Invariance of the parity conjecture for <span class="mathjax-formula">$p$</span>-Selmer groups of elliptic curves in a <span class="mathjax-formula">$D_{2p^{n}}$</span>-extension

Bulletin

SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Publié avec le concours du Centre national de la recherche scientifique

de la SOCIÉTÉ MATHÉMATIQUE DE FRANCE

Tome 139 Fascicule 4

2011

INVARIANCE OF THE PARITY CONJECTURE

Thomas de La Rochefoucauld

INVARIANCE OF THE PARITY CONJECTURE FORp-SELMER GROUPS OF ELLIPTIC CURVES IN AD2pⁿ-EXTENSION

by Thomas de La Rochefoucauld

Abstract. — We show ap-parity result in aD2pⁿ-extension of number fieldsL/K (p≥5) for the twist1⊕η⊕τ: W(E/K,1⊕η⊕τ) = (−1)h^{1⊕η⊕τ ,X}p(E/L)i, where E is an elliptic curve overK,η and τ are respectively the quadratic character and an irreductible representation of degree2ofGal(L/K) =D2pⁿ, andXp(E/L)is the p-Selmer group. The main novelty is that we use a congruence result betweenε0-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to thep-parity conjecture (using the machinery of the Dokchitser brothers).

Résumé (Invariance de la conjecture de parité des p-groupes de Selmer de courbes elliptiques dans uneD2pⁿ-extension)

On démontre un résultat dep-parité, dans une extension galoisienne de corps de nombre de groupeD2pⁿ, pour le twist1⊕η⊕τ:

W(E/K,1⊕η⊕τ) = (−1)h^{1⊕η⊕τ ,X}p(E/L)i,

oùE est une courbe elliptique définie surK,η etτ sont respectivement le caractère quadratique et une représentation irréductible de degré 2 deGal(L/K) =D2pⁿ, et Xp(E/L)est lep-groupe de Selmer. La principale nouveauté est le fait que l’on uti- lise un résultat de congruence (dû à Deligne) pour déterminer les « root numbers » locaux dans les mauvais cas (les places additives au-dessus de2et3). On donne aussi, en utilisant la machinerie des frères Dokchitser, deux applications à la conjecture de p-parité.

Texte reçu le 18 février 2010, révisé et accepté le 21 février 2011.

Thomas de La Rochefoucauld, 4 place Jussieu, 75005 Paris •

E-mail :thomas@math.jussieu.fr • Url :http://people.math.jussieu.fr/~thomas/

2000 Mathematics Subject Classification. — 11G05, 11G07, 11G40.

Key words and phrases. — Elliptic curves, Birch and Swinnerton-Dyer conjecture, parity conjecture, regulator constants, epsilon factors, root numbers.

BULLETIN DE LA SOCIÉTÉ MATHÉMATIQUE DE FRANCE 0037-9484/2011/571/$ 5.00

©Société Mathématique de France

1. Introduction

1.1. The conjecture of Birch and Swinnerton-Dyer and the parity conjecture. — Let Kbe a number field andEan elliptic curve defined overK. Denote byKv the completion of Kat a placev.

We recall a few definitions:

Definition 1.1 (Tate Module). — Thel-adic Tate module ofEis the inverse limit of the system of multiplication by l mapsE[lⁿ⁺¹]−→E[lⁿ], whereE[m]

denotes the kernel of multiplication by mon E. Set Tl(E) = lim

←−E[lⁿ], Vl(E) =Ql⊗_ZlTl(E) and

σ⁰_E/K

v,l: Gal(K_v/K_v)−→GL(V_l(E)^∗).

Fix an embedding, ι : Ql ,→ C; we can then associate to σ⁰_E/K

v,l a complex representation σ⁰_E/K

v,l,ι of the Weil-Deligne group (see [9] §13).

Remark 1.2. — One can show that the isomorphism class of σ⁰_E/K

v :=

σ⁰_E/K

v,l,ι is independent of the choice ofl andι(see [9]§13, §14, §15).

Denote byL(E/K, s)the globalL-function, product of localL-functions:

L(E/K, s) = Q

vfinite

L(E/Kv, s) Å

= Q

vfinite

L(σ⁰_E/K

v, s) ã

defined for Re(s)>³₂ (see [9] §17 for the correspondence between the classical definition of L(E/Kv, s)and the one using σ⁰_E/K

v) and by

Λ(E/K, s) =A(E/K)^s/2L(E/K, s)(2(2π)^−sΓ(s))^[K:Q],

the “complete” L-function where A(E/K) is a constant depending on the disciminant and the conductor ofE/K (see [9] §21).

Recall the following classical conjectures:

Conjecture 1.3 (Birch and Swinnerton-Dyer: BSD). — We have ords=1Λ(E/K, s) =rk(E/K).

Conjecture 1.4 (Functional equation ofΛ :FE). — L(E/K, s) has a holo- morphic continuation to Cand there is a number

W(E/K) =Q

v

W(E/Kv)∈ {±1}

such that:

Λ(E/K, s) =W(E/K)Λ(E/K,2−s)

(see [9]§12 and §19 for the definition of W(E/Kv) :=W(σ⁰_E/K

v)and[9]§21 p. 157 for the functional equation ofΛ).

This conjecture is known in a few cases:

– For elliptic curves overQthanks to modularity results on elliptic curves due to Wiles, Taylor, Breuil, Diamond and Conrad.

– For elliptic curves over a totally real field K, we only know a meromor- phic continuation and the functional equation ofΛthanks to a potential modularity result of Wintenberger (see [16]) together with an argument of Taylor.

In general, Conjecture 1.4is not known.

