Arithmetic of “units” in $\protect \mathbb{F}_q[T]$

by

Bruno Anglès & Mohamed Ould Douh

Abstract. — The aim of this note is to study the arithmetic of Taelman’s unit module for A:=F^q[T].This module is the A-module (via the Carlitz module) generated by1. LetP be a monic irreducible polynomial inA, we show that the "P-adic behaviour" of1 is connected to some isotypic component of the ideal class group of the integral closure of A in the Pth cyclotomic function field. The results contained in this note are applications of the deep results obtained by L. Taelman in [10].

Résumé. — SoitF^q un corps fini ayantqéléments et de caractéristiquep, q≥3.Nous mon- trons que siP est un premier deF^q[T]de degréd,lep-rang de la composante isotypique associée au caractère de Teichmuller dup-sous-groupe de Sylow des pointsF^q-rationnels de la jacobienne du P-ième corps de fonctions cyclotomique est entièrement déterminé par le "comportement P-adique" de1.

1. Background on the Carlitz module

Let F^q be a finite field having q elements, q ≥ 3, and let p be the characteristic of F^q. Let T be an indeterminate over Fq, and set: k := Fq(T), A := Fq[T], A₊ := {a ∈A, amonic}. A prime in A will be a monic irreducible polynomial in A. Let ∞ be the unique place of k which is a pole ofT, and set: k_∞:=Fq((_T¹)).LetC∞ be a completion of an algebraic closure of k_∞,then C∞ is algebraically closed and complete and we denote by v_∞ the valuation on C∞ normalized such that v_∞(T) = −1.We fix an embedding of an algebraic closure of kin C∞,and thus all the finite extensions of kconsidered in this note will be contained inC∞. LetL/k be a finite extension, we denote by:

• S_∞(L): the set of places of Labove∞,if w∈S_∞(L) we denote the completion ofL at w byL_w and we viewL_w as a subfield of C∞,

• O_L: the integral closure of AinL,

2000 Mathematics Subject Classification. — 11R58, 11R60.

Key words and phrases. — Arithmétique des corps de fonctions, module de Carlitz, nombres de Bernoulli-Goss.

The authors thank L. Taelman and D. Thakur for fruitful discussions.

• Pic(O_L): the ideal class group of L,

• L_∞: thek_∞-algebraL⊗kk_∞,recall that we have a natural isomorphism ofk_∞-algebras:

L_∞'Q

w∈S_∞(L)L_w.

1.1. The Carlitz exponential. — Set D₀ = 1 and for i≥ 1, D_i = (T^qi −T)D_i^q₋₁. The Carlitz exponential is defined by:

e_C(X) =X

i≥0

X^qi

D_i ∈k[[X]].

Since ∀i≥0, v_∞(D_i) = −iqⁱ, we deduce thate_C defines an entire function on C∞ and that e_C(C∞) =C∞.Observe that:

e_C(T X) =T e_C(X) +e_C(X)^q.

Thus, ∀a ∈ A, there exists a Fq-linear polynomial φ_a(X) ∈ A[X] such that e_C(aX) = φ_a(e_C(X)). The map φ : A → End_Fq(A), a 7→ φ_a, is an injective morphism of F^q-algebras called the Carlitz module.

Let ε_C =^q−1√

T−T^qY

j≥1

1− T^qj −T T^qj+1−T

!

∈C∞. Then by [4] Theorem 3.2.8, we have the following equality in C∞[[X]]:

e_C(X) =X Y

α∈εCA\{0}

1−X

α

. Note that v_∞(ε_C) =− q

q−1. Let log_C(X) ∈ k[[X]] be the formal inverse of e_C(X), i.e.

eC(logC(X)) =logC(eC(X)) =X. Then by [4] page 57, we have:

log_C(X) =X

i≥0

X^qi L_i ,

where L₀= 1,and fori≥1, L_i = (T−T^qi)L_i−1.Observe that ∀i≥0, v_∞(L_i) =−qⁱ⁺¹−q q−1 . Therefore log_C converges on {α∈C∞, v_∞(α)>− q

q−1}. Furthermore, for α in C∞ such that v_∞(α)>− q

q−1, we have:

• v_∞(e_C(α)) =v_∞(log_C(α)) =v_∞(α),

• e_C(log_C(α)) =log_C(e_C(α)) =α.

