Critical slope p adic L functions of CM modular forms

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Lei, Antonio, Loeffler, David and Zerbes, Sarah Livia. (2012) Critical slope p-adic

L-functions of CM modular forms. Israel Journal of Mathematics, 198 (1). pp. 261-282.

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CRITICAL SLOPE

P

-ADIC

L

-FUNCTIONS OF CM

MODULAR FORMS

BY

Antonio Lei∗

Department of Mathematics and Statistics Burnside Hall

McGill University Montreal QC Canada H3A 2K6 e-mail: [email protected]

AND

David Loeffler

Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL, UK e-mail: [email protected]

AND

Sarah Livia Zerbes

Mathematics Research Institute Harrison Building University of Exeter Exeter EX4 4QF, UK e-mail: [email protected]

ABSTRACT

For ordinary modular forms, there are two constructions of ap-adicL -function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the the critical-slope L -function arising from Kato’s Euler system and comparing this with results of Bella¨ıche on the critical-slopeL-function defined using overconvergent modular symbols.

1. Setup

1.1. _{Introduction.} Letfbe a cuspidal new modular eigenform of weight≥2, andpa prime not dividing the level of f. It has long been known that if αis any root of the Hecke polynomial off atpsuch thatvp(α)< k−1, then there

is ap-adic L-function Lp,α(f) interpolating the critical L-values of f and its

twists by Dirichlet characters ofp-power conductor; see [12, 1, 16].

Iff is non-ordinary (the Hecke eigenvalue off at phas valuation>0) then both roots of the Hecke polynomial satisfy this condition, but iff is ordinary, then there is one root with valuation k−1 (“critical slope”), to which the classical modular symbol constructions do not apply. Two approaches exist to rectify this injustice to the ordinary forms by constructing a critical-slope p-adicL-function. Firstly, there is an approach using p-adic modular symbols [15, 14, 2]. Secondly, there is an approach using Kato’s Euler system [9] and Perrin-Riou’s p-adic regulator map [13] (cf. [4, Remarque 9.4]). Although it is natural to conjecture that the objects arising from these two constructions coincide (cf. [14, Remark 9.7]), and the results of [10] are strong evidence for this conjecture, prior to the present work this was not known in a single example.

In this paper, we show that the two critical-slope L-functions coincide for modular forms of CM type. In this case, Bella¨ıche has shown [3] that the “modular symbol” critical-slopep-adicL-function is related to the Katz p-adic L-function for the corresponding imaginary quadratic field. We show here that

∗_{The first author is grateful for the support of a CRM-ISM postdoctoral fellowship.}

the same relation holds for the Kato critical slopep-adic L-function, by com-paring Kato’s Euler system with another Euler system: that arising from el-liptic units. Using the results of [18] and [5] relating elel-liptic units to Katz’s L-function, we obtain a formula (Theorem 3.2) for the KatoL-function, which coincides with Bella¨ıche’s formula for its modular symbol counterpart (up to a scalar factor corresponding to the choice of periods). This establishes the equality of the two critical-slopep-adicL-functions for ordinary eigenforms of CM type (Theorem 3.4).

1.2. _Notation. Let K be a finite extension of either Qor Qp, where pis an

odd prime. We writeK∞=K(µp∞),K for an algebraic closure ofK andKab for the maximal abelian extension of K in K. Ap-adic representation of the absolute Galois group Gal(K/K) is a finite-dimensional Qp-vector space with

a continuous linear action of Gal(K/K).

A Galois extension L of K will be called a p-adic Lie extension if G = Gal(L/K) is a compact p-adic Lie group of finite dimension. In this case, we denote by Λ(G) its Iwasawa algebra; it is defined to be the completed group ring

Λ(G) = lim

←−Zp[G/U],

whereU runs over all open normal subgroups ofG. We writeQ(G) for the total quotient ring of Λ(G). If R is ap-adically complete_Zp-algebra, we shall write

ΛR(G) forR⊗bΛ(G), the Iwasawa algebra with coefficients inR.

If L is a complete discretely valued subfield of _Cp, we write HL(G) for the

algebra of L-valued distributions on G (the continuous dual of the space of locally L-analytic functions). This naturally contains ΛL(G) as a subalgebra.

WhenGis the cyclotomic Galois group Γ (isomorphic to Z×p), and i∈ Z, we

shall write`i for the element log(γ)

logχ(γ)−iofHQp(Γ) (where γis any element of

Γ of infinite order, and χis the cyclotomic character).

Assume now thatKis a number field, and letS be a finite set of places ofK (which we shall always assume to contain the infinite places). Let KS _{be the}

maximal extension of K which is unramified outsideS, and let V be a p-adic representation of Gal(KS_{/K). For an extensionLofK contained inK}S_{, write}

H1

S(L, V) for the Galois cohomology group H1(Gal(KS/L), V). Let T be a

Gal(K/K)-stable lattice inV. IfL⊂KS is ap-adic Lie extension ofK, define

H_Iw1 _,S(L, T) = lim ←−H

1

whereLn is a sequence of finite Galois extensions of K such that L=SnLn

and the inverse limit is taken with respect to the corestriction maps. Note thatH_Iw1 _,S(L, T) is equipped with a continuous action ofG= Gal(L/K), which extends to an action of Λ(G). We also defineH1

Iw,S(L, V) =H 1

Iw,S(L, T)⊗ZpQp,

which is independent of the choice of latticeT.

