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Original citation: Loeffler, David and Zerbes, Sarah Livia. (2011) Wach modules and critical slope p-adic L-functions. Journal für die reine und angewandte Mathematik (Crelles Journal), 2013 (679). pp. 181-206. Permanent WRAP url: http://wrap.warwick.ac.uk/43664 Copyright and reuse: The Warwick Research Archive Portal (WRAP) makes this work of researchers of the University of Warwick available open access under the following conditions. Copyright © and all moral rights to the version of the paper presented here belong to the individual author(s) and/or other copyright owners. To the extent reasonable and practicable the material made available in WRAP has been checked for eligibility before being made available. Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. Publisher’s statement: http://dx.doi.org/10.1515/crelle.2012.012 A note on versions: The version presented in WRAP is the published version or, version of record, and may be cited as it appears here. For more information, please contact the WRAP Team at: [email protected]

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J. reine angew. Math. 679 (2013), 181—206

DOI 10.1515/crelle.2012.012

Journal fur die reine undangewandte Mathematik( Walter de Gruyter

Berlin � Boston 2013

Wach modules and critical slope p-adicL-functions

By David Loe¿er at Coventry and Sarah Livia Zerbes at Exeter

Abstract. We study Kato and Perrin-Riou’s critical slope p-adic L-function at-tached to an ordinary modular form using the methods of A. Lei, D. Loe¿er and S. L.Zerbes, Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (2010),475–528. We show that it may be decomposed as a sum of two bounded measures multi-plied by explicit distributions depending only on the local properties of the modular format p. We use this decomposition to prove results on the zeros of the p-adic L-function, andwe show that our results match the behaviour observed in examples calculated by Pollackand Stevens in ‘‘Overconvergent modular symbols and p-adic L-functions’’, Ann. Sci. Ec.Norm. Super. (4) 44 (2011), no. 1, 1–42.

1. Introduction

1.1. Background. Let pf 3 be prime, and let f be a normalised, new modular ei-genform of level N, character � and weight k f 2, with N prime to p. Then a classical con-struction of Amice–Velu and Visik ([1], [15]) gives rise to p-adic L-functions for f , whichare distributions on Z�p interpolating the critical values of the L-functions of f and itstwists by Dirichlet characters of p-power conductor.

The construction depends on a choice of root of the Hecke polynomial of f at p, andrequires that the root be of ‘‘non-critical slope’’, i.e. that its p-adic valuation should bestrictly less than k � 1. The L-function corresponding to the root a is the unique distribu-tion Lp;a on Z�p of order h ¼ ordpðaÞ whose values at ‘‘special’’ characters of Z�p , i.e. thoseof the form z 7! z joðzÞ where 0e j e k � 2 and o is a finite-order character of conductorpn, are given by

ÐZ�p

z joðzÞ dLp;a ¼ð1� p ja�1Þ

�1� �ðpÞpk�2�ja�1

�~LLð f ; 1; j þ 1Þ if n ¼ 0;

a�npnð jþ1Þ~LLð f ;o�1; j þ 1Þ

Gðo�1Þ if nf 1;

8><>:ð1:1Þ

The first author is supported by EPSRC Postdoctoral Fellowship EP/F04304X/1. The second author is

supported by EPSRC Postdoctoral Fellowship EP/F043007/1.

Brought to you by | University of WarwickAuthenticated

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where ~LLð f ;o�1; j þ 1Þ is the complex L-value Lð fo�1 ; j þ 1Þ of the twisted form fo�1 di-vided by certain explicit transcendental factors (see equation (6.1) below), and Gðo�1Þ isthe Gauss sum.

If the Hecke eigenvalue apð f Þ is not a p-adic unit, then both roots have non-criticalslope, and one obtains two p-adic L-functions, both of which are uniquely determined bythe corresponding interpolation formula (1.1). If apð f Þ is a p-adic unit (the ordinary case),then one root has non-critical slope (and in fact gives rise to a bounded measure) but theother does not, so one can only construct one p-adic L-function by these methods. Twoconstructions exist that redress the balance by constructing a ‘‘critical slope L-function’’for ordinary eigenforms: a p-adic analytic approach via the theory of overconvergent mod-ular symbols ([13]), and an algebraic approach via p-adic Hodge theory, using Kato’s Eulersystem ([9]). Both approaches give a distribution of order k � 1 on Z�p with the same inter-polation property at special characters, which depends on the restriction of the Galois rep-resentation of f to a decomposition group at p. If the local representation is non-split, thenthe values of both of these critical-slope L-functions at special characters are given by (1.1);if the local representation is split, the L-functions vanish at all such characters. However,these values do not uniquely determine a distribution of order k � 1, and we cannot neces-sarily deduce that the L-functions arising from the two approaches are equal.

Since Z�p GD� Zp, where D is cyclic of order p� 1, the D-isotypical components ofthese L-functions may be interpreted as distributions on Zp and hence (via the Amice trans-form) as power series on the p-adic open unit disc. In [13], §9, Pollack and Stevens calculatethe Newton polygon of the trivial D-component of the analytic critical-slope L-functionLPS

p;b in some explicit examples. They observe that the distribution of the zeros follows inter-

esting patterns which seem to be governed by the Iwasawa m and l-invariants of the unit-root L-function Lp;a: when p ¼ 3 and f is the twist of X0ð11Þ by a quadratic character ofconductor D prime to 3, for example, the numerical values suggest that the number of zeros

inside the open disc of radius rn ¼1

pnðp� 1Þ is pnðp� 1Þ þ lD, where lD is the l-invariant

of Lp;a. The same behaviour occurs when p ¼ 5 and the discriminant of the character isnegative. However, when p ¼ 5 and the discriminant is positive, the number of zeros insidethe open disc of radius rn is pn � 1þ �, for some mysterious non-negative integer � depend-ing on f .