1.2. Torsion points. — We recall some basic properties of cyclotomic function fields. For a nice introduction to the arithmetic properties of such fields, we refer the reader to [7]

Chapter 12. Let P be a prime of A of degree d. Set Λ_P := {α ∈ C∞, φ_P(α) = 0}. Note that the elements of ΛP are integral over A, and that ΛP is a A-module via φ which is isomorphic to A

P A. Set λP =eC

εC

P

, then λP is a generator of the A-module ΛP. Let KP =k(ΛP) =k(λP).We have the following properties:

• K_P/kis an abelian extension of degree q^d−1,

• K_P/kis unramified outside P,∞,

• let RP =OK_P,then RP =A[λP],

• if w ∈ S_∞(KP), the completion of KP at w is equal to k_∞(εC), in particular the de- composition group at w is equal to the inertia group at w and is isomorphic to F^∗q, furthermore|S_∞(K_P)|= q^d−1

q−1 ,

• K_P/k is totally ramified at P and the unique prime ideal of R_P above P is equal to λ_PR_P.

Let∆ = Gal(K_P/k). For a∈A\P A,we denote byσ_athe element in ∆such thatσ_a(λ_P) = φa(λP).The map: A\P A→∆, a7→σa induces an isomorphism of groups:

A P A

_∗ '∆.

1.3. The unit module and the class module. —

Let R be an A-algebra, we denote by C(R) the Fq-algebra R equipped with the A-module structure induced by φ, i.e. ∀r ∈ C(R), T.r = φ_T(r) = T r+r^q. For example, the Carlitz exponential induces the following exact sequence of A-modules:

0−→ε_CA−→C∞−→C(C∞)−→0.

LetL/K be a finite extension, then B. Poonen has proved in [6] thatC(O_L) is not a finitely generatedA-module. Recently, L. Taelman has introduced in [8] a natural sub-A-module of C(O_L) which is finitely generated and called the unit module associated to L and φ. First note that the Carlitz exponential induces a morphism of A-modules: L_∞→C(L_∞),and the kernel of this map is a freeA-module of rank| {w∈S_∞(L), ε_C ∈L_w} |.Now, let us consider the natural map of A-modules induced by the inclusionC(O_L)⊂C(L_∞):

α_L: C(O_L)−→ C(L_∞) e_C(L_∞).

L. Taelman has proved the following remarkable results ([8], Theorem 1, Corollary 1):

• U(O_L) := Ker(α_L)is a finitely generated A-module of rank [L:k]− | {w∈S_∞(L), ε_C ∈L_w} |,

theA-module (via φ) U(O_L) is called the unit module attached toL andφ,

• H(O_L) := Coker(α_L) is a finite A-module called the class module associated to L and φ.

Set:

ζO_L(1) := X

I6=(0)

h 1

OL

I

i

A

∈k_∞,

where the sum is taken over the non-zero ideals ofO_L,and where for any finiteA-module M, [M]A denotes the monic generator of the Fitting ideal of the finite A-module M. Then, we have the following class number formula ([9], Theorem 1):

ζ_OL(1) = [H(O_L)]_A[O_L:e⁻_C¹(O_L)],

where[O_L:e⁻_C¹(O_L)]∈k^∗_∞ is a kind of regulator (see [9] for more details).

2. The unit module for F^q[T]

2.1. Sums of polynomials. — In this paragraph, we recall some computations made by G. Anderson and D. Thakur ([2] pages 183, 184).

Let X, Y be two indeterminates over k. We define the polynomial Ψ_k(X) ∈ A[X] by the following identity:

e_C(Xlog_C(Y)) =X

k≥0

Ψ_k(X)Y^qk.

We have that Ψ₀(X) =X and fork≥1:

Ψ_k(X) = Xk

i=0

1

D_i(L_k−i)^qiX^qi. For a=a0+a1T+· · ·+anTⁿ, a0,· · ·, an∈F^q,we have:

φa(X) = Xn i=0

[^a_i]X^qi,

where [^a_i] ∈ A for i = 0,· · · , n, [^a₀] = a and [^a_n] = a_n. But since e_C(aX) = φ_a(e_C(X)), we deduce that for k≥1:

Ψ_k(X) = 1 D_k

Y

a∈A(d)

(X−a),

whereA(d)is the set of elements in A of degree strictly less than k.In particular:

Ψ_k(X+T^k) = Ψ_k(X) + 1 = 1 D_k

Y

a∈A+,k

(X+a),

whereA_+,k is the set of monic elements in A of degreek.Now for j∈N and fori∈Z,set:

S_j(i) = X

a∈A+,j

aⁱ ∈k.