Similarly, let F be a finite extension of _Qp, V a p-adic representation of

Gal(F /F) andT a Gal(F /F)-invariant lattice inV. For ap-adic Lie extension LofF such thatL=S_L

n withLn/F finite Galois, define

H_Iw1 (L, T) = lim ←−H

1_(L

n, T) and HIw1 (L, V) =H 1

Iw(L, T)⊗ZpQp.

For a finite extension K of _Q, denote by _AK the ring of ad`eles of K. If f is an integral ideal of K, write K(f) for the ray class field modulo f. Let K(fp∞) =S

nK(fp

n_{), and define the Galois groupG}

fp∞ = Gal(K(fp∞)/K).

1.3. _{Gr¨ossencharacters.} LetK be an imaginary quadratic field. We fix an embeddingK ,→C. An algebraic Gr¨ossencharacter ofKof infinity-type (m, n) is a continuous homomorphism ψ:K×\A×K −→C× whose restriction to C× is

given byz7→zmz¯n.

Let θ be the Artin map Kb×/K× −→Gal(Kab/K). We choose the normal-izations such that

θ($q) = [q]−1modIq,

where $q is a uniformizer at the prime q, Iq is the inertia group and [q] is

the arithmetic Frobenius element atq. Then we have the following well-known result:

Theorem 1.1 (Weil, [17]): Letψ be an algebraic Gr¨ossencharacter ofK, and letL be the finite extension of Qinside Cgenerated by ψ(Kb×). Then for any

prime λofL, there is a (clearly unique) continuous character

ψλ: Gal(K/K)−→L×λ

with the property that

ψλ◦θ=ψ|K_b×.

The character ψλ is unramified outside the primes dividing`f, where ` is the

prime ofQbelowλandfis the conductor ofψ.

The choice of normalization for the Artin map implies that

for eacha coprime to`f. With these conventions, the Hodge–Tate weights1 of ψλ are given as follows. Letλbe a prime ofL, andµasplit prime ofK, which

lie above the same prime of L∩K. Then the decomposition groups of µand µ in Gal(Kab_{/K) are each isomorphic to Gal(}

Qabp /Qp), and the Hodge–Tate

weight ofψλism atµandnatµ.

2. Comparison of Euler systems

2.1. Elliptic units. As above, let K be an imaginary quadratic field, with a fixed choice of embedding K ,→ C. We shall fix, for the remainder of this paper, an embedding K ,→ C compatible with this choice. In particular, for each integral idealf, we regard the ray class field K(f) as a subfield of _C, and we write K(f)+ _{for its real subfield}2_.

Definition 2.1: If L is a subfield ofC, a CM-pair of modulus f over L is a

pair (E, α)consisting of an elliptic curveE/Land a pointα∈E(L)tors, such that

• there is an isomorphism EndKL(E)∼=OK, such that the resulting

ac-tion ofEndKL(E)oncoLie(E/KL)∼=KLis the natural action ofK;

• the annihilator of αin OK is exactlyf;

• there is an isomorphism E(C)−→C/fmappingαto1.

Note that we do not assume thatL⊇K here, hence the slightly convoluted statement of the first condition.

Theorem 2.2: Letfbe such thatO_K×∩(1 +f) ={1}, f=f, and the smallest integer infis≥5. Then there exists a CM-pair of modulusfoverK(f)+_{, and for} any fieldL containingK(f)+_{, this CM-pair is the unique CM-pair of modulus}

foverL up to unique isomorphism.

Proof. Consider the canonical CM-pair (C/f,1) overC. This corresponds to a pointPf on the modular curveY1(N)(C), whereN is the smallest integer inf.

1 We adopt the convention that the cyclotomic character has Hodge–Tate weight +1; this is, of course, the Galois character attached to the norm mapA×K −→ R

×_{, which has}

infinity-type (1,1).

2 We stress thatK(f) is not a CM field in general, so the definition ofK(f)+ _{depends on}

Since N ≥5 by assumption, the curveY1(N) has a canonical model overQ such thatY1(N)(L) parametrises elliptic curves overLwith a point of orderN

for eachL⊆C. Our claim is then precisely thatPf ∈Y1(N)(K(f)+).

It is clear that Pf ∈Y1(N)(R), since there is a canonical isomorphism from

C/fto the elliptic curve ER={y

2_{= 4x}3_−g

2x−g3} where g2 andg3 are the

usual weight 4 and 6 Eisenstein series, given by z 7→ (℘(z,f), ℘0(z,f)). Since

f= ¯f, the coefficients g2 and g3 are real, so ER is indeed defined over R; and

as ℘(z,Λ) = ℘(¯z,Λ), this uniformization maps 1¯ ∈ _C/fto a real point of E_R. HencePf∈Y1(N)(R).

On the other hand, it is well known that there exists a CM-pair of mod-ulus f over K(f) (whether or not ¯f = f), so Pf ∈ Y1(N)(K(f)). Hence Pf ∈

Y1(N)(K(f)+).

Remark 2.3: It follows from this construction that the canonical CM pair

(E, α) over K(f)+ _{becomes isomorphic over}

Rto (ER,image of 1∈C). So the

complex conjugation automorphism ofE(_C) arising from thisK(f)+_-model

cor-responds to the natural complex conjugation onC/f.

We recall the theory of elliptic units, as described in [9,§15.5-6].

Theorem 2.4: For each pair(f,a)of ideals ofKsuch thatO×_K∩(1 +f) ={1}

andais coprime to6f, there is a canonical element

aef∈K(f)×,

the elliptic unit of modulusf and twista. Iff has at least two prime factors,

aef ∈ O×_K(f); and for any two idealsa,bcoprime to6f, we have

(1) (N(b)−[b])·aef= (N(a)−[a])·bef,

where[a] = a K(f)/K

∈Gal(K(f)/K)is the arithmetic Frobenius element ata.

