In this section we globalize the constructions of the previous sections. Unfor-tunately, we have to take different approaches depending on B being split or non-split since the dualizing module of arithmetic subgroups of G is trivial (resp. non-trivial) if B is non-split (resp. split).
Let Div(P1(F)) be the free abelian group overP1(F) and let Div0(P1(F)) be the kernel of the map
Div(P1(F))→Z, X
P
mPP 7→X
P
mP.
Note that we have aG(F)-action on Div0(P1(F)) induced by theG(F)-action on P1(F). We write
DG = (
Z if B is non-split and Div0(P1(F)) if B is split.
We fix pairwise disjoint finite sets SSt, Stw and S0 of finite places of F disjoint from ram(B) and put S = SSt ∪ Stw. From now on we assume
that every p in Stw is inert in E. For an R-module N and a compact open subgroup K ⊆G(AS,∞) we consider
A(K, SSt, Stw;N)S0 =C
G(AS∪S
0,∞
)/K, (2.12)
Hom O
p∈SSt
Stp⊗ O
p∈Stw
Sttwp ,Hom(DG, N) with its natural G(F)-action, i.e. for every p ∈ SSt (resp. p ∈ Stw) we view Stp (resp. Sttwp ) as a G(F)-module via the embeddingG(F),→Gp and put
(g.Φ)(x)(fSt⊗ftw) = Φ(g−1x)((g−1fSt)⊗(g−1ftw)) for g ∈G(F), Φ∈ A(K, SSt, Stw;N)S0, x∈ G(AS∪S
0,∞)/K, fSt ∈ N
p∈SStStp and ftw ∈N
p∈StwSttwp . Further, we fix a locally constant character : T∞ −→ {±1}.
We will often viewas a character on T(F) via the embeddingT(F),→T∞. There exists a unique extension : G∞→ {±1} such that the diagram
T∞ {±1}
G∞ ι
is commutative. Again, we viewalso as a character onG(F) via the embed-dingG(F),→G∞. We are interested in the cohomology of theG(F)-modules A(K, SSt, Stw;N)().
Proposition 2.10. Let SSt, Stw be disjoint finite sets of finite places of F disjoint from ram(B) and K ⊆G(AS,∞) a compact open subgroup.
(i) Let N be a flat R-module equipped with the trivial G(F)-action. Then the canonical map
Hq(G(F),A(K, SSt, Stw;R)())⊗RN →Hq(G(F),A(K, SSt, Stw;N)()) is an isomorphism for all q≥0.
(ii) If R is Noetherian, then the groups Hq(G(F),A(K, SSt, Stw;R)()) are finitely generated R-modules for all q≥0.
Proof. This is almost verbatim Proposition 4.6. of [Spi14].
Definition 2.11. The space ofN-valued, (SSt, Stw)-special modular symbols on Gof level K and sign is defined to be
M(K, SSt, Stw;N) = Hd(G(F),A(K, SSt, Stw;N)()).
Let n ⊆ OF be a non-zero ideal coprime to ram(B). We fix an Eichler order R(n) ⊆ R of level n contained in the fixed maximal order R. As in the local case, we write Kp (resp. Kp(n)) for the image of R∗p (resp. R(n)∗p) in Gp and set
K = Y
p∈S/ ∞
Kp
resp. K(n) = Y
p/∈S∞
Kp(n)
.
Following Remark 2.6 we can identify R(n) with a b
c d
∈Mat2×2(OF) s.t. c≡0 modn
in case B is split.
We put
M(n, SSt, Stw;N) =M(K(n)S, SSt, Stw;N) and
M(n;N) =M(n,∅,∅;N).
Without loss of generality we will always assume that every p∈S divides n exactly once.
For an open subgroupUe ⊆UTS∪S0,∞ we define
D(U , S;e N)S0,∞= HomR(Cc(U , S, R)e S0,∞, N).
In case Ue = UT(m) with m ⊆ OF a non-zero ideal we write D(m, S;N)S0,∞
for the corresponding distribution space.