Note that the derivative of Ψ_k(X) is equal to _L¹_k.Therefore we get:

1 L_k

1

Ψ_k(X) + 1 = X

a∈A+,k

1 X+a. Thus:

1 L_k

1

Ψ_k(X) + 1 =X

n≥0

(−1)ⁿS_k(−n−1)Xⁿ. But:

Ψ_k(X)≡ 1

L_kX mod X^q. Therefore:

∀k≥0,forc∈ {1,· · · , q−1}, Sk(−c) = 1 L^c_k.

But observe that we also have:

1 L_k

1

Ψ_k(X) + 1 =X

n≥0

(−1)ⁿS_k(n)X⁻ⁿ⁻¹.

But: 1

Ψ_k(X) + 1 ≡0 (mod X^−qk).

Therefore:

∀k≥0,fori∈ {0,· · · , q^k−2}, S_k(i) = 0.

The Bernoulli-Goss numbers,B(i) for i∈N,are elements of Adefined as follows:

• B(0) = 1,

• if i≥1and i6≡0 (mod q−1), B(i) =P

j≥0Sj(i) which is a finite sum by our previous discussion,

• if i≥1, i≡0 (mod q−1), B(i) =P

j≥0jS_j(i)∈A. We have:

Lemma 2.1. — Let P be a prime of A of degreedand let c∈ {2,· · · , q−1}.Then:

B(q^d−c)≡

d−1

X

k=0

1

L^c_k⁻¹ (modP).

Proof. — Note that q^d−cis not divisible by q−1 and that1≤q^d−c < q^d−1.Thus:

B(q^d−c) =

d−1X

k=0

S_k(q^d−c).

Now, fork∈ {0,· · · , d−1},we have:

S_k(q^d−c)≡S_k(1−c) (mod P).

The lemma follows by our previous computations.

We will also need some properties of the polynomialΨ_k: Lemma 2.2. —

1) Let X, Y be two indeterminates over k.We have:

∀k≥0,Ψ_k(XY) = Xk i=0

Ψi(X)Ψ_k−i(Y)^qi.

2) For k≥0,we have:

ψ_k+1(X) = Ψ_k(X)^q−Ψ_k(X) T^qk+1−T .

Proof. —

1) Recall that we have seen that:

∀a∈A, φ_a(X) =X

k≥0

Ψ_k(a)X^qk.

Furthermore, for a∈A:

e_C(aXlog_C(Y)) =φ_a(e_C(Xlog_C(Y))).

Thus, for all a∈A:

∀k≥0,Ψ_k(aX) = Xk

i=0

Ψ_i(a)Ψ_k−i(X)^qi. The first assertion of the lemma follows.

2) For all a∈A,we have:

φa(T X+X^q) =T φa(X) +φa(X)^q. Thus, for all a∈A:

∀k≥0, ψ_k+1(a) = Ψ_k(a)^q−Ψ_k(a) T^qk+1−T .

Lemma 2.3. — Let P be a prime of A of degree d.We have:

φ_P(X) = Xd k=0

[^P_k]X^qk,

where [^P₀] =P and [^P_d] = 1.Then, for k= 0,· · · , d−1, P divides [^P_k]and:

[^P_k] P ≡ 1

L_k (mod P).

Proof. — Since [^P_k] = Ψ_k(P),the lemma follows from the second assertion of Lemma 2.2.

If we combine Lemma 2.1 and Lemma 2.3, we get:

Corollary 2.4. —

Let P be a prime of A of degreed. Then:

φP−1(1)≡P B(q^d−2) (modP²).

Remark 2.5. — D. Thakur has informed the authors that the congruence in Corollary 2.4 was already observed by him in [11].