Vital for our purposes is the following complex conjugation symmetry of the elliptic units:

Proposition 2.5: Iffsatisfies the hypotheses of Theorem 2.2, then we have

aef =a¯ef.

Proof. This follows from the construction of the elliptic units. We have

where (E, α) is the canonical CM pair overK(f), andaθE is the element of the

function field ofE constructed in [9,§15.4].

By Theorem 2.2,E admits a model overK(f)+, and it is clear that ifιis the nontrivial element of Gal(K(f)/K(f)+_{) arising from complex conjugation, we}

haveι(aE) =¯aE and hence (by the uniqueness of aθE) we have (aθE)ι =¯aθE.

Sinceα∈E(K(f)+_{), we deduce that}

aef= (aθE)ι(α)−1=¯aθE(α)−1=¯aef

as required.

Remark 2.6: Modulo differing choices of conventions, this is the formula

la-belled “Transport of Structure” in§2.5 of [7].

2.2. _{Elliptic units in Iwasawa cohomology.} Let p be a rational prime which splits inK. For fixed f (which we shall assume prime to p), the ideal

g=fpn _{satisfies the conditionO}×

K∩(1 +g) ={1}for alln0, so if (a,6pf) = 1

we may define the elementsaefpn. These arenorm-compatible (c.f. [9,§15.5]),

and we may extend their definition to alln≥0 using the norm maps.

Remark 2.7: Since fpn has at least two prime factors for n ≥ 1, we have

aefpn∈ O_K×_(fpn).

Let S be a set of places ofK containing the infinite places and the primes abovep. Then we have the Kummer maps

κL:Zp⊗ZO

×

L,S

∼

=

- H_S1(L,_Zp(1)).

Since the sequence of elements aefp∞ = (_ae_fpn)_n≥0 is a norm-compatible

se-quence of units, their images under the Kummer maps are corestriction-compatible, so we obtain an element

aefp∞ ∈H_Iw1 _,S(K(fp∞),_Zp(1)) = lim ←−

n

H_S1(K(fpn),Zp(1)).

Theorem 2.8: Iffis Galois-stable, then we have

ι∗(aefp∞) =¯aefp∞,

whereι∗ is the involution ofHIw1 ,S(K(fp∞),Zp(1))induced by complex

conju-gation.

Proof. Immediate from Proposition 2.5, since fpn _{satisfies the conditions of}

Definition 2.9: We also define the element

efp∞ = (N(a)−[a])−1·_ae_fp∞ ∈Q(G_fp∞)⊗_Λ(G fp∞)H

1

Iw,S(K(fp

∞_), Zp(1)),

whereΛ(Gfp∞)is the Iwasawa algebra ofG_fp∞ = Gal(K(fp∞)/K)andQ(G_fp∞)

its total ring of quotients.

Remark 2.10: The elementefp∞is independent of the choice ofa, by equation (1).

Corollary 2.11: We haveι∗(efp∞) =e_fp∞.

Proof. The automorphism ι∗ of HIw1 ,S(K(fp∞),Zp(1)) is Λ(Gfp∞)-semilinear, with the action ofιonGfp∞ being given by conjugation in Gal(K/_Q); henceι_∗ extends canonically to the tensor product withQ(Gfp∞); and since ι[a]ι= [¯a], this finishes the proof by Theorem 2.8 and Remark 2.10.

LetW be any continuous representation ofGfp∞ on a one-dimensional vector space over some finite extensionLof_Qp. Then we have an isomorphism

(2) H_Iw1 _,S(K(fp∞),_Zp(1))⊗ZpW

∼

=

- H_Iw1_,S(K(fp∞), W(1)).

Definition 2.12: For an elementw∈W, letefp∞(w)be the image ofe_fp∞⊗w

under (2), which is an element of

Q(Gfp∞)⊗_Λ(G fp∞)H

1

Iw,S(K(fp

∞_{), W(1)).}

Define

e∞(w)∈Q(Γ)⊗Λ(Γ)HIw1 ,S(K∞, W(1))

to be the image ofefp∞(w)under the corestriction map

H_Iw1 _,S(K(fp∞), W(1)) - H_Iw1_,S(K_∞, W(1)).

Lemma 2.13: IfW has no fixed points underGal(K(fp∞)/K∞), then we have

e∞(w)∈HIw1,S(K∞, W(1)).

Proof. SupposeGfp∞ acts on W via the character τ :G_fp∞ −→L×. Then we must show that the ideal in Λ(Γ) generated by the elements

{(Na−τ([a])−1[a]) :a is an integral ideal coprime to 6f}

vanishes for everya. By the Chebotarev density theorem, we must haveτ = χ−1η, which contradicts the assumption thatτdoes not factor through Γ.

We write ιW for the representation ofGfp∞ that acts on{ιw:w∈W} via g·(ιw) =ι(ιgι)·w.

Theorem 2.14: IfW has no fixed points underGal(K(fp∞)/K∞), the element

e∞(w)∈HIw1 ,S(K∞/K, W(1))

satisfies

ι∗(e∞(w)) =e∞(ιw)

whereι∗ is induced from the maps

H_S1(K(fpn), W(1)) - H_S1(K(fpn),(ιW)(1))

sending a cocycleτ to the cocycleg7→ιτ(ιgι), for eachn≥0.

We split the proof of the theorem into a number of steps.