Since we assume that every prime p in Stw is inert in E the local maps (2.6) and (2.7) (resp. (2.10)) induce the semi-local maps
δSSt =⊗p∈SStδp: O
p∈SSt, p split
Cc(Fp,Z)⊗O
p∈SSt, p non-split
Cc(Tp,Z)/Z−→ O
p∈SSt
Stp
respectively
δStwtw =⊗p∈Stwδptw: O
p∈Stw
Cc(Tp,Z)/Z−→ O
p∈Stw
Sttwp .
For every compact open subgroup K ⊆ G(AS∪S
0,∞) and every element g ∈G(AS∪S
0,∞) we get a T(F)-equivariant homomorphism
∆Sg,S0
St,Stw: A(K, SSt, Stw;N)S0 −→ D(ι−1(gKg−1), S;N)S0 given by
∆Sg,S0
St,Stw(Φ)(x)(fSSt ⊗fStw)
=
(Φ(ι(x)g)(δSSt(fSSt)⊗δStwtw(fStw)) ifB is non-split and Φ(ι(x)g)(δSSt(fSSt)⊗δStwtw(fStw))(0− ∞) if B is split
(2.13)
for Φ ∈ A(K, SSt, Stw;N)S0, x∈T(AS∪S
0,∞)/ι−1(gKg−1) andfSSt, as well as fStw, in the appropriate semi-local function spaces.
Composing ∆g,SSt,Stw with the restriction map
M(K, SSt, Stw;N) −→Hd(T(F),A(K, SSt, Stw;N)()) on cohomology yields a map
M(K, SSt, Stw;N) −→Hd(T(F),D(ι−1(gKg−1), S;N)∞()), which we will also denote by ∆g,SSt,Stw.
Keep in mind that by Remark 2.1 there is an up to orientation unique G(Aram(B)∪S,∞)-equivariant isomorphism
G(Aram(B)∪S,∞)/K(n)ram(B)∪S ∼=Y0
p6∈ram(B)∪S∪S∞
E~p,ordp(n). (2.14) Definition 2.12. A non-zero ideal m ⊆ OF is called n-allowable if m is coprime to ram(B) and ordp(m)−ordp(n)> ηp for all p∈/ ram(B).
Let us fix an n-allowable ideal m. For a finite place p of F that is not inS∪ram(B) we defineep = (wordp(m)−ordp(n), . . . , wordp(m)), where the wi are the vertices chosen in Section 2.1. Let gm = (gp)p ∈ G(AS,∞)/K(n)S be the element that is equal to one at places in ram(B) and corresponds to (ep)p under the above isomorphism for all placesp∈/ S∪ram(B). In this case, the equality
UT(m) = ι−1(gmK(n)gm−1) holds and hence, we have a map
∆m,SSt,Stw = ∆gm,SSt,Stw:M(n, SSt, Stw;N) −→Hd(T(F),D(m, S;N)∞()).
(2.15)
As always, we drop SSt and Stw from the notation if they are empty.
For every p∈/ ram(B) the Hecke operator Tp as defined in (2.1) acts on M(n;N) via the isomorphism (2.14). Similarly, for n0 |n the global Atkin-Lehner involutionWn0 is given by applying the local Atkin-Lehner involutions Wpordp(n0) at the places p | n0. Also, for every p ∈ ram(B) the local Atkin-Lehner involution Wp is given by interchanging the two elements in the set Gp/Kp.
Remark 2.13. Once again, as in Remark 2.6 we can give an explicit descrip-tion of the elements gm ifB is a split quaternion algebra in terms of matrices by identifyingB with Mat2×2. We choose gm as the projection of the matrix (hp)p ∈PGL2(A∞) given by
hp =
1 0 0 1
!
if ordp(m) = 0,
$ordp p(m) 1
0 1
! else,
(2.16)
where $p∈ OFp is a local uniformizer at p.