The conjecture of Birch and Swinnerton-Dyer implies the following weaker conjecture:

Conjecture 1.5 (BSD(mod 2)). — We have

rk(E/K)≡ords=1Λ(E/K, s) (mod 2). Combining it with the conjectural functional equation we get:

Conjecture 1.6 (Parity conjecture). — We have (−1)^rk(E/K)=W(E/K).

Tim and Vladimir Dokchitser showed that this conjecture is true assuming that the6^∞-part of the Tate-Shafarevich group ofE overK(E[2])is finite (see [5] Th 7.1 p. 20).

Definition 1.7 (Selmer group). — Let

Xp(E/K) := Hom_Zp(S(E/K, p^∞),Qp/Zp)⊗_ZpQp

where S(E/K, p^∞) := lim

−→

n

S(E/K, pⁿ) is the p^∞-Selmer group, sitting in an exact sequence:

0−→E(K)⊗Qp/Zp−→S(E/K, p^∞)−→XE/K[p^∞]−→0.

If we letrkp(E/K) := dim_QpXp(E/K) = rk(E/K)+cork_ZpXE/K[p^∞], a more accessible form of the Conjecture1.6is the following:

Conjecture 1.8 (p-parity conjecture). — We have (−1)^rkp(E/K)=W(E/K).

IfL/K is a finite Galois extension andτ is a self-dualQp-representation of Gal(L/K)then there is an equivariant form of Conjecture1.8:

Conjecture 1.9 (p-parity conjecture for (self-dual) twists) We have

(−1)^{hτ ,Xp(E/L)i}=W(E/K, τ), whereW(E/K, τ) =Q

v

W(σ⁰_E/K

v⊗ResD_vτ),Dv⊂Gal(L/K)is the decompo- sition group at v andhτ ,∗i is the usual representation-theoretic inner product of τ and the complexification of ∗.

It is this last conjecture in a particular setting that will interest us for the rest of the paper.

1.2. Statement of the main theorem and applications to thep-parity conjecture. — Let K be a number field, E/K an elliptic curve and L/K a finite Galois ex- tension such thatGal(L/K)'D2pⁿ, withp≥5 a prime number.

D2pⁿ admits the following irreducible representations overQp:

• 1 the trivial representation

• η the quadratic character

• ^pn₂⁻¹ irreducible representations of degree 2; they are of the form, I(χ) := Ind^D_C^2pn

pn (χ) =I(χ⁻¹),

where χis a non-trivial character ofCpⁿ (I(1) = 1⊕ηis reducible). See for example [12] for the description of irreducible representations ofD2pⁿ. Letτ=I(χ)be such an irreducible representation of degree 2.

Theorem 1.10. — With the notation above andp≥5, we have the following equality:

W(E/K, τ)

W(E/K,1⊕η) = (−1)^{hτ ,Xp(E/L)i} (−1)^{h1⊕η,Xp(E/L)i}

In other words, thep-parity conjecture forE/K tensored by1⊕η⊕τ holds:

W(E/K,1⊕η⊕τ) = (−1)^{h1⊕η⊕τ ,Xp(E/L)i}

Remark 1.11. — The Dokchitser brothers have shown that this equality holds in two different cases:

• In the case when p is any prime number but the extension L/K has a cyclic decomposition group at all places of additive reduction of E/K above 2 and3(see [3]Th.4.2 (1) p. 65).

• In the case when p ≡ 3 (mod 4) (without any additional assumption) using a strong global p-parity result over totally real fields due to Nekovář [8](see [5]Prop. 6.12 p. 18).

Remark 1.12. — The statement of Thm.1.10 holds for p = 3 (see previous remark). This case can be proved without using the “painful calculation” ([3]

p. 53) in the case of additive reduction (see the appendix below).

Here we prove the equality for allp≥5(without any additional assumption).

Corollary 1.13. — W(E/K,I(χ))

(−1)hI(χ),Xp(E/L)i does not depend onχ:C_pn −→C^∗. Theorem 1.10is equivalent to the fact that Hypothesis 4.1 of [3] holds for any elliptic curve and any p > 3 (using a result of the Dokchitser brothers it is also true for p= 3, see Remark1.11 above). Now using the machinery of the Dokchitser brothers (see Th.4.3 and Th.4.5 in [3]) we have the following theorems:

Theorem 1.14. — LetKbe a number field,p6= 2, andE/Kan elliptic curve.

SupposeF is a p-extension of a Galois extensionM/K, Galois over K.If the p-parity conjecture (−1)^rkpE/L=W(E/L) holds for all subfieldsK⊂L⊂M, then it holds for all subfields K⊂L⊂F.

Theorem 1.15. — Let K be a number field, p 6= 2, E/K an elliptic curve and F/K a Galois extension. Assume that the p-Sylow subgroup P of G= Gal(F/K) is normal andG/P is abelian. If thep-parity conjecture holds forE overK and its quadratic extensions inF, then it holds for all twists of E by orthogonal representations ofG.

2. Invariance of the parity conjecture in aD2pⁿ-extension

2.1. Reduction to the case of aD2p-extension. — Here we reduce the demonstra- tion of Theorem 1.10by an induction argument together with the Galois in- variance of root numbers due to Rohrlich (see [11] Theorem 2), to the following statement:

Proposition 2.1. — It is sufficient to prove Theorem 1.10 in the case when n= 1 (i.e. Gal(L/K)'D_2p).

Proof. — Suppose Theorem 1.10 is true for n = N −1. We will show that theorem is true forn=N.

Consider L/K a finite Galois extension such that Gal(L/K) 'D_2pN and τ =I(χ)an irreducible representation of degree2 ofD_2pN.

• If χ is not injective, then the statement is known by the induction hy- pothesis.