Definition 2.15: Let Λ](Gfp∞)(1) denote Λ(G_fp∞)(1) endowed with the

ac-tion of Gal(KS/K) via the product of the cyclotomic character with the in-verseof the canonical character Gal(KS/K)Gfp∞ ,→Λ(G_fp∞)×, i.e. g.ω= χ(g)¯g−1_{ω for anyg ∈Gal(K}S_{/K) andω ∈Λ}]_{(G). Here, ¯g denotes the image}

ofgin Gfp∞.

Lemma 2.16: We have a commutative diagram

(3)

H_Iw1 _,S(K(fp∞),Zp(1))⊗ZpW

∼ =

- H_Iw1 _,S(K(fp∞), W(1))

H_Iw1 _,S(K(fp∞),_Zp(1))⊗ZpιW ι∗⊗ι

? ∼₌

- H_Iw1 _,S(K(fp∞),(ιW)(1))

ι∗

?

where the left-hand vertical map is the tensor product of the automorphismι∗

ofH_Iw1 _,S(K∞,Zp(1))and the canonical mapι:W −→ιW, and the right-hand

vertical map is as defined in the statement of Theorem 2.14.

easier to handle. By Shapiro’s lemma, we have a canonical isomorphism of Λ(Gfp∞)-modules

H_Iw1 _,S(K(fp∞), M(1))∼=H_S1(K, M⊗_ZpΛ] Gfp∞)(1)

for any Gal(KS_{/K)-moduleM which is finite-rank over}

Zp or Qp.

Let τ be the character by whichGfp∞ acts on W, and define τ_∗ : Λ](G)→ Λ]_{(G) to be the map induced byg→τ(g)}−1_{g. Then the natural twisting map}

j:H_S1 K,Λ](G)(1)

⊗W ∼=- H_S1 K,Λ](G)(1)⊗W,

is explicitly given as follows: if c : Gal(KS_{/K) → Λ}]_{(G)(1) is a cocycle and}

w∈W, define

j(c⊗w)(g) =τ∗(c(g))⊗w.

We check thatj(c⊗w) is a cocycle. Leth, g∈Gal(KS_{/K). Then}

j(c⊗w)(gh) =τ∗(c(gh))⊗w

=τ∗(g.c(h))⊗w+τ∗c(g)⊗w

=χ(g)τ∗(g−1c(h))⊗w+τ∗c(g)⊗w

=χ(g)τ(g) g−1[τ∗(c(h))]⊗w+τ∗(c(g))⊗w

=g.[j(c⊗w)(h)] +j(c⊗w)(g)

Rewrite the diagram (3) as

(4)

H_S1(K,Λ](G)(1))⊗_ZpW jW- H_S1(K,Λ](G)(1)⊗W)

H_S1(K,Λ](G)(1))⊗_ZpιW ι∗⊗ι

? _j

ιW

- H_S1(K,Λ](G)(1)⊗ιW)

ι∗

?

It is then immediate from the description of j that the diagram commutes, which finishes the proof.

Proof of Theorem 2.14. By Corollary 2.11 and Lemma 2.16, we have

The action ofι∗ is clearly compatible with corestriction, so we have a commu-tative diagram

H_Iw1 _,S(K(fp∞), W(1)) - H_Iw1 _,S(K_∞, W(1))

HIw1,S(K(fp∞),(ιW)(1))

ι∗

?

- H_Iw1_,S(K∞, ιW(1)) ι∗

?

which implies thatι∗(e∞(w)) =e∞(ιw), completing the proof.

Lemma 2.17: Let V be any p-adic representation of Gal(KS/Q). Then the

restriction map induces an isomorphism

H_Iw1 _,S(_Q∞, V) - H_Iw1 _,S(K_∞, V)Gal(K∞/Q∞).

Proof. The restriction map is induced from the restriction maps on finite level, which fit into the exact sequence

0 - H1 Gal(Kn/Qn), VGal(K

S_/K n)

- H_S1(_Qn, V)

- H_S1(Kn, V)Gal(Kn/Qn) - H2 Gal(Kn/Qn), VGal(K

S_/K n)_.

SinceQp has characteristic 0, the higher cohomology groups of anyQp-linear

representation of the cyclic group of order 2 are zero. This gives the claim at each finite level, and hence in the inverse limit.

Letαbe the unique nontrivial element of Gal(K∞/Q∞).

Lemma 2.18: We have α=δι, where δis the unique element of Gal(K∞/K)

which acts on _Q∞ as complex conjugation. In particular, δis of order2.

Corollary 2.19: Ifαis the unique nontrivial element ofGal(K∞/Q∞), then

for any w∈W,

α∗(e∞(w)) =δ·e∞(ιw).

Proof. As above, writeα=δι. By Lemma 2.17, we haveι∗·e∞(w) =e∞(ιw). Hence α∗(e∞(w)) =δ·ι∗(e∞(w)) =δ·e∞(ιw).

Let p be one of the two primes of K above p. We choose an embedding K ,→ Qp inducing the p-adic valuation on K. Then for any finite extension

L/K, and any Gal(K/K)-moduleM, we may define

Z_p1(L, M) =M

q|p

H1(Lq, M) =H1(Kp,IndKL M).

which is a Gal(L/K)-module. We also define

Z_Iw1 _,p(K(fp∞), M) = lim ←−

L

Z_p1(L, M)

where the limit is taken over finite extensionsL/K contained in K(fp∞). We now recall the theory of two-variable Coleman series, as introduced, under certain additional hypotheses, by Yager [18], and generalized to the semi-local situation here by de Shalit [5, §II.4.6]. Let ζ = (ζpn)_n≥0 be a compatible

system ofp-power roots of unity inK; and letFb∞be the completion ofK(f¯p∞) with respect to the primePofK abovep induced by our choice of embedding K ,→Qp, andOb∞ the ring of integers ofFb∞. (Thus Ob∞ is a complete discrete valuation ring with maximal ideal generated byp, and its residue field is a finite extension of the uniqueZp-extension ofFp.)