3 Stickelberger elements for modular sym-bols
In this section we study Stickelberger elements coming from the modular symbols defined in Section 2. They are constructed by taking the cap product of the pullback of the modular symbols via the map (2.15) with the homology class defined in Section 1.3 associated to the Artin reciprocity map. As an immediate consequence of the results of Section 1 we can bound their order of vanishing from below and, furthermore, prove a functional equation.
Throughout this section we fix a ringR, anR-moduleN, a non-zero ideal n ⊆ O, which is coprime to ram(B), and a characteras before. In addition, we fix a modular symbol κ∈ M(n;N).
Definition 3.1. Assume thatE is a field. Let Lbe a finite Galois extension of F. We sayL isE-anticyclotomic if it contains E, is abelian overE and if τ στ−1 =σ−1 holds for all σ ∈Gal(L/E).
Let us fix a finite Galois extension L of F which is E-anticyclotomic (resp. abelian) if B is non-split (resp. split). WriteG for the Galois group of LoverE (resp.F). The Artin reciprocity map ofLoverE (resp.F) induces a group homomorphism
rL:T(A)/T(F)−→ G.
In addition, we fix an n-allowable idealmofOF that bounds the ramification of Lover E (resp. LoverF), i.e.UT(m) is contained in the kernel of rL. Let
cL =crL ∈Hd(T(F),C(m,Z[G])∞())
be the image of rL under (1.9) with S =∅. We adopt similar notations if S is not the empty set, e.g. we set cL(m, S, ) =crL(m, S, ).
The natural pairing
Cc(m,Z[G])∞× D(m;N)∞−→Z[G]⊗N induces a cap-product pairing
Hd(T(F),Cc(m,Z[G])∞())×Hd(T(F),D(m;N)∞())−→Z[G]⊗N.
Definition 3.2. The Stickelberger element of modulus massociated with κ and L/F is defined as the cap-product
Θm(L/F, κ) = ∆m(κ)∩cL∈Z[G]⊗N.
As a direct consequence of functoriality of the Artin reciprocity map we get the following compatibility property:
Proposition 3.3. LetL0 be an intermediate extension ofLoverF containing E ifE is a field. We will denote byG0 the Galois group of LoverE (resp.F) in case B is non-split (resp. split). Then we have
πL/L0(Θm(L/F, κ)) = Θm(L0/F, κ), where
πL/L0: Z[G]⊗N −→Z[G0]⊗N is the canonical projection.
Let k be an R-algebra and χ: G → k∗ a character. The character also induces anR-linear mapχ: Z[G]⊗N −→k⊗RN. Via the Artin reciprocity map we can view χ as a character of T(A). We write χ∞: T(F∞) → µ2(k) for the component at infinity of χ.
Proposition 3.4. Let k be an R-algebra which is a field and let χ: G →k∗ be a character. If χ∞6=, we have
χ(Θm(L/F, κ)) = 0.
Proof. We have
χ(Θm(L/F, κ)) = χ(cL)∩∆m(κ)
=cχ(m,∅, )∩∆m(κ).
Making the construction of cχ(m,∅, ) (in particular (1.6)) present, one sees that orthogonality of characters yields the vanishing of cχ(m,∅, ) for χ∞ 6=
.
LetSSt and Stw be finite disjoint sets of finite places of F with - p divides n exactly once for all p∈S =SSt∪Stw,
- S is disjoint from ram(B) and - every prime in Stw is inert in E.
The local evaluation maps (2.8) and (2.11) induce a map EvSSt,Stw: M(n, SSt, Stw;N) −→ M(n;N).
For a place v of F we let Gv ⊆ G be the decomposition group at v. Note that even if E is a field and v is split in E this is well defined since L is E-anticyclotomic. If p∈ S, we define Ip ⊆Z[G] as the kernel of the projection Z[G]Z[G/Gp]. If v ∈S∞ is real and split in E, we let σv be a generator of Gv and define Iv±1 ⊆Z[G] as the ideal generated by σv∓1. For complex and non-split real Archimedean places we define Iv±=Z[G].