• Ifχ is injective:

Letσ= res(I(χ)) := res^D_D^2pN

2pN−1(I(χ)).

Thenσ=I(χ⁰), where χ⁰:=χ|^C_pN−1 :C_pN−1 →Qp is injective.

We have: Ind^D_D^2pN

2pN−1(σ) =L

χ₀

I(χ₀), where the sum is taken over theχ₀such that χ₀|^CpN−1 =χ|^CpN−1.

For each such χ₀ there is an element ofAut(C)sendingχinto χ₀ andI(χ) into I(χ₀).

By inductivity of root numbers in Galois extension:

W(E/K, σ) =W(E/K,Ind^D_D^2pN

2pN−1(σ)).

By Galois invariance of root numbers:

W(E/K, I(χ⁰)) =W(E/K, I(χ₀)), ∀χ₀ such thatχ₀|^C_pN−1 =χ|^C_pN−1. SoW(E/K, σ) =W(E/K,Ind^D_D^2pN

2pN−1(σ)) =W(E/K, τ)^p=W(E/K, τ).

On the other hand, hσ, Xp(E/L)i=D

Ind^D_D^2pN

2pN−1(σ), Xp(E/L)E

=p.hτ , Xp(E/L)i,

becauseX_p(E/L)is aQp-representation. So(−1)^{h1⊕η⊕σ,Xp(E/L)i}= (−1)^{h1⊕η⊕τ ,Xp(E/L)i}. By the induction hypothesis, (−1)^{h1⊕η⊕σ,Xp(E/L)i}=W(E/K, σ). As a result, W(E/K,1⊕η⊕τ) = (−1)^{h1⊕η⊕τ ,Xp(E/L)i}.

2.2. The case of aD2p-extension. — We first restate Theorem1.10in the case of a D_2p-extension.

LetKbe a number field,E/Kan elliptic curve andL/K a Galois extension such that Gal(L/K)'D_2p 'C_poC₂, with p≥5 a prime number. We are going to use the notation D2 instead of C2 to avoid confusion with the local Tamagawa factorsCv defined below.

Recall the irreducible representations of D2p overQp:

• 1 the trivial representation

• η the quadratic character

• I(χ) = Ind^G_C

p(χ) irreducible representations of degree 2, where χ is a non-trivial character ofCp.

Theorem 2.2. — With the notation above and p≥5, we have the following equality:

W(E/K, τ)

W(E/K,1⊕η) = (−1)^{hτ ,Xp(E/L)i} (−1)^{h1⊕η,Xp(E/L)i}.

In other words, the p-parity conjecture forE/K tensored by1⊕η⊕τ holds:

W(E/K,1⊕η⊕τ) = (−1)^{h1⊕η⊕τ ,Xp(E/L)i}.

The proof of Theorem2.2will occupy the rest of section2.

We use the following notations:

• v a finite place ofK

• Kv the completion ofK atv

• q=l^r_v the cardinality of the residue field ofK_v

• z |v a finite place ofL

• w|v a finite place ofL^H (whereH is a subgroup ofGal(L/K) =D2p)

• δ= ord_v(the minimal discriminant ofE/K_v)

• δ_H= ord_w the minimal discriminant ofE/ L^H

w

• eH the ramification index of L^H

w/Kv

• fH the residue degree of L^H

w/Kv

• ω⁰_E/K

v =a minimal invariant differential ofE/K_v

• Cw(E/L^H) =cw(E/L^H)ω(H),

where











cw(E/L^H) = local Tamagawa factor of E/ L^H

w

ω(H) =

ω⁰

E/Kv

ω⁰

E/(LH)_w

_(LH)

w

.

A minimal invariant differential of E/K_v and one ofE/ L^H

w differ by an element of L^H

w.If we chooseω⁰_E/K⁰

v (respω⁰⁰_E/(LH)_w) a different minimal in- variant differential ofE/Kv(respE/ L^H

w), we have ^ω

00 E/Kv

ω⁰⁰

E/(LH)_w

=α ^ω

0 E/Kv

ω⁰

E/(LH)_w

, where αis a unit in L^H

w(see [14] p. 172). Therefore ω(H)is well defined.

Furthermore, ifl_v>3 then (see [3] p. 53):

ω⁰_E/K

v

ω⁰_E/(LH)

w

(L^H)_w

=q^{δ.eH12−δHfH} Å

=q _δ.eH

12

f_H

in the case of potentially good reduction ã

. ForD2p, there is the following equality:

Ind^D_{1}^2p1−2.Ind^D_D^2p

2 1−Ind^D_C^2p

p 1 + 2.1 = 0

of virtual representations ofG, this gives theG-relationΘ :{1}−2D2−Cp+2G in the sense of [3] (Def 2.1 p. 34).

We recall two definitions in our setting (i.e. withΘ :{1}−2D2−Cp+2D_2p), for general definitions see [3].

Definition 2.3 ([3], Def. 2.13 p. 36). — Let ρbe a self-dualQp[G]-represen- tation.

Pick a G-invariant non-degenerateQp-linear pairingh,ionρand set C_Θ(ρ) = det(h,i

ρ^{1}) det(¹₂h,i

ρ^D2)⁻²det(¹_ph,i

ρ^Cp)⁻¹det(_2p¹ h,i ρ^D2p)² as an element ofQ^×p/Q^×2p , wheredet(h,i

ρ^A)isdet((hei, eji_i,j)in anyQp-basis {ei} ofρ^A.

Remark 2.4. — C_Θ(ρ)is well defined and does not depend on the choice of the pairing (see[3]Theorem 2.17 p. 37).

Definition 2.5 ([3], Def. 2.50 p. 46). — We define:

T_Θ,p=

( σa self-dual Qp[G]- representation

hσ, ρi ≡ordpCΘ(ρ) (mod 2)

∀ρa self-dual Qp[G]-representation.