2.2. The unit module forFqⁿ[T]. — Setk_n=Fqⁿ(T)andA_n=Fqⁿ[T].In this paragraph we will determine U(A_n)and H(A_n).We have:

k_n,∞=k_n⊗kk_∞=F^qn((1 T)).

Let ϕ be the Frobenius of Fqⁿ/Fq,recall that k_n/k is a cyclic extension of degree nand its Galois group is generated byϕ.SetG= Gal(kn/k) and letα∈F^qn which generates a normal basis ofFqⁿ/Fq.ThenA_n is a freeA[G]-module of rank one generated byα. Note that:

k_n,∞=A_n⊕ 1 TFqⁿ[[1

T]].

By the results of Paragraph 1.1:

log_C(α)∈Fqⁿ[[1 T]]^∗, and:

e_C 1

TFqⁿ[[1 T]]

= 1 TFqⁿ[[1

T]].

Now:

k_n,∞=

n−1

M

i=0

k_∞log_C(α^qi).

Thus:

k_n,∞= 1 TF^qn[[1

T]]⊕

n−1

M

i=0

A log_C(α^qi).

LetS_n(A) be the sub-A-module of C(A_n) generated by Fqⁿ,thenS_n(A) is a freeA-module of rankngenerated by {α, α^q,· · ·, α^qn−1}.We have:

e_C(k_n,∞) =S_n(A)⊕ 1 TF^qn[[1

T]].

Thus:

U(An) =An∩e_C(kn,∞) =Sn(A), and:

H(An) = C(kn,∞)

C(A_n) +e_C(k_n,∞) ={0}.

In particular, for n = 1,we get U(A) =S₁(A) = the freeA-module of rank one generated (viaφ) by1 andH(A) ={0}.

LetF ∈k_∞[G]be defined by:

F =

n−1

X

i=0



 X

j≡i (modn)

1 Lj



ϕⁱ.

Then:

e⁻_C¹(A_n) =

n−1M

i=0

A log_C(α^qi) =F A_n.

Write n = mp<sup>`</sup>, where ` ≥ 0 and m 6≡ 0 (modp). Let µ_m = {x ∈ C∞, x^m = 1} which is a cyclic group of order m. Then we can compute Taelman’s regulator (just calculate the

"determinant" of F):

[A_n:e⁻¹_C (A_n)] =



(−1)^m−1 Y

ζ∈µm





n−1

X

i=0



 X

j≡i (modn)

1 Lj



ζⁱ









p^`

. Thus, Taelman’s class number formula becomes in this case:

ζ_An(1) =



(−1)^m−1 Y

ζ∈µm



ⁿ⁻¹X

i=0



 X

j≡i (modn)

1 L_j



ζⁱ









p^`

.

In particular, we get the following formula already known by Carlitz:

ζ_A(1) =log_C(1).

2.3. The P-adic behavior of "1". — Let P be a prime of A of degree d. Let CP be a completion of an algebraic closure of the P-adic completion ofk. Let vP be the valuation on CP such thatv_P(P) = 1. Forx ∈R, we denote the integer part ofx by [x]. Let i∈N\ {0} and observe that v_P(T^qi−T) = 1 if ddivides iand v_P(T^qi −T) = 0otherwise. Therefore:

• fori≥0, v_P(L_i) = [i/d],

• fori≥0, v_P(D_i) = qⁱ−q^i−[i/d]d q^d−1 .

This implies thatlog_C(α)converges forα∈CP such thatv_P(α)>0,and thate_C(α)converges for α ∈ C^P such that v_P(α)> 1

q^d−1. Furthermore, for α ∈ C^P such that v_P(α)> 1 q^d−1, we have:

• v_P(e_C(α)) =v_P(log_C(α)) =v_P(α),

• e_C(log_C(α)) =log_C(e_C(α)) =α.

Lemma 2.6. — Let A_P be the P-adic completion of A. There exists x ∈ A_P such that φP(x) =φP−1(1) if and only ifφP−1(1)≡0 (modP²).

Proof. — First assume that φ_P−1(1) 6≡ 0 (modP²). By Lemma 2.3, we have that v_P(φ_P−1(1)) = 1, and therefore φ_P(X)−φ_P−1(1) ∈A_P[X]is an Eisenstein polynomial. In particular φP−1(1)6∈φP(AP).