Proposition 2.20: There is a unique morphism of Λ(Gfp∞)-modules

Colζ :Z_Iw1 _,p(K(fp∞),_Zp(1))−→ΛO∞b (Gfp

∞)

with the following property:

For each finite-order character η of Gfp∞ which is not unramified at p, we

have

Colζ(u)(η) =τ(η, ζ)−1η( ˜ϕ)n  

X

σ∈Gfpm

η(σ)−1log_P(uσ_m)  .

Hereϕ˜is the unique lifting of the arithmetic Frobenius of Gal(K(f¯p∞)/K)to

Gal(K(fp∞)/K∞), m is any integer such that η factors through the quotient Gfpm = Gal(K(fpm)/K),log_P is the logarithm map

O×_K(fpn_),P - K(fp

n₎ P,

and

τ(η, ζ) = X

σ∈Gal(K(f¯p∞_)(µ

pn)/K(f¯p∞))

ω(σ)−1ζ_pσn,

Definition 2.21: We let

Lfp∞ = Colζ(e_fp∞)∈Ob∞⊗bZpQ(Gfp∞).

Proposition 2.22: The elementLfp∞lies inΛ

b

O∞(Gfp∞), and it coincides with

the measureµ(f¯p∞)in [5, Theorem II.4.14].

Proof. We have (Na−[a])·Lfp∞ ∈ Λ

b

O∞(Gfp∞) for all a. Since the ideal generated by Na−[a] for all integral ideals a coprime to 6fhas height 2, this implies that Lfp∞ ∈Λ

b

O∞(Gfp∞) (cf. [5,§II.4.12]).

To show that the resulting measure coincides with de Shalit’s µ(f¯p∞), we compare the defining property of the map Col above with [5, Theorem II.5.2]. For a finite-order character η of Gfpn, whose conductorgis divisible by pand

satisfies O_K×∩(1 +g) ={1}, de Shalit shows that

η(µ(f¯p∞)) = −1 12gG(η)

X

c∈Cl(g)

η−1([c]) logφg(c),

whereg is the smallest rational integer ing,φg(c) is Robert’s invariant and the

quantityG(η) coincides with what we have calledτ(η, ζ)−1_{η( ˜ϕ)}n_{. Since}

(N(a)−[a])φg(c) = [c]·(aeg)− 12g

,

this shows that the two measures coincide at every finite-order character, and hence they are equal in Λ

b

O∞(Gfp∞).

Remark 2.23: If one identifies G(fp∞) with the ray class group modulo fp∞ via the Artin map, normalized as in §1.3 above, then this measure coincides with the pullback of the Katz two-variable L-function ofK (cf. [8,§4]) up to a difference of signs. This remark will be important in the proof of Theorem 3.4 below.

2.4. Kato’s zeta element. Letf =P

anqn be a modular form of CM type,

corresponding to a Gr¨ossencharacter ψofK with infinity-type (1−k,0) where k is the weight off. It is clear that the coefficient fieldF =_Q(an:n≥1) off

is contained in the finite extension L/K contained in_Cgenerated byψ(Kb×). Following [9,§6.3], we writeS(f) andV(f) for the subspaces of the de Rham and Betti cohomology of the Kuga–Sato variety attached tof. Note that both of these are F-vector spaces, and S(f) is 1-dimensional over F while V(f) is 2-dimensional. For a commutative ring A over F, define SA(f) = S(f)⊗FA

with thep-adic representation associated to f of Deligne [6] and SFλ(f) may

be identified with Fil1Dcris(VFλ(f)).

Definition 2.24: Letχ be a Dirichlet character of conductor pn. We define the mapsθ_χ,f± by

θ±_χ,f : S(f)⊗_QQ(µpn) −→ V_C(f)±

x⊗y 7−→ P

σ∈Gnχ(σ)σ(y) perf(x)

±

whereGn = Gal(Q(µpn)/_Q),per_f :S(f) - V_C(f)is the period map as

de-fined in[9,§6.3]andγ7→γ±is the projection fromV_C(f)to its (1-dimensional)

±1-eigenspace for the complex conjugation.

Theorem 2.25([9, Theorem 12.5(1)]): We have aLλ-linear map

VLλ(f) −→ H 1

Iw,S(Q∞, Vλ(f))

γ 7−→ zKato_γ

which satisfies the following. Let χ be a Dirichlet character of conductor pn_,

γ∈VL(f)and1≤r≤k−1, then

θ±_χ,f◦exp∗zKato_γ ⊗(ζpn)⊗(k−r)

= (2πi)k−r−1L{p}(f∗, χ, r)·γ±

where±= (−1)k−r−1_χ(−1).

Let f be an ideal of OK satisfying the conditions in Theorem 2.2 which is

contained in the conductor ofψ. Let (E, α) be the canonical CM-pair overK(f). Following [9, §15.8], we define VL(ψ) =H1(E(C),Q)⊗(k−1)⊗K Land S(ψ) =

H0_{(Gal(K(f)/K),coLie(E)}⊗(k−1)_⊗

KL), where the action of Gal(K(f)/K) on

the space coLie(E)⊗(k−1)_⊗

KL is as described in op.cit.. Both of these are

1-dimensionalL-vector spaces. For any commutative ring A overL, we write VA(ψ) =VL(ψ)⊗LAand SA(ψ) =S(ψ)⊗LA. The Galois group Gal(K/K)

acts onVL(ψ)⊗LLλ viaψλ, and there exists a period map

per_ψ:S(ψ) - V_C(ψ)

induced by passing to the (k−1)-st tensor power from the comparison isomor-phism per_∞described above.