)

Following the approach of the Dokchitser brothers, we have the following theorem

Theorem 2.6 (Theorem 1.14 of [3]). — Let L/K be a Galois extension of number fields with Galois group G = D2p, where p > 2 is a prime num- ber. Let Θ : {1} −2D2−Cp + 2D2p. For every elliptic curve E/K, the Qp[G]-representionX_p(E/L)is self-dual, and

∀σ∈T_Θ,p, (−1)^hσ,Xp(E/L)i= (−1)^ordp(C),

where C = Y

v-∞

Cv with Cv = Cv({1})Cv(D2)⁻²Cv(Cp)⁻¹Cv(G)² and C_v(H) = Y

w|v wplaces ofL^H

C_w(E/L^H).

Now, since1⊕η⊕τ ∈TΘ,p (see [3], example 2.53 p. 46), we only need to prove that :

(1) W(E/K, τ)

W(E/K,1⊕η)= (−1)^ordpC.

Furthermore, since we are only interested in the parity of ordp(C), we do not have to determine C_v(D₂)and C_v(G), because these terms only bring an even contribution (since they appear with an even exponent).

Both sides of(1)are of local nature.

As W(E/K, τ) =Q

v

W(E/Kv, τv), whereσv := resGal(L_z/K_v)σ, all we need to do is to prove the following local equality:

(2) W(E/K_v, τ_v)

W(E/Kv,(1⊕η)v) = (−1)^ordp(Cv),

for each finite placev ofK (v| ∞ do not contribute, sincep6= 2).

Denote byGv := Gal(Lz/Kv)the decomposition group of v. The proof of Theorem2.2 splits in several cases:

•G_v={1}(there are2pplaces above vin L) see section2.2.2

•Gv=D2 (there arepplaces above v inL) see section2.2.3

•Gv=Cp (there are2places abovev in L) see section2.2.2

•G_v=D_2p (there is a unique place abovev inL) see section2.2.4 The case where Gv is cyclic is treated in Dokchitser’s work but the proof given here is slightly different and specific to our particular choices ofGandΘ.

We first recall a few facts about the local Tamagawa factors of elliptic curves.

2.2.1. Local Tamagawa factors of elliptic curves. — The assumptions and no- tation from above are in force.

The local Tamagawa factor at v, c(E/K_v) = # E(K_v)/E⁰(K_v) , whereE⁰(K_v) ={Points of non-singular reduction}

is determined by Tate’s algorithm (see [13] IV §9):

c(E/K_v) =























1 ifEhas good reduction atv 1, 2, 3 or4 ifEhas additive reduction atv n ifEhas split multiplicative reduction

of typeIn atv

1 or2 ifEhas non-split multiplicative reduction of typeIn atv

IfE acquires semi-stable reduction overL_z, then:

1. IfE has split multiplicative reduction of typeIn overKv, then:

c(E/ L^H

w) =n.eH.

2. IfE has non-split multiplicative reduction of typeIn overKv, then:

c(E/ L^H

w) =





 n.eH

ifE has split multiplicative reduction over L^H

w

1 or2 otherwise.

3. IfE has potentially good reduction, thenc(E/ L^H

w) = 1,2,3 or4.

4. IfE has additive and potentially multiplicative reduction then:

c(E/ L^H

w) =







n.e_H ifE has split multiplicative reduction of typeI_n over L^H

w andl_v6= 2.

1,2,3or 4 otherwise.

The following proposition will be used in the subsequent computations.

Proposition 2.7. — 1. If w1 andw2 are two places of L above the same v, then: cw1(E/L) =cw2(E/L). In particular:

(Cv({1}) =Cw(E/L)^r Cv(Cp) =Cw⁰(E/L^Cp)^r0,

wherer=the number of placeswofLsuch thatw|vandr⁰=the number of placesw⁰ of L^Cp such thatw⁰ |v.

2. IfE/K has potentially good reduction atv, then:∀w(resp. w⁰) place ofL (of L^Cp),cw(E/L) (cw⁰(E/L^Cp))∈ {1, ..,4}, and thereforeordp(cv) = 0 and(−1)^ordp(Cv)= (−1)^ordp

ω({1}) ω(Cp)

.

3. If the reduction of E/K at v is semi-stable, then ∀H subgroup of D2p, δH=δ.eH and thereforeω(H) = 1 and(−1)^ordp(Cv)= (−1)^ordp(cv). 4. If v -p(i.e. p6=lv,p is fixed,lv is variable), then ordp(ω(H)) = 0 and

(−1)^ordp(Cv)= (−1)^ordp(cv).

Remark 2.8. — By points3and4of the proposition, ifE/K has good reduc- tion at v, then: (−1)^ordp(Cv) = 1. As _W^W(E/K_(E/K ^v,τv)

v,(1⊕η)v) = _det(1⊕η)^detτv(−1)

v(−1) = 1 in the case of good reduction, we have the desired equality (2)in the case of good reduction atv.

Remark 2.9. — From 2 and 4 we deduce that the only case that needs the calculation of bothω(H)andc_w(E/L^H)is the case of additive potentially mul- tiplicative reduction atv|p.

2.2.2. The cases Gv = {1} and Gv = Cp. — In these cases, Cv({1}) and Cv(Cp)are squares, so ordp(Cv)≡0 (mod 2).

• IfGv={1},resGal(L_z/K_v)τ= 1⊕1 = (1⊕η)v, hence _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)v)= 1.

• IfGv=Cp,(1⊕η)v= 1⊕1andτv=χ⊕χ^∗, so

W(E/K_v, τ_v) = 1 =W(E/K_v,(1⊕η)_v)(see [3] lemma A.1 p. 69).