1.2. Statement of the main results. Let p and f be as above, and assume thatk e p� 1 and that f is ordinary. Let a and b be the roots of the Hecke polynomial, andfix a to be the unit root. We choose a prime of the coe‰cient field of f above p. In thisintroduction, let us assume for simplicity that the completion of the coe‰cient field atthis prime is Qp. Let V �f be the dual of the p-adic representation attached to f , so it is a2-dimensional Qp-vector space which is crystalline with Hodge–Tate weights 0 and k � 1.Let G ¼ Gal

�QpðmpyÞ=Qp

�, HðGÞ the algebra of Qp-valued distributions on G, and write

LV �f: H 1

IwðQp;V�

f Þ !HðGÞnQpDcrisðV �f Þ

for the Perrin-Riou regulator map. (Here H 1IwðQp;V

�f Þ is the Iwasawa cohomology group,

whose definition we recall in §1.3.3 below. For a definition of LV �f

, see [11], Definition 3.4.)For any z A H 1

IwðQp;V �f Þ, we write LaðzÞ (resp. LbðzÞ) for the projection of LV �fðzÞ into the

a (resp. b) eigenspace of j�1. If zKato A H 1IwðQp;V

�f Þ is Kato’s zeta element, then (for appro-

182 Loeffler and Zerbes, Wach modules and critical slope p-adic L-functions



priate normalisations of the Frobenius eigenvectors) we have LaðzKatoÞ ¼ Lp;a, and it is

conjectured that LbðzKatoÞ agrees with the critical slope p-adic L-function LPSp;b constructed

by Pollack and Stevens (cf. [14], Remark 8.5).

To simplify the notation, write Lp;b for LbðzKatoÞ. In this paper, we study Lp;b usingthe description via Wach modules developed in [10] and [11]. This gives rise to a canonicalsubspace �

j�NðV �f Þ�c¼0

LHðGÞnQpDcrisðV �f Þ;

stable under G and of rank 2 as a LðGÞ-module, through which the map LV �f

factors. InSection 4, we explicitly construct a basis n1, n2 of the Wach module NðV �f Þ. By comparing

this basis with the j-eigenvector basis of DcrisðV �f Þ, we obtain the following result:

Theorem A. If V �f is not locally split, then there exist Lp;1;Lp;2 A LQpðGÞ such that

aLp;a ¼ Lp;2;

bLp;b ¼ Lp;1M�1 ð1þ pÞj t

p

� �k�1 !

� Lp;2M�1�ð1þ pÞjðaÞ

�;

8>><>>:

where, as an element of QpJtK, we have

a ¼ ðk � 2Þ!a 1

1� p1�kmþ ð�1Þk

Pnfk

n� 1

k � 2

� �Bntn

n!ð1� mpn�kþ1Þ

!:

(Here M denotes the Mellin transform HðGÞ !G ðBþrig;QpÞc¼0, whose definition we re-

call in §1.3 below.)

Since Lp; i is bounded for i ¼ 1; 2, the distribution of the zeros of Lp;b is determined by

the Newton polygons oft

p

� �k�1

and a. In Section 7, we consider the case k ¼ 2 and deter-

mine which of these two terms dominates, depending on the behaviour of the m-invariantof Lp;a.

Theorem B. Let h be a character of D and let lh1 , mh

1 , lh2 , mh

2 be the Iwasawa l- and

m-invariants of the h-isotypical components of Lp;1 and Lp;2. Suppose that V �f is non-split at p.

(a) If mh2 <

1

ðp� 1Þ2þ m

h1 , then for ng 0, L

hp;b has pnðp� 1Þ2 zeros of valuation

1

pnðp� 1Þ2, and the total number of zeros of valuation > rn is pnðp� 1Þ þ l

h2 .

(b) If mh2 >

1

p� 1þ m

h1 , then for ng 0, L

hp;b has pnðp� 1Þ zeros of valuation

1

pnðp� 1Þ2, and the number of zeros of valuation > rn is pn � 1þ l

h1 .

Under the assumption that Kato’s zeta element is integral, which is known in manycases, this explains the numerical phenomena observed by Pollack and Stevens (see the endof Section 7).

183Loeffler and Zerbes, Wach modules and critical slope p-adic L-functions



1.3. Notation. As above, fix a prime pf 3, and let G ¼ Gal�QpðmpyÞ=Qp

�. Note

that the cyclotomic character w gives an isomorphism GGZ�p . We write

G ¼ D� G1;

where D is cyclic of order p� 1 (corresponding to the roots of unity in Z�p ) and

G1 ¼ Gal�QpðmpyÞ=QpðmpÞ

�is the preimage of 1þ pZp (non-canonically isomorphic to Zp). We denote the absoluteGalois group of Qp by GQp

.

We write Bþrig;Qpfor the ring of power series f ðpÞ A QpJpK such that f ðXÞ converges

everywhere on the open unit p-adic disc. Equip Bþrig;Qpwith actions of G and a Frobenius

operator j by g:p ¼ ðpþ 1ÞwðgÞ � 1 and jðpÞ ¼ ðpþ 1Þp � 1. We can then define a left in-verse c of j satisfying

j � c�

f ðpÞ�¼ 1

p

Pz p¼1

f�zð1þ pÞ � 1

�:ð1:2Þ

Inside Bþrig;Qp, we have subrings AþQp

¼ ZpJpK and BþQp¼ Qp nZp

AþQp. Moreover, the

actions of j, c and G restrict to these rings. Finally, we write t ¼ logð1þ pÞ A Bþrig;Qpand

q ¼ jðpÞ=p A AþQp. A formal power series calculation shows that gðtÞ ¼ wðgÞt for g A G and

jðtÞ ¼ pt.

If E is a finite extension of Qp, with ring of integers OE , we extend the actions of j, cand G to E nQp

Bþrig;Qpby E-linearity; these obviously preserve the subrings E nQp

BþQpand

OE nZpAþQp

.