Lemma 2.26([9, Lemma 15.11]): Fix a choice of isomorphism ofL-vector spaces

s:S(ψ) ∼- SL(f).

(a) There exists a unique isomorphism of representations ofGal(Q/Q)over Lλ

^

VLλ(ψ) - VLλ(f)

such that the isomorphismSLλ(ψ) - SLλ(f)induced by the functo-riality of DdR is compatible with s.

(b) There exists a unique isomorphism of representations ofGal(C/R)over L

^

VL(ψ) - VL(f)

for which the diagram

S(ψ) per-ψ V^_C(ψ)

SL(f) ? per_f

- V_C(f) ?

commutes.

Note that the isomorphism of part (b) implies an isomorphism VLλ(ψ)

∼

=

−→

VLλ(f) on extending scalars toLλ, but one does not know that this coincides

with the isomorphism of part (a), as remarked in [9,§15.11].

Definition 2.27: We writeΦψ,f for the canonical map

H_Iw1 _,S(K(fp∞), VLλ(ψ)) - H 1

Iw,S(Q∞, VLλ(f))

as defined in[9, (15.12.1)].

Concretely, this map can be defined as follows:

H_Iw1 _,S(K(fp∞), VLλ(ψ)) - H 1 S(K,Λ

]_(Γ)⊗V

Lλ(ψ))

-H_S1(_Q,IndQ

K Λ

]_(Γ)⊗V Lλ(ψ)

) ∼=- H_S1(_Q,Λ](Γ)⊗V_Lλ(f)).

Theorem 2.28: Letγ∈VL(ψ) and writeγ0 for its image in VL(f)under the

map given by Lemma 2.26(b). Then we have

Φψ,f

e∞(γ)⊗(ζpn)⊗(−1)

Proof. This is [9, (15.16.1)]; it is immediate from a comparison the interpolating properties of the two zeta elements, since an element ofH_Iw1 (Q∞/Q, VLλ(f)) is

uniquely determined by its images under the dual exponential maps at each finite level in the tower_Q∞/_Q.

Proposition 2.29: We have a commutative diagram

H_Iw1 _,S(K∞, VLλ(ψ))

Φψ,f

- H_Iw1 _,S(_Q∞, VLλ(f))

H_Iw1 _,S(K∞, VLλ(ψ)⊕ιVLλ(ψ))

α=1

?

∼₌

where the left-hand vertical map sends x to x⊕δ·ι∗(x), and the diagonal

isomorphism is given by restriction.

Proof. Clear.

3. Critical-slopeL-functions

Let f be a modular form of CM type, as above, and ψ the corresponding Gr¨ossencharacter. We choose a basis γ of VL(ψ), and let γ0 be its image in

VL(f) under the isomorphism of Lemma 2.26(b).

We fix an embeddingK ,→Qpwhich induces theλ-adic valuation onL. This

gives an embedding Gal(Qp/Qp),→Gal(K/Q), whose image is contained in the subgroup Gal(K/K). This gives a localization map

locp:HIw1 ,S(Q∞, M) - H_Iw1 (_Qp,∞, M)

for each Gal(KS_/

Q)-moduleM. Moreover, we have a map

locp:HIw1 ,S(K∞, M) - H_Iw1 (_Qp,∞, M)

for each Gal(KS/K)-moduleM, and we clearly have locp= locp◦resK/_Q.

Via the isomorphism of Lemma 2.26(a), the space VLλ(f) is isomorphic as

a representation of Gal(_Qp/_Qp) to VLλ(ψ)⊕ι(VLλ(ψ)). Note that ι does not

normalize the image of Gal(_Qp/_Qp), so the two factors are non-isomorphic;

indeed VLλ(ψ) has Hodge–Tate weight 1−k, while ι(VLλ(ψ)) has Hodge–Tate

weight 0. Hence we have

locp(zKatoγ0 )∈H_Iw1 (_Qp,∞, VLλ(ψ))⊕H 1

Let us write pr1 and pr2 for the projections to the two direct summands

above. By Corollary 2.28, the projection pr1locp(zKatoγ0 ) to H_Iw1 (Qp,∞, VLλ(ψ))

is

locp

e∞(γ)⊗(ζpn)⊗(−1)

.

By Proposition 2.29, we see that the projection of locp(zKatoγ0 ) to the other direct summand is

δ·locp

h ι∗

e∞(γ)⊗(ζpn)⊗(−1)

i

= [δ·locp(ι∗(e∞(γ)))]⊗(ζpn)⊗(−1).

We have

ι∗(e∞(γ)) =e∞(ιγ),

so this simplifies to

pr₂ locpzKatoγ0 =δ·[locp(e∞(ιγ))]⊗(ζpn)⊗(−1).

Definition 3.1: LetLγp,1∈Λ(Γ)⊗ZpDcris(VLλ(ψ)(k−1))andL

γ

p,2∈Λ(Γ)⊗Zp

Dcris(ιVLλ(ψ)(k−1))be the unique elements such that

LΓ

V_Lλ(f)(k−1)

zKato_γ0 ⊗(ζpn)⊗(k−1)

=Lγ_p,1⊕Lγ_p,2.

We shall see below that if g = ¯f is the complex conjugate of f, then Lγ_p,1 will be the ordinaryp-adicL-function ofg, and Lγ_p,2 is the critical-slope p-adic L-function ofg.