Lemma 3.5. Assume that N is Z-flat and that there exists an (SSt, Stw )-special modular symbolκ0 ∈ M(n, SSt, Stw,;N) liftingκ, i.e. EvSSt,Stw(κS) = κ holds. Then we have
Θm(L/F, κ)∈ Y
v∈S∞
Iv−v(−1)· Y
p∈Sm
Ip
!
⊗N.
In particular, if N =R is a Z-flat ring and is trivial, we have 2−deΘm(L/F, κ)∈R[G]
and
ordR(2−deΘm(L/F, κ))≥#Sm,
where deis the number of real Archimedean places of F which are split in E.
Proof. By Lemma 2.8 (i) and Lemma 2.9 we have Θm(L/F, κ) = ∆m(κ)∩cL(m,∅, )
=± ∆m,SSt,Stw(κ0)∩cL(m, S, ).
We set I = Q
v∈S∞Iv−v(−1) ·Q
p∈SmIp and consider the ring A = Z[G]/I together with the projection maps π: Z[G]→Aand πN: Z[G]⊗N →A⊗N. We have
πN(Θm(L/F, κ)) =± ∆m,SSt,Stw(κ0)∩π∗(cL(m, S, )) = 0
since the homology classπ∗(cL(m, S, )) = cπ◦rL(m, S, ) vanishes by applying Proposition 1.6 with a=A.
Lemma 3.6. Suppose that every p ∈ ram(B) is inert in E and that we can decompose n = n1n2 such that n1 is coprime to m and n2 | m. Write n1 = Qr
i=1pnii, with ni ≥ 1 for 1 ≤ i ≤ r. Let κ be an eigenvector of Wpni i
with eigenvalue εi ∈ {±1}for1≤i≤r and ofWp with eigenvalue εp ∈ {±1}
for everyp∈ram(B). Further, write εn1 =Qr
i=1εi for the eigenvalue ofWn1. Then
Θm(L/F, κ)∨ = (−1)d+δ·(−1)·εn1 Y
p∈ram(B)
εp·Θm(L/F, κ) holds up to multiplication with an element in G with
δ=
(0 if B is non-split and 1 if B is split.
Proof. Let Φ be an element in A(K(n);N)(). By Lemma 2.5 we have
∆m(Wn1 · Y
p∈ram(B)
Wp·Φ) = (−1)δ(−1)(inv◦∆m)(JΦ))
up to multiplication with an element in T(F). For the placesp∈ram(B) we use that we have J /∈Kp (see the beginning of Section 2). The factor (−1) is the contribution of the Archimedean places and the factor (−1)δ is coming from (2.13), i.e. it follows from
J(0− ∞) =−(0− ∞) if B is split.
Hence, by passing to cohomology we get εn1 · Y
p∈ram(B)
εp·Θm(L/F, κ) = Θm(L/F, Wn1 · Y
p∈ram(B)
Wp·κ)
= (−1)δ(−1)cL∩inv(∆m(κ))
= (−1)δ(−1) inv(cL)∩∆m(κ)
= (−1)d+δ(−1)(cL)∨∩∆m(κ),
up to multiplication with an element inG. The last equality holds for the fol-lowing reason: TheT(F)-action onC(m,Z[G])∞() is inverted by applying inv and inverting the US+-action induces multiplication by (−1)d on Hd(US+,Z).
Thus we get
inv(cL) = (−1)d(cL)∨ and the claim follows.
Remark 3.7. As a consequence of the choices made in Remark 2.6 or rather by the equations (2.4) and (2.5), we have a more precise statement of Lemma 3.6 in case B is split:
Θm(L/F, κ)∨ = (−1)d+1·(−1)·εn1 ·σ−1n1 ·Θm(L/F, κ) Here σn1 = Q
p|n1σpordp(n1) and σp ∈ G is the (arithmetic) Frobenius at p (note that in particular the p | n1 are unramified by the definition of n1).