We also let AQpbe the p-adic completion of AþQp

½1=p�, and BQp¼ AQp

½p�1�. Theserings appear in Fontaine’s theory of ðj;GÞ-modules (see [8]): if T is any finite-rank freeOE-module with a continuous action of GQp

, its ðj;GÞ-module DðTÞ is a finite-rank freeOE nZp

AQp-module with commuting semilinear actions of j and G, satisfying a certain

etaleness condition; and the functor Dð�Þ is an equivalence of categories. Similarly, for arepresentation of GQp

on an E-vector space V , one obtains a ðj;GÞ-module DðVÞ over thering E nQp

BQp.

1.3.1. The Mellin transform. We write LZpðGÞ for the Iwasawa algebra of G over

Zp,

LZpðGÞ ¼ lim �

ULGU open

Zp½G=U �:

We define LQpðGÞ ¼ Qp nZp

LZpðGÞ, and similarly for any finite extension E=Qp we define

LOEðGÞ and LEðGÞ by extending scalars. We use also the corresponding notations for G1 in

place of G.

Let HðGÞ be the algebra of distributions on G (the continuous dual of the space oflocally analytic functions on G, with multiplication given by convolution). If we choose a




generator g1 of G1, then the Amice transform allows us to identify HðGÞ with the algebra offormal power series

f f A Qp½D�JXK : f converges everywhere on the open unit p-adic discg;

where X corresponds to g1 � 1. We may identify LQpðGÞ with the subring of HðGÞ consist-

ing of power series with bounded coe‰cients.

The action of G on Bþrig;Qpgives an isomorphism of HðGÞ with ðBþrig;Qp

Þc¼0, theMellin transform

M : HðGÞ ! ðBþrig;QpÞc¼0;

f ðg� 1Þ 7! f ðg� 1Þðpþ 1Þ:ð1:3Þ

In particular, LZpðGÞ corresponds to ðAþQp

Þc¼0 under M. Similarly, we define HðG1Þ as thesubring of HðGÞ defined by power series over Qp, rather than Qp½D�. Then, HðG1Þ (respec-tively LZp

ðG1Þ) corresponds to ð1þ pÞjðBþrig;QpÞ (respectively ð1þ pÞjðAþQp

Þ) under M.

1.3.2. Crystalline representations. Let E again denote a finite extension of Qp, withring of integers OE . Fix a uniformizer $E . For a crystalline E-linear representation V ofGQp

, denote its Dieudonne module by DcrisðVÞ; this is an E-vector space of the same di-mension as V with an E-linear Frobenius j and a filtration, whose jumps are the negativesof the Hodge–Tate weights.

The following result is shown in [3], §II.1 and §III.4: if V is an E-linear representation,then V is crystalline with Hodge–Tate weights in ½a; b� if and only if there exists a (nec-essarily unique) E nQp

BþQp-module NðVÞ contained in the ðj;GÞ-module DðVÞ of V such

that the following conditions are satisfied:

(1) NðVÞ is free of rank d ¼ dimEðVÞ over E nQpBþQp

.

(2) The action of G preserves NðVÞ and is trivial on NðVÞ=pNðVÞ.

(3) j�pbNðVÞ

�H pbNðVÞ and pbNðVÞ=j�

�pbNðVÞ

�is killed by qb�a where

q ¼ jðpÞp

. (If M is an R-module equipped with a Frobenius j where R is any ring, then

j�ðMÞ denotes the R-module generated by jðMÞ.)

If the Hodge–Tate weights of V are e0, we may take b ¼ 0 above, so j preservesNðVÞ. In this case, if we endow NðVÞ with the filtration

Fil i NðVÞ ¼ fx A NðVÞ : jðxÞ A qiNðVÞg;

then NðVÞ=pNðVÞ is a filtered E-linear j-module, and as shown in [3], §III.4, we have anisomorphism of filtered j-modules NðVÞ=pNðVÞGDcrisðVÞ. Also, as shown in [3], II.2.1,DcrisðVÞ is contained in NðVÞnBþ

QpBþrig;Qp

, and in particular we can recover DcrisðVÞ as

DcrisðVÞ ¼�NðVÞnBþ

QpBþrig;Qp

�G:




Moreover, we have a comparison isomorphism

NðVÞnBþQpBþrig;Qp

½t�1�GDcrisðVÞnQpBþrig;Qp

½t�1�;ð1:4Þ

which holds independent of the Hodge–Tate weights of V .

If T is a GQp-stable lattice in V , then NðTÞ ¼ NðVÞXDðTÞ is an OE nZp

AþQp-lattice

in NðVÞ, and by [3], §III.4, the functor T ! NðTÞ gives a bijection between the GQp-stable

OE-lattices T HV and the OE nZpAþQp

-lattices in NðVÞ satisfying the conditions (1)–(3)above.

Finally, it is easy to see from the construction that for all j A Z, NðVÞ and N�Vð jÞ

�are related by

N�Vð jÞ

�¼ p�jNðVÞn ej;

where ej is a basis for Qpð jÞ.

1.3.3. Iwasawa cohomology. If V is a p-adic representation of GQp, and T is a GQp

-stable Zp-lattice in V , we define the Iwasawa cohomology of T to be

H 1IwðQp;TÞ :¼ lim �

n

H 1�QpðmpnÞ;T

�:

As shown in [12], this is a LZpðGÞ-module of finite rank. Define

H 1IwðQp;VÞ :¼ Qp nZp

H 1IwðQp;TÞ

for any GQp-stable lattice T ; this definition is independent of the choice of T .

By [2], Theorem A.3, if V is crystalline with Hodge–Tate weightsf 0, and V hasno quotient isomorphic to the trivial representation, there is a canonical isomorphism ofLQpðGÞ-modules

H 1IwðQp;VÞGNðVÞc¼1;ð1:5Þ

which also identifies H 1IwðQp;TÞ with NðTÞc¼1 for each lattice T . These constructions

clearly commute with the additional OE-linear and E-linear structures when V is anE-linear representation and T is a GQp

-stable OE-lattice.

2. Setup

Let p be prime and let f be a normalised new modular eigenform of level N, charac-ter � and weight k f 2, with N prime to p and k e p� 1. Let F be the coe‰cient field of f ;we fix a choice of prime of F above p and let E be the completion of F at that prime, whichwe regard as a subfield of Qp.