Theorem 3.2: For every characterη ofΓ, we have

Lγ_p,1(η) =_Lfp∞(η ψ_λχk−2 −1

)·tk−1γ,

and

Lγ_p,2(η) = (`0. . . `k−2δLfp∞) (η ψ_λιχk−2 −1

)·ιγ.

Proof. For brevity, we shall writeej for (ζpn)⊗j, considered as a basis vector of

Qp(j).

It is easy to see that if ξ is a character ofGfp∞ of the form χjτ, whereτ is unramified andj ≥0, andV is any crystalline representation with non-negative Hodge-Tate weights, then for anyx∈H_Iw1 (K(fp∞), V) and any choice of basis eξ ofQp(ξ) we have

LGfp∞

V(ξ) (x⊗eξ)(η) = (`0. . . `j−1)(η)· L Gfp∞

V (x)(ηξ

−1_)⊗t−j_e ξ.

Note that if ξ takes values in the finite extension L/Qp, this is an equality

transforms viaτunderG_Qp, and hence lies inFb∞⊗Dcris(L(ξ)), since the periods

of unramified characters lie inFb∞⊆Bcris.

We apply this result withV =Qp(the trivial representation),x=efp∞⊗e₋₁, and various values ofξ. Firstly, takingξto be the cyclotomic character, we have

Lfp∞ =`−₀1LGfp ∞

Qp(1)(efp

∞),

and thus

(5) _Lfp∞(η) =LGfp ∞

Qp (efp∞⊗e−1)(χ

−1_η)⊗t−1_e 1.

On the other hand we have

Lγ_p,1(η) =LΓ

V_Lλ(ψ)(k−1) pr1(z Kato

γ0 )⊗ek−1)(η)

=LGfp∞

V_Lλ(ψ)(k−1)(e∞(γ)⊗ek−2) (η)

The groupG_Qp acts onVLλ(ψ)(k−1) via the unramified characterχ

k−1_ψ λ, so

this is

Lγ_p,1(η) =LGfp∞

Qp (e∞⊗e−1) (χ

k−1_ψ λ)−1η

⊗(tk−1γ)⊗(t1−kek−1).

Comparing this with (5), we deduce that

Lγ_p,1(η) =_Lfp∞ (χk−2ψ_λ)−1η⊗(tk−1γ)⊗(t2−ke_k−2).

If we identifyDcris(Qp(k−2)) with Qp in the usual way, t2−kek−2 is sent to

1. As remarked above, the elementtk−1γ ∈ Bcris⊗QpVLλ(ψ) lies in Fb∞⊗Qp

Dcris(VLλ(ψ)). So if ω is a K-basis of S(ψ), then the image of ω under the

crystalline comparison isomorphism is a basis ofDcris(VLλ(ψ)), and if we define

Ωp= (γ⊗e1−k)/ω, this will lie inFb∞ and our result becomes

Lγ_p,1(η) =Lfp∞ (χk−2ψ_λ)−1η·Ω_pω.

We now turn to Lγ_p,2. We have

Lγ_p,2(η) =LΓ

ι(V_Lλ(ψ))(k−1) pr2(z Kato

γ0 )⊗ek−1

(η)

=LGfp∞

ι(V_Lλ(ψ))(k−1)((δ·e∞(ιγ))⊗ek−2) (η)

= (−1)k−2η(δ)LGfp∞

The groupG_Qp acts onι(VLλ(ψ)) by the character ψ

ι

λ, which is unramified; so

this is

Lγ_p,2(η) = (−1)k−2η(δ)(`0. . . `k−2)(η)· L Gfp∞

Qp (e∞⊗e−1) ((χ

k−1_ψι λ)−

1_η)

⊗t1−kek−1⊗ιγ.

= (−1)k−2η(δ)(`0. . . `k−2)(η)·Lfp∞ (χk−2ψι_λ)−1η⊗t2−ke_k−2⊗ιγ.

As above, we identify t2−k_e

k−2 ∈Dcris(Qp(k−2)) with 1∈ Qp; and if ω is a

basis ofSL(ψ), the image ofιωunder the comparison isomorphism is a basis of

Dcris(ι(VLλ(ψ))), so if we define Ω

ι

p= (ιγ)/(ιω) this becomes

Lγ_p,2(η) = (−1)k−2η(δ)(`0. . . `k−2)(η)·Lfp∞ (χk−2ψ_λι)−1η·Ωι_pιω.

Definition 3.3: Letω be a basis ofSL(ψ)as above, letg= ¯f, and letLp,α(g)

andLp,β(g)be the elements ofHLλ(Γ)defined by

Lγ_p,1=Lp,α(g)·ω

and

Lγ_p,2=Lp,β(g)·ιω.

ThenLp,α and Lp,β are thep-adicL-functions attached to g, where αand β

are respectively the unit and non-unit roots of the Hecke polynomial ofg.

As shown in [9,§16], this is consistent with the classical Amice–Velu–Vishik construction of the ordinaryp-adicL-functionLp,α(g), and thus it is natural to

regardLp,β(g) as a candidate for a critical-slopep-adicL-function. This is the

definition of the Kato critical-slope L-function used in [11].

Theorem 3.4: Up to multiplication by two nonzero scalars, one for each sign,

Lp,β(g) coincides with the modular symbol critical-slope L-function LMSp,β(g)

attached to the non-ordinary p-stabilization ofg in[3].