Now, let us assume that φ_P−1(1) ≡ 0 (modP²). Then v_P(log_C(φ_P−1(1))) = v_P(φ_P−1(1)).

Therefore, there exists y∈P A_P such that:

log_C(φ_P−1(1)) =P y.

Set x=e_C(y)∈P A_P.We have:

φ_P(x) =e_C(P y) =e_C(log_C(φ_P−1(1))) =φ_P−1(1).

Remark 2.7. — Since 1 is an Anderson’s special point for the Carlitz module, the above lemma can also be deduced by Corollary 2.4 and the work of G. Anderson in [1].

3. Hilbert class fields and the unit module for Fq[T]

LetP be a prime ofAof degreed.Recall thatK_P is thePth-cyclotomic function field, i.e. the finite extension ofkobtained by adjoining to kthePth-torsion points of the Carlitz module.

LetR_P be the integral closure ofAinK_P and let∆be the Galois group ofK_P/k.Recall that

∆is a cyclic group of order q^d−1 (see Paragraph 1.2). Recall that the unit moduleU(A) is the freeA-module (via φ) generated by 1(see Paragraph 2.2).

3.1. Kummer theory. — We will need the following lemma:

Lemma 3.1. — The natural morphism of A-modules: U(A)

P.U(A) −→ C(KP)

P.C(K_P) induced by the inclusion U(A)⊂C(K_P),is an injective map.

Proof. — Recall that K_P,∞ = K_P ⊗kk_∞. Let T r :K_P,∞ → k_∞ be the trace map. Now let x ∈ U(A)∩P.C(K_P). Then there exists z ∈ K_P such that φ_P(z) = x. Since e_C(K_P,∞) is A-divisible, we get thatz∈U(R_P).Thus T r(z)∈U(A).But:

−x=φ_P(T r(z)).

Therefore x∈P.U(A).

LetU={z∈C∞, φ_P(z)∈U(A)}.ThenUis anA-module (viaφ) andP.U=U(A).Therefore the multiplication byP gives rise to the following exact sequence of A-modules:

0−→Λ_P ⊕U(A)−→U−→ U(A)

P.U(A) −→0.

Set γ = e_C(^P_P⁻¹log_C(1)). Then γ ∈ U. Set L = K_P(U). By the above exact sequence, we observe that:

L=K_P(γ).

FurthermoreL/k is a Galois extension and we set: G= Gal(L/K_P) andG= Gal(L/k).Let δ ∈ ∆and select eδ ∈ G such that the restriction of eδ to K_P is equal to δ. Let g ∈ G, then eδgδe⁻¹ ∈Gdoes not depend on the choice of eδ. Therefore Gis aFp[∆]-module.

Lemma 3.2. — We have a natural isomorphism of Fp[∆]-modules:

G'Hom_A

U(A) P.U(A),Λ_P

.

Proof. — Recall that the multiplication byP induces anA-isomorphism:

U

Λ_P ⊕U(A) ' U(A) P.U(A). Forz∈Uand g∈G,set:

< z, g >=z−g(z)∈ΛP. One can verify that:

• ∀z1, z2∈U,∀g∈G, < z1+z2, g >=< z1, g >+< z2, g >,

• ∀z∈U,∀g₁, g₂ ∈G, < z, g₁g₂>=< z, g₁ >+< z, g₂ >,

• ∀z∈U,∀a∈A,∀g∈G, < φ_a(z), g >=φ_a(< z, g >),

• ∀z ∈ U,∀g ∈ G,∀δ ∈ ∆, < eδ(z), δ.g >= δ(< z, g >), where eδ ∈ G is such that its restriction toK_P is equal toδ,

• letg∈G then: < z, g >= 0 ∀z∈Uif and only ifg= 1.

Let z∈Ube such that< z, g >= 0 ∀g∈G. Then z∈U^G.Thus z∈K_P and φ_P(z)∈U(A).

Thus, by Lemma 3.1, we get φP(z)∈P.U(A),and thereforez∈ΛP ⊕U(A).

We deduce from above that< ., . >induces a non-degenerate and∆-equivariant bilinear map:

U(A)

P.U(A) ×G−→Λ_P.