We assume that f is ordinary, i.e. apð f Þ is a p-adic unit. Thus the roots of theHecke polynomial X 2 � apð f ÞX þ pk�1e, where e ¼ �ðpÞ, are elements of E with valua-




tions 0 and k � 1. Let a be the unit root and b ¼ pk�1ea�1 the non-unit root. We definem :¼ pk�1a=b ¼ e�1a2, which is a p-adic unit. By the Deligne–Ramanujan–Peterssonbound on japj, all embeddings of m into C have complex absolute value pk�1; in particular,m3 1.

We define ðrf ;Vf Þ to be the ‘‘cohomological’’ p-adic representation attached to f , sothe characteristic polynomial of geometric Frobenius at primes lFNp is

X 2 � alX þ lk�1�ðlÞ.

Then it is well known that

rf jGQpG

lðaÞ �0 w1�klðea�1Þ

� �

where lðxÞ denotes the unramified character of GQpmapping geometric Frobenius to x. In

particular, rf jGQphas an unramified subrepresentation.

Since f has level prime to p, rf jGQpis crystalline, and the characteristic polynomial

of j on Dcrisðrf Þ is X 2 � apX þ pk�1e. The unramified subrepresentation of rf jGQpcorres-

ponds to the j ¼ a eigenspace.

We shall mostly work with the ‘‘homological’’ representation ðr�f ;V �f Þ, the linear dualof Vf . This is crystalline at p and the arithmetic Frobenius at primes lFNp has the Heckepolynomial as its characteristic polynomial. On the decomposition group it is given by

r�f jGQpG

wk�1lðe�1aÞ �0 lða�1Þ

� �;

so the j-eigenvalues on DcrisðV �f Þ are b�1 ¼ p1�ke�1a (corresponding to the subrepresenta-tion with Hodge–Tate weight k � 1) and a�1.

We let Tf and T �f denote the canonical GQ-stable OE-lattices in Vf and V �f definedusing the cohomology of the modular curve X1ðNÞQ with coe‰cients in Zp, as in [9], §8.3.

3. A sequence of polynomials

In this section we define a certain sequence of polynomials which are related to theEulerian polynomials.

Definition 3.1. Let the polynomials hjðXÞ, for j f 0, be defined by h0ðXÞ ¼ 1 and

hjðXÞ ¼ ð1þ XÞ �Xd

dXþ j

� �hj�1ðX Þ:ð3:1Þ

It is immediate that hj is a monic polynomial of degree j with integral coe‰cients andconstant term hjð0Þ ¼ j!.




Proposition 3.2. For any j f 0, the following identity of formal power series holds in

QJtK:

t

et � 1

� �jþ1

hjðet � 1Þ ¼ j!

1þ ð�1Þ j P

nfjþ1

n� 1

j

� �Bntn

n!

!;

where Bn is the n-th Bernoulli number.

Proof. The case j ¼ 0 of the proposition is the definition of the Bernoulli numbers,

t

et � 1¼ 1þ

Pnf1

Bntn

n!:

If Dj is the operator �td

dtþ j

� �, then we compute that

Dj �t

et � 1

� �j

hj�1ðet � 1Þ ¼ t

et � 1

� �jþ1

hjðet � 1Þ

and

Dj � ð j � 1Þ!

1þ ð�1Þ j�1Pnfj

n� 1

j � 1

� �Bntn

n!

!¼ j!

1þ ð�1Þ j P

nfjþ1

n� 1

j

� �Bntn

n!

!:

So the proposition holds for all j by induction. r

4. Calculating the (j,G)-module

Let V1 ¼ E�wk�1lðe�1aÞ

�and V2 ¼ E

�lða�1Þ

�. The extensions

0! V1 ! V ! V2 ! 0

in the category of E-linear continuous representations of GQpare parametrised by

H 1ðQp;V1 nE V �2 Þ ¼ H 1�Qp;E

�wk�1lðmÞ

��:

By [6], Theorem 0.2 (ii), this cohomology group is 1-dimensional over E. Hence up to iso-morphism there are exactly two such extensions: one split and one non-split. In this sectionwe give an explicit description of the ðj;GÞ-module corresponding to the non-split exten-sion V .

Since the ðj;GÞ-module functor is exact, DðVÞ is a free module of rank 2 overE nBQp

with a basis v1, v2, where v1 is a basis for the ðj;GÞ-module of V1 and the imageof v2 is a basis for the ðj;GÞ-module of the quotient V2. Thus in this basis ðv1; v2Þ, j acts viathe matrix

P ¼ e�1a x

0 a�1

� �




and an element g A G acts via

G ¼ wðgÞk�1 ay

0 1

!

for some x; y A E nBQp. In the split case, we may clearly take x ¼ y ¼ 0; so let us as-

sume we are in the non-split case. Since the actions of j and G commute, we must havePjðGÞ ¼ GgðPÞ, so x and y satisfy

ðmj� 1ÞðyÞ ¼�wðgÞk�1g� 1

�ðxÞ:

This, of course, is exactly the requirement that ðx; yÞ forms a 1-cocycle in the Herr complex(see e.g. [4], §I.4) calculating H 1

�Qp;E

�wk�1lðmÞ

��. We shall construct an explicit choice of

ðx; yÞ realising the non-split extension.

Proposition 4.1. If xk ¼ p1�khk�2ðpÞ, where hj are the polynomials of Definition 3.1,then we have

�wðgÞk�1g� 1

�xk A AþQp

for all g A G; and hk�2 is the unique monic polynomial of degree k � 2 such that this holds.