As a result, in both cases we have: _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)v) = 1 = (−1)^ordp(Cv). 2.2.3. The case Gv=D2. — We haveτv= (1⊕η)v, so _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)v)= 1.

On the other hand, in this case, ∀w⁰ |v place ofL^Cp and ∀w |w⁰ place of L,[ L^Cp

w⁰ :Kv] = 2and L^Cp

w⁰ =Lw. In particular, Cv({1}) =Cv(Cp)^p, thereforeC_v=C_v(C_p)^p−1 andord_p(C_v) = 0.

Finally, we get: _W(E/K^W(E/Kv,τv)

v,(1⊕η)v) = 1 = (−1)^ordp(Cv).

2.2.4. The case Gv =D2p. — Denote by w (respz) the unique place of L^Cp (resp L) abovev.

In this case, there are two possibilities for the inertia group of Gv,Iv =Cp

or D2p (becauseIv is a normal subgroup ofGv =D2p andGv/Iv is cyclic).

Furthermore, iflv6=pthenIv =Cp:

– For l_v 6= 2 because the inertia group of a tamely ramified extension is cyclic.

– Forl_v= 2because the caseI_v =D_2p,I_v^wild=D₂(the wild inertia group) is impossible since I_v^wildis normal inIv.

2.2.4.1. Computation of(−1)^ordp(Cv)

1. IfE/Kv has potentially multiplicative reduction:

(a) IfE/Kv acquires split multiplicative reduction of type In over Lz

(and therefore over L^Cp

w), then:

Cv({1}) =cw(E/Lz) =e_L

z/(^LCp)_w×cw⁰(E/ L^Cp

w)

=e_{1}

e_Cp ×cv(E/K)Cv(Cp)

=e_{1}

e_Cp ×Cv(Cp) but

(ifI_v=C_p thene_{1}=pande_Cp= 1 ifIv=D2p thene_{1}= 2pandeC_p= 2.

In both cases we get: C_v=pand(−1)^ordp(Cv)=−1.

(b) If E/Kv does not acquire split multiplicative reduction of typeIn

overLz (and therefore nor over L^Cp

w), then:

cv({1}), cv(Cp)∈ {1,2,3,4} andordp

Åω({1}) ω(C_p) ã

≡0 (mod 2).

The second claim is a consequence of Proposition2.7.4in the case lv 6=p.

In the casel_v =p, we have to distinguish two cases:

(i) If E/Kv acquires non-split multiplicative reduction of type In overLz (and therefore over L^Cp

w), thenδ_{1}=δC_p. Furthermore,fC_p=f_{1} = 1or2and ^ω({1})_ω(C

p) =q^{δf(e{1}−eCp)}, soordp

Ä_ω({1})

ω(C_p)

ä≡0 (mod 2)(becausep−1| e_{1}−eC_p ).

(ii) If E/Kv, E/ L^Cp

w and E/Lz have additive reduction (of typeI_n^∗):

• IfI_v =C_p, thenf_Cp=f_{1}= 2 and the result follows.

• if Iv =D2p, since p≥ 5, E becomes of type I_2n^∗ over L^Cp

w andI_2pn^∗ overLz and we get:

ord_p(ω({1})) = ord_p(ω(C_p))≡0 (mod 2). To sum up, in the case of potentially multiplicative reduction:

(−1)^ordp(Cv)=

(−1 ifE/(L^Cp)has split multiplicative reduction 1 otherwise.

2. IfE/Kv has potentially good reduction, then:

(a) If I_v = C_p (i.e. e_{1} = pand e_Cp = 1), we get: f_{1} =f_Cp = 2 so ordp(ω(Cp)) ≡ ordp(ω({1})) ≡ 0 (mod 2) and therefore (−1)^ordp(Cv)= 1 (see Proposition2.7.2).

(b) If Iv =D2p (i.e. e_{1}= 2p,eC_p= 2andlv=p), we get:

Cv({1})

C_v(C_p) = ω({1}) ω(C_p) =q

ö_δ.e

{1}

12

ù

−

ö_δ.eCp

12

ù

=qb^δ.2p12 c⁻b^δ.212c.

(i) Ifqis an even power ofp, then (−1)^ordp(Cv)= (−1)^ordp

ω({1}) ω(Cp)

= 1.

(ii) Ifqis an odd power ofp: A computation of ö_δ.2p

12

ù and δ.2 12

depending on pmodulo 12gives the following table:

Table of values of (−1)^ordp(Cv) depending on the Kodaira symbol of the curve (and the value of e = _pgcd(δ,12)¹² ) and pmod 12:

pmod 12 1 5 7 11

II, II^∗(e= 6) 1 -1 1 -1 III, III^∗(e= 4) 1 1 -1 -1 IV, IV^∗(e= 3) 1 -1 1 -1

I_o^∗(e= 2) 1 1 1 1

In relation to the above table it may be useful to recall the following fact: if the residue characteristic ofK_v is>3, then we have the following correspondence betweene= _pgcd(δ,12)¹² , the valuation of the minimal discriminantδand the Kodaira symbols:

e= 1 ⇔δ= 0 ⇔E is of typeI0

e= 2 ⇔δ= 6 ⇔E is of typeI₀^∗

e= 3 ⇔δ= 4or 8 ⇔E is of typeIV orIV^∗ e= 4 ⇔δ= 3or 9 ⇔E is of typeIII or III^∗ e= 6 ⇔δ= 2or 10 ⇔E is of typeII orII^∗. For the meaning of the Kodaira symbols see [13] p. 354.