See [BG17], Proposition 1.15, for more details.
As a consequence, we can compute the parity of the order of vanish-ing of Stickelberger elements. With the same hypothesis as in the previous proposition we get:
Corollary 3.8. Suppose that N =R, r = ordR(Θm(L/F, κ))< ∞ and that 2 acts invertibly on IR(G)r/IR(G)r+1. Then we have
(−1)r = (−1)d+δ·(−1)·εn1.
Proof. The involution (·)∨ induces multiplication by (−1) on IR(G)/IR(G)2 and thus, it induces multiplication by (−1)r on IR(G)r/IR(G)r+1. Since the equality in Lemma 3.6 holds up to multiplication with an element in G, we have
(−1)rΘm(L/F, κ) = Θm(L/F, κ)∨ = (−1)d(−1)·n1 ·Θm(L/F, κ) in IR(G)r/IR(G)r+1. Assume that (−1)r 6= (−1)d(−1)·n1. Then the above equation would imply
2Θm(L/F, κ)≡0 mod IR(G)r+1 and thus we get
Θm(L/F, κ)≡0 modIR(G)r+1
since by assumption 2 acts invertibly on IR(G)r/IR(G)r+1. But this contra-dicts our assumption that the order of vanishing of Θm(L/F, κ) is exactly r.
4 Automorphic Stickelberger elements
We will apply the results of the previous sections to cohomology classes coming from automorphic forms. After constructing Stickelberger elements associated to automorphic representations and proving norm relations for them (Section 4.1) we get lower bounds for the order of vanishing. Moreover, we prove interpolation formulae (Section 4.2) and a leading term formula (Section 4.3). We close our discussion by giving a positive answer to the conjecture of Mazur and Tate in the analytic rank zero situation formulated in the introduction and by giving a construction of p-adic L-functions in Section 4.4.
4.1 Stickelberger elements associated to automorphic representations
Let π = ⊗vπv be a cuspidal automorphic representation of PGL2(A) with the following properties:
- πv is a discrete series representation of weight 2 for all real Archimedean places v of F,
- πv is isomorphic to the principal series representation π(µ1, µ2) with µ1(z) =z12z−12 and µ2(z) = z−12z12 for all complex Archimedean places v of F and
- πp is special, i.e. a twist of the Steinberg representation, for all p ∈ ram(B).
A result of Clozel (cf. [Clo90]) tells us that there exists a smallest subfield Qπ ⊆ C, which is a finite extension of Q such that π∞ = Q
p∈S/ ∞πp can be defined over Qπ. Qπ is called the field of definition of π. We write Γ0(n) ⊆ PGL2(A) for the usual adelic congruence subgroup of level n. By the automorphic formulation of Atkin-Lehner theory due to Casselman (see [Cas73]) there exists a unique non-zero idealf(π)⊆ OF such that (π∞)Γ0(f(π)) is one-dimensional. Thus, the standard Hecke operator Tp (resp. the Atkin-Lehner involutions Wp) acts on (π∞)Γ0(f(π)) via multiplication by a scalar which we denote by λp (resp. ωp). More precisely, the Hecke eigenvalues λp are elements of the ring of integers Rπ of Qπ.
If B is a non-split quaternion algebra, our assumptions on π provide a transfer of π to B, proven by Jacquet and Langlands in [JL70], i.e. there exists an automorphic representation πB of G(A) such that
- πB,v ∼=πv for all places v at which B is split,
- πB,v is the trivial one-dimensional representation for all v ∈ S∞ at which B is non-split and
- πB,p is the trivial (resp. non-trivial) smooth one-dimensional represen-tation ofGpfor everyp∈ram(B) for whichπpis the (twisted) Steinberg representation. In particular, the eigenvalue ofWpacting on πB,p is the negative of the root number of πp.
As before, we identify G and PGL2 if the fixed quaternion algebra B is split. In this situation we will write πB=π etc. to unify the notation.