3.2. Class groups. — Let ω_P : ∆ ' (A/P A)^∗ be the cyclotomic character, i.e. ∀a ∈ A \P A, ω_P(σ_a) ≡ a (modP). Let W = Zp[µ_qd−1], and fix ρ : A/P A → W/pW a Fp- isomorphism. We still denote by ω_P the morphism of groups ∆' µ_qd−1 which sends σ_a to the unique root of unity congruent to ρ(ω_P(a))modulopW.Observe that∆ := Hom(∆, Wb ^∗) is a cyclic group of order q^d−1 generated by ωP.Forχ∈∆,b we set:

• e_χ= _qd¹

−1

P

δ∈∆χ(δ)δ⁻¹ ∈W[∆],

• [χ] ={χ^pj, j≥0} ⊂∆,b

• e_[χ]=P

ψ∈[χ]e_ψ ∈Zp[∆].

Let Pic(R_P) be the ideal class group of the Dedekind domainR_P.

Corollary 3.3. — TheZp[∆]-module: e_[ωP](Pic(R_P)⊗_ZZp)is a cyclic module. Furthermore, it is non trivial if and only if B(q^d−2)≡0 (mod P).

Proof. — Recall that H(A) = {0}. Note that the trace map induces a surjective morphism of A-modules H(R_P)→H(A).Therefore:

H(R_P)^∆={0}.

Now, note that, ∀χ∈∆,b we have an isomorphism of W-modules:

e_χ(Cl⁰(K_P)⊗ZW)'e_χ^p(Cl⁰(K_P)⊗ZW).

Thus by [3] we get thate_[ωP](Cl⁰(K_P)⊗ZZp) is a cyclicZp[∆]-module. Furthermore, by [5], this latter module is non-trivial if and only if B(q^d−2)≡0 (mod P).We conclude the proof by noting that:

e_[ωP] Cl⁰(K_P)⊗ZZp

'e_[ωP](Pic(R_P)⊗ZZp).

Recall that L = K_P(γ) where γ =e_C

P−1

P log_C(1)

. Since γ ∈ O_L, the derivative of φ_P(X)−φ_P−1(1) is equal to P, and e_C(K_P,∞) is A-divisible, we conclude that L/K_P is unramified outside P and every place of K_P above∞ is totally split inL/K_P.Furthermore, by Lemma 2.6:

• ifφ_P−1(1)≡0 (modP²), L/K_P is unramified,

• if φ_P−1(1)6≡0 (mod P²), L/K_P is totally ramified at the unique prime ofR_P aboveP (see the proof of Lemma 2.6).

Let H/K_P be the Hilbert class field of R_P, i.e. H/K_P is the maximal unramified abelian extension ofK_P such that every place in S_∞(K_P) is totally split in H/K_P.Then the Artin symbol induces a∆-equivariant isomorphism:

Pic(R_P)'Gal(H/K_P).

Note that e_[ωP]G = G, where G = Gal(L/K_P). Thus the Artin symbol induces a F^p[∆]- morphism:

ψ:e_[ωP]

Pic(R_P) pPic(R_P)

−→Gal(L∩H/K_P).

Therefore, by Corollary 3.3 and Lemma 3.2, we get the following result which explains the congruence of Corollary 2.4:

Theorem 3.4. — The morphism of F^p[∆]-modules induced by the Artin map:

ψ:e_[ωP]

Pic(R_P) pPic(R_P)

−→Gal(L∩H/K_P), is an isomorphism, where L=K_P

e_C

P−1

P log_C(1)

and H is the Hilbert class field of R_P.

3.3. Prime decomposition of units. — A natural question arises: are there infinitely many primes P such thatφ_P−1(1)≡0 (mod P²)?

We end this note by some remarks centered around this question.

Lemma 3.5. — Let N(d) be the number of primes P of degree d such that φ_P−1(1) 6≡ 0 (modP²). Then:

N(d)> 1

d(q−1)q^d−1− q

d(q−1)q^d/2. Proof. — LetN_q(d) be the number of primes of degreed. Then:

Nq(d)> 1

dq^d− q

d(q−1)q^d/2.

LetM(d)be the number of primes P of degreedsuch thatφ_P−1(1)≡0 (mod P²).Set:

V(d) =

d−1X

i=0

L_d−1 L_i ∈A.