2.2.4.2. Computation of _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)_v)

1. The case of potentially multiplicative reduction:

We have an explicit formula of Rohrlich (see [10] Th.2 (ii) p. 329):

W(E/K_v, σ) = detσ(−1)χ(−1)^dimσ(−1)^hχ,σi,

where χis the character ofK_v^∗ associated to the extensionKv(√

−c6)of Kv (c6 is the classical factor, see [14] p. 46).

Since dimτ_v = dim 1⊕η = 2, det(τ_v) = det(1⊕η)and hχ, τvi= 0, we get:

W(E/Kv, τv)

W(E/K_v,(1⊕η)_v)= (−1)^hχ,τvi

(−1)h^χ,(1⊕η)vi = 1

(−1)h^χ,(1⊕η)vi = (−1)h^χ,(1⊕η)vi. (a) If the reduction ofE/Kv is split multiplicative (i.e. χ= 1):

Then(−1)hχ,(1⊕η)_vi=−1.

(b) If the reduction of E/K_v is non-split multiplicative (i.e. χ is an unramified quadratic character):

(i) IfE acquires split multipl. reduction overLz (and therefore over L^Cp

w), thenη_v=χ, hence(−1)h^χ,(1⊕η)vi=−1.

(ii) If E acquires non-split multiplicative reduction over Lz (and therefore over L^Cp

w), then η_v 6= χ, hence (−1)hχ,(1⊕η)_vi= 1.

(c) If the reduction ofE/K_v is additive (i.e. χis a ramified quadratic character)

(i) IfE acquires split multipl. reduction overLz (and therefore over L^Cp

w), thenη_v=χ, hence(−1)h^χ,(1⊕η)vi=−1.

(ii) If E acquires non-split multiplicative reduction over Lz (and therefore over L^Cp

w), then η_v 6= χ, hence (−1)hχ,(1⊕η)_vi= 1.

To sum up, in the case of potentially multiplicative reduction:

W(E/K_v, τ_v) W(E/Kv,(1⊕η)_v) =

(−1 ifE/(L^Cp)has split multiplicative reduction 1 otherwise.

= (−1)^ordp(Cv), by2.2.4.1.1 2. The case of potentially good reduction:

Here we have to distinguish the caseslv=pandlv6=p.

(a) The casel_v=p.

We have again an explicit formula of Rohrlich, since p ≥ 5 (see [10], Th.2 (iii) p. 329). We use the following notation:

• q =p^r the cardinality of the residue field residue degree of Kv

• e= _pgcd(δ,12)¹²

• =















1 ifr is even ore= 1 Ä−1

p

ä ifr is odd ande= 2 or6 Ä−3

p

ä

ifr is odd ande= 3 Ä−2

p

ä

ifr is odd ande= 4.

Then∀σ a self-dual representation ofGal(K_v/K_v)with finite im- age:

W(E/Kv, σ) =







α(σ, ) ifq≡1[e]

α(σ, )(−1)^{h1+ηnr+ˆσe,σi} ifq≡ −1[e]

ande= 3,4,6,

where η_nr is the unramified quadratic character, σˆe is an ir- reductible representation of degree 2 of D2e and α(σ, ) :=

(detσ)(−1)^dimσ.

Sincedimτv= dim (1⊕η)_v= 2 anddetτv= det (1⊕η)_v, α((1⊕η)_v, ) =α(τv, )and we get:

W(E/Kv, τv) W(E/K_v,(1⊕η)_v) =

(1 ifq≡1[e]

(−1)^{h1+ηnr+ˆσe,1+ηv+τvi} ifq≡ −1[e]ande= 3,4,6,

=

(1 ifq≡1[e]

(−1)^{h1+ηnr,1+ηvi} ifq≡ −1[e]ande= 3,4,6, (hˆσe, τvi= 0sincee= 3,4,6andp≥5).

(i) Ifris even, thenq≡1[e]∀e∈ {2,3,4,6}and therefore W(E/K_v, τ_v)

W(E/Kv,(1⊕η)_v) = 1 = (−1)^ordp(Cv),

by 2.b.i (in section 2.3.4.1).

(ii) Ifris odd, thenq≡1[e]⇐⇒p≡1[e]and:

W(E/K_v, τ_v) W(E/Kv,(1⊕η)_v) =

(1 ifq≡1[e]

(−1)^{h1+ηnr,1+ηvi} ifq≡ −1[e]ande= 3,4,6.

(A) IfI_v =C_p, thenη_nr=η_v and _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)_v)= 1.

(B) IfIv =D2p, thenη_nr6=η_v and:

W(E/Kv, τv) W(E/K_v,(1⊕η)_v) =

(1 ifq≡1[e]

−1 ifq≡ −1[e] ande= 3,4,6.

In both cases, we obtain for the values of _W(E/K^W(E/Kv,τv)

v,(1⊕η)_v)

exactly the same table as for the values of (−1)^ordp(Cv), depending on p modulo 12. Here is the table of values of

W(E/K_v,τ_v)

W(E/Kv,(1⊕η)_v) depending on the Kodaira symbol of the curve (and the value ofe= _pgcd(δ,12)¹² ) andpmod 12:

pmod 12 1 5 7 11

II, II^∗(e= 6) 1 -1 1 -1 III, III^∗ (e= 4) 1 1 -1 -1 IV, IV^∗(e= 3) 1 -1 1 -1

I_o^∗(e= 2) 1 1 1 1

(b) The caselv6=p:

In this case, the explicit formula of Rohrlich cannot be used, since l_v can be2 or3.

