≈0.0513 0.0518 6 mod 27 2/39≈0.0513 0.0512 15 mod 27 2/39≈0.0513 0.0513 21 mod 27 2/39≈0.0513 0.0507
Example 28. LetK = Q(ζ3)×, and letGbe a finitely generated and torsion-free subgroup of Q(ζ3)×. Consider the group G0 = G× hζ6i. We study the density of primes p of K such that ordp(G0) ≡ amod 10, as considered in Section 4. For a = 1,3,5,7,9, we have
dens0K(G,6;amod 10) = 0. Fora= 4we have dens0K(G,6; 4 mod 10)
= dens0K(G,2; 24 mod 30) + dens0K(G,2; 8 mod 10)−dens0K(G,2; 18 mod 30)
= densK(G2,12 mod 15) + densK(G2,4 mod 5)−densK(G2,9 mod 15), and also
dens0K(G,6; 4 mod 10) = dens0K(G2,3; 2 mod 5)
= densK(G2,12 mod 15) + densK(G2,4 mod 5)−densK(G2,9 mod 15), where the difference in the two calculations consists only in whether we consider the prime 2 or the prime3 first for the method described in Section 4. Fora = 2,6,8we can make a similar computation. Finally, fora= 0we have
dens0K(G,6; 0 mod 10) = densK(G,0 mod 5),
as2always dividesordp(G0), andordp(G0)≡0 mod 5if and only ifordp(G)≡0 mod 5.
REFERENCES
[1] CHINEN, K. - MURATA, L.,On a distribution property of the residual order ofa (modp)IV. In: Papers from the 3rd China-Japan Seminar on Number Teory, Xi’an, China, February 12–16, 2004. (Zhang, Wenpeng, et al. eds), Number Theory. Tradition and Modernization. Developments in Math. Vol. 15, Springer, New York, 2006.
[2] DEBRY, C. - PERUCCA, A.,Reductions of algebraic integers, J. Number Theory167(2016), 259–283.
[3] MOREE, P.,Artin’s primitive root conjecture – a survey, Integers12(2012), no. 6, 1305–1416.
[4] MOREE, P.,On the distribution of the order and index ofg (modp)over residue classes III. J. Number Theory120(2006), no. 1, 132–160.
[5] PERUCCA, A.,Multiplicative order and Frobenius symbol for the reductions of number fields, J. S. Balakrish- nan et al. (eds.), Research Directions in Number Theory, Association for Women in Mathematics, Series 19 (2019), 161–171.
[6] PERUCCA, A.,Prescribing valuations of the order of a point in the reductions of abelian varieties and tori. J.
Number Theory129(2009), no. 2, 469–476.
[7] PERUCCA, A.,Reductions of algebraic integers II. I. I. Bouw et al. (eds.), Women in Numbers Europe II, Association for Women in Mathematics, Series 11 (2018), 10–33.
[8] PERUCCA, A. - SGOBBA, P.,Kummer theory for number fields and the reductions of algebraic numbers, Int.
J. Number Theory,15(2019), no. 8 , 1617–1633.
[9] SageMath, the Sage Mathematics Software System (Version 8.9), The Sage Developers, 2019,https://
www.sagemath.org.
[10] ZIEGLER, V.,On the distribution of the order of number field elements modulo prime ideals, Unif. Distrib.
Theory1(2006), no. 1, 65–85.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OFLUXEMBOURG, 6AV. DE LAFONTE, 4364 ESCH-SUR- ALZETTE, LUXEMBOURG
Email address:antonella.perucca@uni.lu; pietro.sgobba@uni.lu