In either case, let f(πB) be the maximal divisor of f(π) which is coprime to ram(B). We define
M(f(πB);Qπ),π ⊆ M(f(πB);Qπ)
to be the common eigenspace of the operators Tp forp∈/ ram(B) with eigen-values λp. The formalism of (g, K)-cohomology together with the strong multiplicity one theorem implies thatM(f(πB);Qπ),π is one-dimensional for every sign character . It follows by Proposition 2.10 (i) that
M(f(πB);Rπ)⊗Rπ Qπ −→ M(f(πB);Qπ)
is an isomorphism. Therefore, the intersection of M(f(πB);Qπ),π with the image of M(f(πB);Rπ) in M(f(πB);Qπ) is a locally free Rπ-module of rank one. We choose a maximal element κπB, of this module.
Remark 4.1.(i) If Rπ is a PID, the generator κπB, is unique up to multi-plication by an element in R∗π. In particular, if the automorphic repre-sentation π corresponds to a modular elliptic curve over F, then Qπ is equal to Qand thus, κπB, is unique up to sign.
(ii) We could weaken the assumptions on πp for p∈ ram(B). It is enough to assume that πp is either special or supercuspidal. But in the super-cuspidal case there is no canonical local new vector for πB,p. To ease the exposition, we stick to the special case.
(iii) In [BG17] we have chosen a different approach for the construction of a modular symbol. Starting with an automorphic form Φ of parallel weight 2 we construct an Eichler-Shimura homomorphism to get an element κΦ ∈ M(f(πB);C) (see Section 2.1 of [BG17]). Using this approach, one has to choose Φ properly such thatκΦ∈ M(f(πB);Rπ). As in Section 3, we fix a finite Galois extension L of F which is E-anticyclotomic (resp. abelian) if B is non-split (resp. split) and write G for the Galois group of LoverE (resp.F). Further, letmbe anf(πB)-allowable ideal of OF that bounds the ramification of L overE (resp. F).
Definition 4.2. The Stickelberger element of modulus mand sign associ-ated to πB and L/F is defined by
Θm(L/F, πB) = Θm(L/F, κπB,)∈Rπ[G].
Remark 4.3. The element Θm(L/F, πB) depends on the choice of an UTp -stable vertex and an end of the Bruhat-Tits tree for every primep∈/ ram(B).
If we take different choices, Θm(L/F, πB) is multiplied by an element of G.
Therefore, the element
Lm(L/F, πB) = Θm(L/F, πB)·(Θm(L/F, πB))∨ ∈Rπ[G]
is independent of these choices.
Next, we study the behaviour of Stickelberger elements under change of modulus.
Theorem 4.4 (Norm relations).(i) Letpbe a finite place ofF that does not divide m. Write σP for the the image of the uniformizer $P under the Artin reciprocity map rL of L over E (resp. F). Then the equality
Θmp(L/F, πB) = (λp−(∗))Θm(L/F, πB) holds with
(∗) =
0 if p is inert in E,
1p(f(πB))σP if p is ramified in E, σ−1P +1p(f(πB))σP if p is split in E, where
1p(f(πB)) =
(1 if ordp(f(πB)) = 0 and 0 else.
(ii) Let p be a finite place of F that does divide m and write m = ordp(m).
Then we have a decomposition
Θmp(L/F, πB) =λpΘm(L/F, πB)+1p(f(πB))vm(Θmp−1(L/F, πB)), where the elements vm(Θmp−1(L/F, πB)) can be characterized by the following properties:
- πL/L0(vm(Θmp−1(L/F, πB)) = vm(Θmp−1(L0/F, πB)) for all inter-mediate extensions L0 of L over F (which contain E if E is a field)
- vm(Θmp−1(L/F, πB)) = [UT(m−1)p : UT(m)p ](Θmp−1(L/F, πB)) in case the Artin reciprocity map for L over E (resp. F) is trivial on UT(mp−1)
- Let k be a field which is an Rπ-algebra andχ: G →k∗ a character such that χp has conductor pm. Then we have
χ(vm(Θmp−1(L/F, πB))) = 0.