Let σ be a representation σ : Gal(Kv/Kv) → GL(Vσ) with fi- nite image; let σ⁰_E/K

v : W D(Kv/Kv)→ GL(V) be the represen- tation of the Weil-Deligne group associated to the elliptic curve given by σE/K_v, N

= σE/K_v,0

(because the reduction is po- tentially good). This is simply a representation of the Weil group W( ¯Kv/Kv)(becauseN = 0) and

σ⁰_E/K

v⊗σ=σ_E/Kv⊗σ:W(K_v/K_v)→GL(W),

whereW =V ⊗Vσ, is also a representation of the Weil group.

We first recall the definition of root numbers viaε-factors (see [9]

§11 and §12):

W(E/Kv, σ) = ε(σ_E/Kv ⊗σ, ψ, dx) ε(σ_E/Kv ⊗σ, ψ, dx)

=ε(σ⁰_E/K

v ⊗σ, ψ, dxψ),

where dxis any Haar measure, ψ is any additive character of Kv

anddx_ψ the self-dual Haar measure with respect toψonK_v.

Here, we choose an additive characterψfor which the Haar measure dxψtakes values (on open compact subsets ofKv) inZp[ζ_p], where ζ_p is a primitivep-th root of unity. For example, if the conductor ofψis trivial, then the values ofdxψ lie in l^Z_v∪ {0} ⊂Zp[ζ_p].

In one of his articles ([2] p. 548), Deligne gives a description of the ε-factors in terms ofε₀-factors; in our settings this gives:

ε(σ_E/Kv⊗σ, ψ, dxψ) =ε0(σ_E/Kv⊗σ, ψ, dxψ) det(−ν(φ)|W^I(v)), whereφis the geometric Frobenius atvandI(v) = Gal( ¯K_v/K_v^ur).

Recall that, sincelv 6=p, the inertia group ofD2p isIv =Cp. (i) IfEhas additive reduction, denote byF the smallest Galois

extension ofK_v^ursuch thatEhas good reduction overF and set Φ = Gal(F/K_v^ur); then the restiction of σ_E/Kv to I(v) factors throughΦ.

It is known that:

• Forlv≥5,Φis cyclic of ordere= _pgcd(δ,12)¹² .

• Forl_v= 3,|Φ| ∈ {2,3,4,6,12}.

• Forlv= 2,|Φ| ∈ {2,3,4,6,8,24}.

For a more precise description ofΦ, see, for example, [1] or [6].

The representationσ_E/K⊗σ(σ=τv or(1⊕η)_v) restricted toI(v)factors through a quotientH ofI(v)which admitsΦ andC_p as quotients.

We have:

(V ⊗Vσ)^I(v)= (V ⊗Vσ)^H = HomH(V^∗, Vσ) = Hom((V^Φ)^∗, V_σ^Cp) because H acts on V (resp. on Vσ) through its quotient Φ (resp. C_p) and|Φ|is prime top.

Futhermore,V^H =V^Φ={0}sinceEhas additive reduction, hence

(V ⊗V_σ)^I(v)= 0, detÄ

−Ä σ⁰_E/K

v ⊗σä

(φ)|(V ⊗V_σ)^I(v)ä

= 1 and

(3) W(E/K_v, σ) =ε₀(σ_E/Kv⊗σ, ψ, dx_ψ) (σ=τ_v,(1⊕η)_v).

Deligne also gives congruence results for theseε₀ ([2] p. 556- 557). Sinceχ≡1 mod(1−ζ_p), we deduce

I(χ)≡I(1) mod(1−ζ_p)andσ⁰_E/K

v⊗τv ≡σ⁰_E/K

v⊗(1⊕η)_v mod(1−ζ_p). So according to Deligne,ε0(σ⁰_E/K

v⊗τv, ψ, dxψ) and ε0(σ⁰_E/K

v ⊗(1⊕η)_v, ψ, dxψ) are two elements of {±1}

(by (3)), which are congruent modulo (1−ζ_p), hence they are equal. As a result,

W(E/Kv, τv)

W(E/K_v,(1⊕η)_v) = 1.

(ii) IfE has good reduction, thenσE/K_v is unramified.

Then we have:

ε(σ_E/Kv⊗τv, ψ, dx) =ε(τv, ψ, dx)^dimσE/Kvdetσ_E/Kv(Φ^m(τv,ψ)), where m(τ_v, ψ)∈ N depends on conductors of bothτ_v and ψ, and the dimension ofτ_v (see [15] 3.4.6 p. 15), therefore:

W(E/Kv, τv) =W(σE/K_v ⊗τv) = ε(σE/K_v ⊗τv, ψ, dx) ε(σ_E/Kv ⊗τ_v, ψ, dx)

= 1, since detσ_E/Kv = 1, W(τ_v) = _|ε(τ^{ε(τv,ψ,dx)}

v,ψ,dx)| = ±1 (because detτv = 1, see Proposition p. 145 [9]) anddimσ_E/Kv = 2.

Similarly,W(E/Kv,(1⊕η)_v) = 1, so _W^W_(E/K^(E/Kv,τv)

v,(1⊕η)_v)= 1.

In both cases i) and ii) we also have(−1)^ordp(Cv) = 1by 2.a. (in section 2.3.4.1).

To sum up, we have, for each finite primev ofK, W(E/Kv, τv)

W(E/Kv,(1⊕η)_v) = (−1)^ordp(Cv). This completes the proof of Theorem2.2.

Remark 2.10. — This proof can be adjusted to work in the case Gal(L/K)'D2pⁿ, the computations are almost the same. The idea to reduce the proof to the case of a D_2p-extension, using Galois invariance of Rohrlich[11], was suggested to me by Tim Dokchitser.

3. Appendix

The purpose of this appendix is to make a small improvement on Theorem 6.7 of [5]. The interest of this improvement is that Proposition 6.12 of [5]

(which is the same statement as Theorem1.10forp≡3 mod 4)will no longer rely on the “truly painful case of additive reduction” anymore (see [3] p. 53).