Thendeg_TV(d) =q^d−1,and if P is a prime of degreed,we have by Lemma 2.3 : φ_P−1(1)≡0 (modP²) if and only ifV(d)≡0 (mod P).Therefore:

M(d)≤ 1 dq^d−1.

Remark 3.6. — We have:

V(2) = 1 +T−T^q.

ThusV(2)is (up to sign) the product ofq/pprimes of degreep.Therefore there exist primes P of degree2such thatφ_P−1(1)≡0 (modP²)if and only ifp= 2,and in this case there are exactly q/2such primes.

Set H(X) = Pp−1 i=0 1

i!Xⁱ ∈ F^p[X]. Let S be the set of roots of H(X) in C∞. Note that

|S|=p−1.Let us suppose thatS ⊂Fq.LetP be a prime ofAsuch thatP dividesT^q−T−α for someα∈F^∗q.Observe that such a prime is of degreep.Now, for k= 0,· · ·, p−1,we have:

L_k≡ 1

k!(−α)^k (mod P).

Therefore:

V(p) =

p−1

X

i=0

Lp−1

L_i ≡ −α^p−1H(−1

α ) (modP).

Thus there exist at least(p−1)^q_p primesP inAof degreepsuch thatφ_P−1(1)≡0 (mod P²).

Lemma 3.7. — Let P be a prime of degree A and let n ≥ 1. We have an isomorphism of A-modules:

C A

PⁿA

' A

Pⁿ⁻¹(P −1)A.

Proof. — We first treat the case n = 1. By Lemma 2.3, we have: φ_P(X) ≡ X^qd (modP).

Therefore (P −1)C(A/P A) ={0}. Now let Q∈A such that Q.C(A/P A) ={0}. Then the polynomial φ_Q(X) (mod P) ∈ (A/P A)[X] has q^d roots in A/P A. Thus deg_TQ ≥ d.This implies that C(A/P A) is a cyclicA-module isomorphic to A/(P −1)A.

Now let us assume thatn≥2.By Lemma 2.3, we have:

∀a∈P A, v_P(φ_P(a)) = 1 +v_P(a).

This implies that C(P A/PⁿA) is a cyclic A-module isomorphic to A/Pⁿ⁻¹A and P is a generator of this module. The lemma follows from the fact that we have an exact sequence of A-modules:

0−→C P A

PⁿA

−→C A

PⁿA

−→C A

P A

−→0.

We deduce from the above lemma:

Corollary 3.8. — Let P be a prime of A. Thenφ_P−1(1)≡0 (modP²) if and only if there exists a∈A\P A such that φ_a(1)≡0 (mod P²).

This latter corollary leads us to the following problem:

Question 3.9. — Let b∈ A₊.Is it true that there exists a prime Q of A, Q ≡1 (mod b), such that φ_Q(1) is not squarefree?

A positive answer to that question has the following consequence:

Lemma 3.10. — Assume that for every b ∈ A₊, we have a positive answer to question 1.

Then, there exist infinitely many primes P such that φ_P−1 ≡0 (mod P²).

Proof. — LetS be the set of primesP such thatφ_P−1(1)≡0 (mod P²).Let us assume that S is finite. Write S ={P₁,· · · , P_s}.Set b= 1 +Q_s

i=1(P_i−1)(ifS =∅, b= 1). Let Qbe a prime of A such thatφ_Q(1)is not squarefree and Q≡1 (modb).Then there exists a prime P of Asuch that:

φ_Q(1)≡0 (mod P²).

Since φ_P(1) ≡ 1 (mod P), we have P 6= Q and therefore Q ∈ A\P A. Furthermore, for i = 1,· · ·, s, Q is prime to P_i −1. Therefore, by Lemma 3.7, φ_Q(1) 6≡ 0 (mod P_i²). Thus P 6∈S which is a contradiction by Corollary 3.8.

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October 3, 2012

Bruno Anglès, Université de Caen, CNRS UMR 6139, Campus II, Boulevard Maréchal Juin, B.P. 5186, 14032 Caen Cedex, France • E-mail :bruno.angles@unicaen.fr

Mohamed Ould Douh, Université de Caen, CNRS UMR 6139, Campus II, Boulevard Maréchal Juin, B.P.

5186, 14032 Caen Cedex, France • E-mail :mohamed.douh@unicaen.fr