Proof. Let f be any element of QpJtK with f ð0Þ ¼ 1, and let F ¼ p1�kf . Then wefind that

�wðgÞk�1g� 1

�F ¼ t1�kðg� 1Þðtk�1FÞ ¼ t1�kðg� 1Þ

t

p

� �k�1

f

!:

Hence we have�wðgÞk�1g� 1

�F A QpJtK if and only if ðg� 1Þ

t

p

� �k�1

f

!A tk�1QpJtK, or

equivalentlyt

p

� �k�1

f A 1þ tk�1QpJtK. By construction, f ¼ hk�2ðpÞ ¼ hk�2ðet � 1Þ satis-

fies this (and it is obvious that there is a unique polynomial with this property up to scal-ing). Since F lies in AQp

, we have�wðgÞk�1g� 1

�F A AQp

XQpJtK ¼ AþQp. r

Recall that m :¼ e�1a2 3 1.

Lemma 4.2. There is no z A E nQpBQp

such that ðmj� 1ÞðzÞ ¼ p1�khk�2ðpÞ.

Proof. We assume (for simplicity of notation) that E ¼ Qp. Observe that we havea decomposition AQp

¼ pAþQplAe0

Qp, where Ae0

Qpconsists of those series with only non-

positive powers of p; thus Ae0Qp

is a subring isomorphic to ZphXi, where X ¼ p�1. Also, jpreserves Ae0

Qp, since the series expansion of jðp�1Þ lies in Ae0

Qp; indeed this gives an action

of j on ZphXi lifting the canonical Frobenius on Fp½X �.

If z is such that ðmj� 1ÞðzÞ ¼ p1�khk�2ðpÞ, then we must have z A Be0Qp¼ Ae0

Qp½p�1�,

and the constant term of z is zero. If z3Ae0Qp

, then there is some j > 0 such that z 0 ¼ p jz isin Ae0

Qpand its mod p reduction z 0 is a non-zero element of XFp½X �. But then z 0 must satisfy




ðmj� 1Þðz 0Þ ¼ 0, which is clearly impossible as j increases the degree of any non-constantpolynomial.

Hence z A Ae0Qp

. Then z A Fp½X � and ðmj� 1ÞðzÞ is a non-constant polynomial in X ofdegree k � 1 < p. It is clear that no such polynomial can lie in the image of mj� 1. r

We now define xk ¼ p1�khk�2ðpÞ � dk, where dk ¼ ð�1Þk Bk�1

ðk � 1Þ . Since

k � 1 < p� 1; dk A Zp;

and hence xk A AQp. Moreover, the choice of d implies that tk�1xk ¼ ðk � 2Þ!þOðtkÞ in

QpJtK; hence�wðgÞk�1g� 1

�ðxkÞ A pAþQp

.

Proposition 4.3. There exists yk A OE nAþQpsolving

ðmj� 1ÞðykÞ ¼�wðgÞk�1g� 1

�ðxkÞ:

Proof. It is clear that mj� 1 is surjective on pðOE nAþQpÞ for any m A OE : if

z A pAþQp, then we have jnðzÞ A jnðpÞAþQp

, and jnðpÞ tends to zero in the ðp; pÞ-adic topo-logy of AþQp

. Thus the series expansion �P

nf0

mnjnðzÞ converges, and its limit is clearly a

preimage of z. r

Note that yk depends on k, m, and g, but xk depends only on k. By construction,ðxk; ykÞ determines a class in H 1 of the Herr complex, and Lemma 4.2 shows that this classis not zero. Since H 1

�Qp;E

�wk�1lðmÞ

��is 1-dimensional over E, as noted above, we de-

duce that the ðj;GÞ-module we have constructed corresponds to the unique non-split exten-sion of the two factors.

5. Calculations in Wach modules

We deduce that the ðj;GÞ-module of the unique non-split extension V has anðE nBQp

Þ-basis ðv1; v2Þ for which the matrices of j and the generator g, in the basisðv1; v2Þ, are given by

P ¼ e�1a xk

0 a�1

� �

and

G ¼ wðgÞk�1 ay

0 1

!:

We let n1 ¼ p1�kv1 and n2 ¼ v2. Then the matrices of j and g in the basis ðn1; n2Þ are givenby

P 0 ¼pk�1

jðpÞk�1e�1a pk�1xk

0 a�1

0B@

1CA




and

G 0 ¼pk�1

gðpk�1Þ wðgÞk�1 pk�1ay

0 1

0B@

1CA:

It follows that if N is the E nBþQp-span of n1 and n2, the module N satisfies the con-

ditions (1)–(3) of §1.3.2, so N ¼ NðVÞ, the Wach module of V . Moreover, the OE nAþQp-

span of ðn1; n2Þ is a Wach module over OE nAþQp, and hence is NðTÞ for a GQp

-stableOE-lattice T in V .

Lemma 5.1. For each c; d A Z with ce d, there is a lattice Tc;d A V whose Wach

module is the OE nAþQp-span of $c

En1, $dEn2, and every GQp

-stable OE-lattice in V is one of

these. The residual representation Tc;d=pETc;d is non-split if c ¼ d and split otherwise.

Proof. It is clear that $cEn1, $d

En2 span a Wach module for every ce d, and hencecorrespond to a lattice in V . By construction the mod$E reduction of the cocycle ðxk; ykÞis nontrivial, so T0;0 ¼ T is residually non-split, and hence the same holds for Tc; c forany c.

Let T 0 be any lattice in V ; by scaling we may assume that it is contained in T , andnot contained in $ET . Then the image of T 0 in T=$ET is a nontrivial Galois-stable sub-space; so it is the subspace corresponding to the reduction of n1, as T=$ET is non-split and

thus has a unique 1-dimensional subspace. By devissage, we deduce that T 0 ¼ $jET þ T1

for some j, where T1 ¼ T XV1 is the lattice corresponding to n1. This corresponds to thecocycle ð$ j

Exk; $jEykÞ, and hence is residually split if j f 1. r

Proposition 5.2. The space�j�NðVÞ

�c¼0is free of rank 2 as a LEðGÞ-module, and a

basis is given by ð1þ pÞjðn1Þ, ð1þ pÞjðn2Þ. More specifically, if Tc;d denotes the lattice in V

defined in Lemma 5.1, then $cEð1þ pÞjðn1Þ, $d

Eð1þ pÞjðn1Þ are a basis of�j�NðTc;dÞ

�c¼0

as a free LOEðGÞ-module.