In fact, we use the passage to the global case to avoid all places of additive reduction, not just those above 2 and 3. Since we have proved the result for p≥5(Theorem1.10) without using any global parity results at all, for us this is of interest essentially in the casep= 3.

We start by recalling the definition of an elliptic curve beingclose toanother one:

Proposition 3.1. — Let

E

^{: y2 +a}1xy+a3y = x³+a2x²+a4x+a6 be an elliptic curve over a non archimedean local field

K

(with valuation v and residue characteristic p) and

F

^/

K

a finite Galois extension.

There exists ε > 0 such that every elliptic curve

E

^{0 : y2 +a0}1xy+a⁰₃y = x³+a⁰₂x² +a⁰₄x+a⁰₆ over

K

^{satisfying ∀i|a0}i−ai|_v < ε, has the following properties:

Over all intermediate fields

F

^{0 of}

F

^/

K

^,

E

^and

E

⁰ have the same:

– conductor

– valuation of the minimal discriminant – local Tamagawa factors,C(E/

F

^0,_2y+a^dx_1x+a3⁾

– root numbers

– the Tate module as aGal(

K

^/

K

)-module (for eachl6=p).

We will say that

E

⁰ is close to

E

^/

K

^.

Proof. — This is Proposition 3.3 of [5].

We now state the minor improvement of Theorem 6.7 of [5]:

Theorem 3.2. — Let

K

a local non archimedean field of characteristic0 and

F

^/

K

a finite Galois extension. Let F/K be a Galois extension of totally real fields and v0 a place ofK such that:

• v0 admits a unique place¯v0 of F above it

• K_v0 '

K

^andFv¯0 '

F

^.

Such an extension exists (see Lemma 3.1 of[5]).

Let

E

^/

K

be an elliptic curve with additive reduction.

Then there exists an elliptic curveE/K such that:

• E has semi-stable reduction for allw6=v₀

• j(E)is not an integer (i.e. j(E)∈/

O

K)

• E/Kv0 is close to

E

^/

K

^.

Proof. — We first choose an elliptic curve E/K such that E/Kv₀ isclose to

E

^/

K

(this is possible, by Proposition3.1).

Now the goal is to remove all places of additive reduction by changingE/K to an elliptic curve satisfying the three conditions of the theorem.

LetE:y²+a1xy+a3y=x³+a2x²+a4x+a6with ai ∈

O

K.

If we want a placewnot to be of additive reduction we have to impose one of the two following conditions:

• The valuationw(∆)is zero (in this casewis of good reduction).

• The valuation w(c4) is zero (in this case w is of good or multiplicative reduction depending onw(∆) = 0 or>0).

Letv6=v0 be a place ofKnot above2.

To get the condition “j(E) is not an integer” it is sufficient to make v a multiplicative place (v is multiplicative ⇔ v(j(E)) < 0). We will do this in Step 2 below. But before doing this, we will show in Step 1 how to make semistable all places above2.

Step 1: Make semi-stable all placesw6=v0 above2.

Denote byv_2,1, ..., v_2,r these places.

In this case:

v2,i(a1) = 0⇒v2,i(c4) = 0(c4= (a²₁+ 4a2)²−24a1a3−48a4) . Letp0andp2,ibe the primes ideals associated to v0andv2,i.

By the Chinese remainder theorem, there exists d₁∈

O

K such that:

• d₁≡0modpⁿ₀ (i.e. v₀(d₁)≥n).

• d1≡1−a1 modp2,i ∀i∈ {1, .., r} (i.e. v2,i(a1+d1) = 0).

• d1≡ −a1 modp(passociated to v6=v0).

So, if we leta⁰₁=a1+d1fornbig enough we get the curvey²+a⁰₁xy+a3y= x³+a₂x²+a₄x+a₆ which is close to

E

^/

K

^{, v}2,i(a⁰₁) = v_2,i(a₁ +d₁) = 0

∀i∈ {1, .., r}andv(a⁰₁)>0.

Step 2: Make v semi-stable.

By the Chinese remainder theorem, there exist d2, d3, d4∈

O

K such that:

• d2≡0 modpⁿ₀ (i.e. v0(d2)≥n)d2≡1−a2 modp(sov(a2+d2) = 0).

• d3≡0 modpⁿ₀ (i.e. v0(d3)≥n)d3≡ −a3 modp (sov(a3+d3)>0).

• d4≡0 modpⁿ₀ (i.e. v0(d4)≥n)d4≡ −a4 modp (sov(a4+d4)>0).

So, if we leta⁰_i=ai+di,i∈ {2,3,4}, fornbig enough we get:

E⁰ :y²+a⁰₁xy+a⁰₃y =x³+a⁰₂x²+a⁰₄x+a6 is close to

E

^/

K

(Proposition 3.1).

Futhermore : •c⁰₄= (a⁰²₁ + 4a⁰₂)²−24a⁰₁a⁰₃−48a⁰₄

•v(a⁰₁)>0

•v(a⁰₃)>0

•v(a⁰₄)>0

•v(a⁰₂) = 0, so v(c⁰₄) = 0.

The curve E⁰ : y²+a⁰₁xy+a⁰₃y = x³+a⁰₂x²+a⁰₄x+a6 is close to

E

^/

K

^;

∀w6=v₀ above 2, w(c⁰₄)>0, andv(c⁰₄) = 0. Since c⁰₄ does not depend ona₆, we can modifya6to allow placesw6=v0such thatw(c⁰₄)>0to become places of good reduction (since c⁰₄ will be unchanged, some places of good reduction can become of multiplicative reduction but not of additive reduction) and such