Proof. This follows by comparing the formulae of Theorem 3.2 with Theorem 2 of [3]. Note that Bella¨ıche shows that ifρ1 andρ2 are the two characters by

which Gal(K/K) acts on V_g∗, then

  

Lp,α(g)(η) =Lfp∞(ρ₂η−1)·(constant±),

Here constant± indicates an equality of distributions on Γ up to multiplication by two nonzero constants (one for each sign). On the other hand, since Vg∗ =

Vf(k−1), we have{ρ1, ρ2}={χk−1ψλ, χk−1ψλι}and the result of Theorem 3.2

shows that   

Lp,α(g)(η) =Lfp∞(χρ−₁1η)·(constant),

Lp,β(g)(η) = (`0· · ·`k−2)(η)·Lfp∞(χρ−₂1η)·(constant).

To reconcile these formulae, we note that thep-adicL-function_Lfp∞ satisfies a functional equation [5,§II.6]

Lfp∞(ι(η)) =C(η)·_Lfp∞(χη−1),

for a functionC(η) (involving ap-adic root number and various other correction terms) which depends only on the coset ofηmodulo characters factoring through Gal(_Q+

∞/Q). Sinceι(ρ1) =ρ2and vice versa, we deduce that

Lp,β(g) =LMSp,β(g)·(constant

±_).

Since the modular symbol L-function is only defined up to scalars, this com-pletes the proof.

Remark 3.5: Both Kato’s and Bella¨ıche’s critical-slope p-adicL-functions are only defined up to multiplication by a nonzero constant for characters of each sign; in Kato’s construction these constants correspond to the choice ofγ, whose projection to each of the ± eigenspaces of complex conjugation must be non-zero. It seems natural to ask whether one can choose normalizations for both in a compatible fashion so Theorem 3.4 holds exactly, but the present authors do not feel sufficiently familiar with the modular symbol construction to comment further.

Remark 3.6: Since the Hodge–Tate weights ofψλatpand ¯pare (1−k,0), we

see that ifηis a character of Γ whose single Hodge–Tate weight ist, the Hodge– Tate weights ofη ψλχk−2

−1

andη ψι λχ

k−2−1_{are respectively (t+1, t+2−k)}

second shows that, sadly, none of the values ofLp,β(g)(η), nor its derivatives at

the points where it is forced to vanish, correspond to a classicalL-value for any value ofη. In particular, we cannot rule out the possibility thatLp,β(g) is zero.

Acknowledgements

We would like to thank Marco Streng for several informative conversations regarding CM theory, and John Coates for his interest and encouragement. We would also like to thank the anonymous referee for his helpful remarks.

References

[1] Yvette Amice and Jacques V´elu,Distributionsp-adiques associ´ees aux s´eries de Hecke, Journ´ees Arithm´etiques de Bordeaux (Conf., Univ. Bordeaux, 1974), Ast´erisque, vol. 24-25, Soc. Math. France, Paris, 1975, pp. 119–131. MR 0376534

[2] Jo¨el Bella¨ıche,Criticalp-adicL-functions, to appear in Inventiones Mathematicae, 2011. [3] Jo¨el Bella¨ıche,P-adicL-functions of critical CM forms, preprint, 2011.

[4] Pierre Colmez,La conjecture de Birch et Swinnerton-Dyerp-adique, Ast´erisque (2004), no. 294, ix, 251–319. MR 2111647

[5] Ehud de Shalit,Iwasawa theory of elliptic curves with complex multiplication, Perspec-tives in Mathematics, vol. 3, Academic Press Inc., Boston, MA, 1987. MR 917944 [6] Pierre Deligne,Formes modulaires et repr´esentations`-adiques, S´eminaire Bourbaki11

(1968/69), Exp. No. 355, 139–172.

[7] Benedict H. Gross,On the factorization ofp-adicL-series, Inventiones Mathematicae57

(1980), no. 1, 83–95. MR 564185

[8] Haruzo Hida and Jacques Tilouine,Anti-cyclotomic Katzp-adicL-functions and congru-ence modules, Annales Scientifiques de l’ ´Ecole Normale Sup´erieure (Quatri`eme S´erie)26

(1993), no. 2, 189–259. MR 1209708

[9] Kazuya Kato, P-adic Hodge theory and values of zeta functions of modular forms, Ast´erisque 295 (2004), ix, 117–290, Cohomologies p-adiques et applications arithm´etiques. III. MR 2104361

[10] David Loeffler and Sarah Livia Zerbes,Iwasawa theory andp-adicL-functions overZp2 -extensions, preprint, 2011.

[11] David Loeffler and Sarah Livia Zerbes, Wach modules and critical slope p-adic L -functions, to appear in Journal f¨ur die reine und angewandte Mathematik, 2011. [12] Barry Mazur and Peter Swinnerton-Dyer,Arithmetic of Weil curves, Inventiones

Math-ematicae25(1974), 1–61.

[13] Bernadette Perrin-Riou,FonctionsL p-adiques des repr´esentationsp-adiques, Ast´erisque

229(1995), 1–198. MR 1327803

[15] Robert Pollack and Glenn Stevens, Overconvergent modular symbols and p-adic L -functions, Annales Scientifiques de l’ ´Ecole Normale Sup´erieure (Quatri`eme S´erie) 44

(2011), no. 1, 1–42. MR 2760194

[16] Misha Viˇsik,Nonarchimedean measures associated with Dirichlet series, Matematicheski˘ı Sbornik (Novaya Seriya)99(141)(1976), no. 2, 248–260, 296. MR 0412114

[17] Andr´e Weil, On a certain type of characters of the id`ele-class group of an algebraic number-field, Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, Science Council of Japan, Tokyo, 1956, pp. 1–7. MR 0083523 [18] Rodney Yager,On two variablep-adicL-functions, Annals of Mathematics (Second