(iii) Suppose that pis inert and divides mas well as f(πB)exactly once. Let k be an Rπ-algebra and χ: G → k∗ a character which is unramified at p. Then we have
χ(Θm(L/F, πB)) = 0.
Proof. This is a direct consequences of the local norm relations of Lemma 2.3. For part (iii), note that the local representation at πp is a (twisted) Steinberg representation and thus, the eigenvalue of Tp◦Wp on a local new vector is −1.
In the following we use the same notation as in the discussion before Lemma 3.5. Let SSt (resp. Stw) be the set of finite places p of F which are disjoint from ram(B) (and inert in E) such that the local component πp is the (twisted) Steinberg representation. As always, we setS =SSt∪Stw. Fix subsets SSt ⊆SSt and Stw ⊆Stw and put S=SSt∪Stw.
Let tS ∈ Z be the product of the exponent of the 2-torsion subgroup of M(f(πB);Rπ) and the exponent of the torsion subgroup of
M
p∈S
M(f(πB)p−1;Rπ).
If d >0, we define cS = gcd
( Y
p∈S0
(N(p) + 1)|S0 ⊂S with #S= #S0+ 1 )
.
In the case d = 0 the above cohomology groups are torsion-free and hence, tS= 1. Further, we put cS= 1.
Finally, we definenS =cS· tS.
Lemma 4.5. Let S=SSt∪Stw as above. Then the cokernel of the map EvSSt,Stw: M(f(πB),SSt,Stw;Rπ) → M(f(πB);Rπ)
is annihilated by nS.
Proof. Let κ be a preimage of κπB, in M(f(πB);Rπ). Since π is new at p for all p∈S we know that the image of κ under the trace maps
Trp: M(f(πB);Rπ) −→ M(f(πB)p−1;Rπ)
is torsion for all p ∈ S. Thus, nS·κ lies in the kernel of all trace maps Trp for p ∈ S and is also an eigenvector under the Atkin-Lehner operator Wp with eigenvalue −1 (resp. 1) if p∈SSt (resp.Stw).
We are going to prove the claim by induction on #S. The well-known short exact sequences
0−→c-indG(FK p)
p(p)Z−→
c-indG(FKp p)Z
Wp=−1
−→Stp −→0 for p∈SSt and
0−→c-indG(FK p)
p(p)Z−→
c-indG(FKp p)Z Wp=1
−→Sttwp −→0
for p∈Stw (see for example Section 2.4 of [Spi14]) induce an exact sequence M(f(πB),{p};Rπ) −−−→Ev{p} (M(f(πB);Rπ))Wp=±1 Tr−−→ M(f(πp B)p−1;Rπ) in cohomology for all p∈S. Hence, we can lift the class nS·κ if S={p}.
Now, let #S≥2. We pick an element p∈S and consider the following commutative diagram with exact columns (where, by abuse of notation, we write M(f(πB),S;Rπ) forM(f(πB),SSt,Stw;Rπ)):
M(f(πB),S;Rπ)
(M(f(πB),S− {p};Rπ))Wp=±1
M(f(πB)p−1,S− {p};Rπ)
M(f(πB),{p};Rπ)
(M(f(πB);Rπ))Wp=±1
M(f(πB)p−1;Rπ)
Trp Trp
Ev{p} Ev{p}
EvS−{p}
EvS−{p}
EvS−{p}
By the induction hypothesis we can lift nS−{p} ·κto a class eκ∈(M(f(πB),S− {p};Rπ))Wp=±1. Ifd= 0, the map
EvS−{p}:M(f(πB)p−1,S− {p};Rπ) −→ M(f(πB)p−1;Rπ)
is injective. Therefore, the claim follows from Trp(nS−{p}·κ) = 0.
Ifd >0, we consider the canonical map
ι: M(f(πB)p−1,S− {p};Rπ) −→ M(f(πB),S− {p};Rπ).