Proof. It is easy to check that the functor�j�ð�Þ

�c¼0is exact, so we have a short

exact sequence of LOEðGÞ-modules

0!�j�NðT1Þ

�c¼0 !�j�NðTÞ

�c¼0 !�j�NðT2Þ

�c¼0 ! 0:

Now as shown in [10], Theorem 3.5,�j�NðT1Þ

�c¼0is a free LOE

ðGÞ-module of rank 1,and there exists a basis n 01 of NðT1Þ which is congruent to n1 mod p such that ð1þ pÞjðn 01Þis a LOE

ðGÞ-basis of�j�NðT1Þ

�c¼0. Observe that an element in OE nAþQp

is invertible ifand only if its constant term is a unit in OE . Write n1 ¼ an 01 with a A ðAþQp

Þ�. By Prop-osition 3.10 in op.cit., jðpÞ ið1þ pÞjðn 01Þ A ð1� gÞ

�j�NðT1Þ

�c¼0. It follows that if we

write ð1þ pÞjðn1Þ ¼ a:ð1þ pÞjðn 01Þ, then a A LOEðGÞ�, so ð1þ pÞjðn1Þ is also a basis of�

j�NðT1Þ�c¼0

.

Similarly, we can show that if n2 denotes the image of n2 in NðT2Þ, then ð1þ pÞjðn2Þis a LOE

ðGÞ-basis of�j�NðT2Þ

�c¼0. As ð1þ pÞjðn2Þ is a lift of ð1þ pÞjðn2Þ, this implies the

result. r




We now calculate DcrisðVÞ using Berger’s comparison isomorphism (1.4). Since thehighest Hodge–Tate weight is k � 1, we have

DcrisðVÞ ¼

t

p

� �1�k

Bþrig;QpnBþ

QpNðVÞ

!G:

It is clear thatt

p

� �1�k

n1 is G-stable, and that it is a j-eigenvector with the following eigen-

value: b�1 ¼ p1�ke�1a.

To find an a�1 eigenvector lifting n2 (which is clearly an eigenvector in DcrisðV2Þ) we

must find a; b A E nBþrig;Qpsuch that

t

p

� �1�k

an1 þ bn2 is G-stable and killed by aj� 1,

with b ¼ 1 modulo p. Comparing coe‰cients of n2, we find that b ¼ 1. Writing out the

equations gðvÞ ¼ v and jðvÞ ¼ a�1v, where v ¼ t

p

� �1�k

an1 þ n2, we need to have

ð1� gÞðaÞ ¼ tk�1ay

and

ð1� p1�kmjÞðaÞ ¼ tk�1axk:

The existence and uniqueness of a solution to these equations is a consequence of the factthat V is known to be crystalline. If a is the solution, then we have the following formulaefor the eigenvectors in DcrisðVÞ:

Proposition 5.3. A basis of j-eigenvectors in DcrisðVÞ is given by

vb�1 ¼ �að0Þ t

p

� �1�k

n1;

va�1 ¼ t

p

� �1�k

an1 þ n2;

where a is the unique solution in Bþrig;Qpto ð1� p1�kmjÞðaÞ ¼ tk�1axk.

The motivation for the factor �að0Þ is the following lemma:

Lemma 5.4. The subspace Fil2�k DcrisðVÞ ¼ Fil0 DcrisðVÞ of DcrisðVÞ is spanned by

the vector va�1 þ vb�1 ¼ n2 þ�a� að0Þ

�n1.

Proof. We know that

Fil j DcrisðVÞ ¼ fv A DcrisðVÞ : jðvÞ A q jNrigðVÞg:




Since vb�1 ¼ �að0Þ t

p

� �1�k

n1, we clearly have vb�1 3Fil2�k DcrisðVÞ. Hence we can

find some scalar l such that Fil2�k DcrisðVÞ is spanned by v ¼ va�1 þ lvb�1 . We have

jðvÞ ¼ a�1va�1 þ lb�1vb�1

¼ t

p

� �1�k�aa�1 � lað0Þb�1

�n1 þ a�1n2:

This is in q2�kNrigðVÞ if and only if�aa�1 � lað0Þb�1

�A E n qBþrig;Qp

; in other words,if and only if aðzp � 1Þ ¼ p1�klmað0Þ where zp is a primitive p-th root of unity. Since

a ¼ tk�1axk þ p1�kmjðaÞ, and t vanishes at zp � 1, we have

aðzp � 1Þ ¼ p1�kmjðaÞðzp � 1Þ ¼ p1�kmað0Þ:

Thus the unique solution is l ¼ 1, as claimed. r

Since the Dieudonne module of a lattice T LV is the image of NðTÞ in

DcrisðVÞ ¼ NðVÞ=pNðVÞ;

we see that:

Corollary 5.5. If Tc;d is the lattice in V defined above, then Fil0 DcrisðTc;dÞ is the

OE-span of $dEðva�1 þ vb�1Þ.

Note that there is a well-defined map Bþrig;Qp! QpJtK, which is determined by send-

ing p to et � 1 and whose image is contained in the ring of power series converging forjtj < p�1=ðp�1Þ.

Proposition 5.6. As elements of EJtK, we have

a

ðk � 2Þ!a ¼1


Pnfk

n� 1

k � 2

� �Bntn

n!ð1� mpn�kþ1Þ :

Proof. By definition, we have ð1� p1�kmjÞðaÞ ¼ atk�1xk. Since

xk ¼ p1�khk�2ðpÞ � ð�1ÞkBk�1=ðk � 1Þ;

we have

tk�1xk ¼t

et � 1

� �k�1

hk�2ðet � 1Þ � ð�1Þk Bk�1tk�1

ðk � 1Þ

as elements of QpJtK. Substituting in the formula of Proposition 3.2, the tk�1 terms cancel,and we obtain

tk�1xk ¼ ðk � 2Þ!

1þ ð�1ÞkP

nfk

n� 1

k � 2

� �Bn

n!tn

!:




Since jðtnÞ ¼ pntn, we deduce that

a

ðk � 2Þ!a ¼1


Pnfk

n� 1

k � 2

� �Bntn

n!ð1� mpn�kþ1Þ ;

as claimed. r

We now consider the relation between this abstract representation V and the repre-sentation V �f attached to f . We have V �f GV

fðk � 1Þ, where f is the complex conjugate

of f . We may identify DcrisðVfÞ ¼ DdRðVf

Þ with the f -isotypical component of the de

Rham cohomology of X1ðNÞ with coe‰cients in the standard line bundle ok; in particular,f gives a canonical basis for the nontrivial filtration step.

Proposition 5.7. If the representation V �f is non-split as a GQp-representation, there is

a unique E-linear isomorphism of GQp-representations

V �f GV

inducing the map DcrisðV �f Þ ! DcrisðVÞ that sends t1�kf to va�1 þ vb�1 .

Proof. Clear, since V is the unique non-split extension up to scaling by E�, and wemay choose our scale factor so that t1�kf corresponds to va�1 þ vb�1 . r

Corollary 5.8. Suppose that V �f is non-split. Let T be the lattice of Lemma 5.1. Then

T �f MT , with equality if and only if T �f =$ET �f is non-split as a mod p representation of GQp.

Proof. As in [9], §14.22, the element f is a basis for Fil1 DcrisðTfÞ. Twisting by t1�k,

we see that T �f corresponds to a strongly divisible lattice in DcrisðVÞ whose intersection withFil0 DcrisðVÞ is the OE-span of va�1 þ vb�1 . Applying Corollary 5.5, T �f must be Tc;0 forsome ce 0; this is residually non-split if and only if c ¼ 0. r

Note that the local splitness of T �f =$ET �f at p can be explicitly checked in certaincases, using [14], Proposition 6.9.

If V �f is split, then we have an isomorphism V �f GV1 lV2, where

V1 ¼ E�wk�1lðe�1aÞ

�and V2 ¼ E

�lða�1Þ

�as above. If n1 and n2 are the natural basis vectors of the Wach modules of V1 and V2, then

it is clear that vb�1 ¼ t

p

� �1�k

n1 and va�1 ¼ n2 are a basis of eigenvectors of DcrisðV1 lV2Þ,

and the nontrivial filtration step is DcrisðV2Þ ¼ Eva�1 . Hence we may choose an isomor-phism V �f ! V1 lV2 such that the resulting isomorphism DcrisðV �f Þ ! DcrisðV1 lV2Þsends t1�kf to va�1 . Since V1 and V2 have distinct mod p reductions, the only possible lat-tices are direct sums of lattices in the factors, and hence we may assume that the above iso-morphism maps T �f to the lattice corresponding to the span of n1 and n2. However, thisisomorphism is still not canonical, since it is only determined up to multiplying n1 by anarbitrary element of O�E .




6. Consequences for the L-functions

We now use the results we have collected on the local representation V �f jGQpto de-

scribe the p-adic L-functions of f . We recall the setup from [10], §3.6. Since V �f has non-negative Hodge–Tate weights 0 and k � 1, and the unique 1-dimensional quotient of V �f isnot the trivial representation, the theorem quoted in §1.3.3 shows that

H 1IwðQp;V �f ÞGNðV �f Þ

c¼1:

We let Col denote the composition of this isomorphism with the map

1� j : NðV �f Þc¼1 !

�j�NðV �f Þ

�c¼0:

Note that since the Hodge–Tate weights of V �f aref0, we have

�j�NðV �f Þ

�c¼0L ðBþrig;Qp

Þc¼0 nQpDcrisðV �f Þ:

Let z A H 1IwðQp;V

�f Þ. We thus have an element ColðzÞ A

�j�NðV �f Þ

�c¼0. We consider

it as an element of ðBþrig;QpÞc¼0 nQp

DcrisðVÞ via the above inclusion, and define LaðzÞ andLbðzÞ to be its projections to the eigenspaces, so

ColðzÞ ¼ LaðzÞva�1 þ LbðzÞvb�1 :

We can consider LaðzÞ and LbðzÞ as power series in p lying in ðE nBþrig;QpÞc¼0. Alter-

natively, we may regard them as E-valued distributions on G, via the Mellin transform(1.3). As shown in [11], §3.1, these correspond to the coordinates of LV �

fðzÞ in the basis

ðva�1 ; vb�1Þ, where

LV �f: H 1

IwðQp;V�

f Þ !HðGÞnQpDcrisðVÞ

is Perrin-Riou’s regulator map. Perrin-Riou’s theory implies that LaðzÞ is a distribution oforder 0, while LbðzÞ has order k � 1.

If o is a Dirichlet character (to any modulus), let Lð f ;o; sÞ denote the complexL-function of f twisted by o,

Panð f ÞoðnÞn�s. Recall (e.g. from [5], §3.1.2) that there are

nonzero complex numbers Wþ and W� such that

~LLð f ;o; j þ 1Þ :¼ Gð j þ 1Þð2piÞ jþ1WG

Lð f ;o; j þ 1Þ A Qð f ;oÞð6:1Þ

where WG denotes Wþ if ð�1Þ jþ1wð�1Þ ¼ 1 and W� if ð�1Þ jþ1wð�1Þ ¼ �1. Here Qð f ;oÞdenotes the finite extension of Q generated by the values of o and the coe‰cients anð f Þ.

Theorem 6.1 ([9], Theorem 16.6). There exists an element zKato A H 1IwðQp;V

�f Þ such

that for any finite-order character o of G of conductor pn, and any 0e j e k � 2,




LaðzKatoÞðw joÞ ¼ð1� p ja�1Þð1� epk�2�ja�1Þ~LLð f ; 1; j þ 1Þ if n ¼ 0;

a�npnð jþ1Þ ~LLð f ;o�1; j þ 1ÞGðo�1Þ if nf 1;

8><>:

where Gðo�1Þ is the Gauss sum.