Since Trp◦ι is equal to multiplication by N(p) + 1 we see that the element (N(p) + 1)·eκ−ι(Trp(eκ)) can be lifted to a class in M(f(πB),S;Rπ). The commutativity of the diagram above together with the fact that Trp(nS−{p}·κ) vanishes implies that (N(p)+1)·eκ−ι(Trp(eκ)) is also a lift of (N(p)+1)nS−{p}· κ.
LetIv ⊆Z[G] forv ∈S∪S∞be the ideals defined in the discussion before Lemma 3.5. Then we have
Theorem 4.6 (Order of vanishing). For every f(πB)-allowable modulus m that bounds the ramification of L overE (resp. F) we have
nSmΘm(L/F, πB) ∈ Y
v∈S∞
Iv−v(−1)· Y
p∈Sm
Ip
!
⊗Rπ.
Proof. It is easy to see that the map
EvSSt,Stw: M(f(πB), SSt, Stw;Qπ),π −→ M(f(πB);Qπ),π
is an isomorphism of one-dimensional Qπ-vector spaces (cf. [Dep16], Propo-sition 4.10 for a proof in the GL2-case). Therefore, the map
M(f(πB), SSt, Stw;Rπ),π −→ M(f(πB);Rπ),π
has finite cokernel. Lemma 4.5 tells us that nSm annihilates this cokernel.
Therefore, the claim is a direct consequence of Lemma 3.5.
As a consequence of Lemma 3.6 and Corollary 3.8 we get the following results:
Proposition 4.7 (Functional equation). Suppose that every p ∈ ram(B) is inert in E and that we can decompose f(πB) = n1n2 with n1 coprime to m and n2 dividing m. Let ε be the root number of π and εn2 the product of the local root numbers of primes dividing n2. Then the equality
(Θm(L/F, πB))∨ =(−1)·ε·εn2 ·Θm(L/F, πB) holds up to multiplication with an element in G.
Proof. We have to convince ourself that this is a consequence of Lemma 3.6.
Let us decompose ε into the local ε-factors. The part corresponding to the finite places is given byεn1·εn2·Q
p∈ram(B)εp. In case thatB is non-split, the passage from π to πB yields εp = −εBp, where εBp is the Atkin-Lehner eigenvalue corresponding to πB (see the definition of πB).
For a infinite placev ofF, theε-factorεv is equal to−1. Ifvis real, this is well known (e.g. see [Gel75], Theorem 6.16). For a complex place we state a proof: The local representation is given byπv =π(µ1, µ2) withµ1(z) = z12z−12 and µ2(z) = z−12z12. By Definition, εv is given as a product ε(µ1)· ε(µ2) (see [JL70], p. 118). Using a shifting formula (see [Kud03]), (3.28)) we see that we can replaceµ1 byµe1 =z0z−1 andµ2 byµe2 =z−1z0 without changing the ε-factors. In this situation Proposition 3.8 (iv) of [Kud03] states that ε(µe1) =ε(µe2) = i, so we getεv =−1.
Altogether, for B a non-split quaternion algebra we get ε·εn2 =εn1 · Y
p∈ram(B)
εBp ·(−1)# ram(B)·(−1)#S∞
=εn1 · Y
p∈ram(B)
εBp ·(−1)#{v∈S∞s.t. Bvis split}
=εn1 · Y
p∈ram(B)
εBp ·(−1)d
using the fact that the total number of ramified places in B is even and the conditions on the splitting behaviour of B. On the other hand, if B is split, we have #S∞ =d+ 1. The claim follows.
Corollary 4.8 (Parity). Suppose that every p∈ram(B) is unramified in E and that there is a decomposition f(πB) = n1n2 with n1 coprime to m and n2
dividing m. Moreover, we assume that r= ordRπ(Θm(L/F, πB))<∞ holds and that 2 acts invertibly on IRπ(G)r/IRπ(G)r+1. Then we have
(−1)r =(−1)·ε·εn2.