Hence LaðzKatoÞ is a distribution of order 0 on G whose values at special charactersare given by (1.1); so it is equal to the p-adic L-function Lp;a. We define Lp;b ¼ LbðzKatoÞ; ifV �f is non-split, this satisfies (1.1) for the root b, while if V �f is split, Lp;b vanishes at allspecial characters. We now use the fact that Col factors through

�j�NðV �f Þ

�c¼0, and the

basis of the latter space given by Proposition 5.2, to give a decomposition of these two dis-tributions.

Definition 6.2. Let n1, n2 be the basis of NðT �f Þ defined above. We let Lp;1 and Lp;2

be the unique elements of LEðGÞ such that

ColðzKatoÞ ¼ Lp;1 � ð1þ pÞjðn1Þ þ Lp;2 � ð1þ pÞjðn2Þ:

Proposition 6.3. If the image of GalðQ=QyÞ in GLðT �f Þ contains a conjugate of

SL2ðZpÞ, then Lp;2 lies in LOEðGÞ. If in addition V �f is split at p, or it is non-split and the re-

sidual representation T �f =$ET �f is also non-split, then the same holds for Lp;1.

Proof. If the hypothesis on the image of the global representation is satisfied, thenzKato A H 1

IwðT �f Þ, by [9], Theorem 12.5 (4) and Theorem 12.6. Since the image of NðT �f Þ inthe quotient is the OE nAþQp

-span of n2, this implies that Lp;2 is integral. The additionalassumptions imply the stronger statement that NðT �f Þ is the span of n1 and n2 (Corollary5.8), so we also obtain integrality for Lp;1. r

Using the formulae of the previous section relating va�1 and vb�1 to n1 and n2, we canwrite the functions Lp;a and Lp;b in terms of the Lp; i.

Theorem 6.4. The following relations hold in E nHðGÞ:

(a) If V �f is locally split, then

aLp;a ¼ Lp;2;

bLp;b ¼ Lp;1M�1

ð1þ pÞj t

p

� �k�1!:

8>><>>:

(b) If V �f is not locally split, then

aLp;a ¼ Lp;2;

�að0ÞbLp;b ¼ Lp;1M�1

ð1þ pÞj t

p

� �k�1!� Lp;2M

�1�ð1þ pÞjðaÞ

�:

8>><>>:




Proof. In the non-split case, we use Proposition 5.3 to write

n1 ¼ �1

að0Þt

p

� �k�1

vb�1 ;

n2 ¼ va�1 þ a

að0Þ vb�1 :

Substituting these into the identity

Lp;a � ð1þ pÞva�1 þ Lp;b � ð1þ pÞvb�1 ¼ Lp;1 � ð1þ pÞjðn1Þ þ Lp;2 � ð1þ pÞjðn2Þ;

where the ‘‘�’’ denotes the action of E nHðGÞ on ðBþrig;QpÞc¼0 nQp

DcrisðVÞ, we obtain

Lp;a � ð1þ pÞva�1 þ Lp;b � ð1þ pÞvb�1

¼ Lp;1 ��1

að0Þb ð1þ pÞj t

p

� �k�1

vb�1 þ Lp;2 � ð1þ pÞ 1

ava�1 þ jðaÞ

að0Þb vb�1

� �:

Since ð1þ pÞva�1 and ð1þ pÞvb�1 are clearly a basis for ðBþrig;QpÞc¼0 nQp

DcrisðVÞ as aE nHðGÞ-module, we can project onto each of these to obtain the proposition.

In the split case, one argues identically using the formulae n1 ¼t

p

� �k�1

vb�1 andn2 ¼ va�1 . r

Note that jt

p

� �k�1

has a zero of order k � 1 at z� 1, for any root of unity z of order

pn, nf 2. It is straightforward to see (using Theorem 5.4 and Lemma 5.9 of [10]) that this

is equivalent to M�1

ð1þ pÞj t

p

� �k�1!

vanishing at every character of G of the form w jo,

where w is the cyclotomic character, 0e j e k � 2 and o is a finite-order character not fac-toring through D. Hence the factor multiplying Lp;1 in Proposition 6.4 vanishes at all butfinitely many of the points corresponding to critical values of the complex L-function.

In fact Lp;1 vanishes at most of the remaining points:

Proposition 6.5. If V �f is locally split, then the distribution Lp;1 vanishes at

z 7! zihðzÞ, for any 0e ie k � 2 and any character h of Z�p factoring through D. If V �f is

not locally split, this is true at all characters of this form with h nontrivial.

Proof. If the representation is split, Lb is known to vanish at all special characters.

Since we have bLp;b ¼ Lp;1M�1

ð1þ pÞj t

p

� �k�1!

, and the second factor on the right-

hand side does not vanish at the characters z 7! zihðzÞ, Lp;1 must do so.

In the non-split case, we suppose that h is nontrivial. Then we must have

bLp;bðw ihÞ ¼ aLp;aðw ihÞ;




since both sides are equal to p jþ1~LLð f ; h�1; 1þ jÞ=Gðh�1Þ. Substituting the formulae ofProposition 6.4, we obtain

Lp;2ðw jhÞM�1�ð1þ pÞj

�a� að0Þ

��ðw jhÞ ¼ Lp;1ðw jhÞM�1

ð1þ pÞj t

p

� �k�1!:

By construction b :¼ a� að0Þ vanishes to order k at 0, so ð1þ pÞjðbÞ vanishes to order k atzp � 1 for any nontrivial p-th root of unity z. Hence the distributions q jð1þ pÞjðbÞ vanish

at zp � 1, for i ¼ 0; . . . ; k � 2, where q ¼ ð1þ pÞ d

dp; equivalently, ð1þ pÞjðbÞ pairs to zero

with any function on Zp whose restriction to each coset of pZp is a polynomial ofdegreee k � 2. In particular, it pairs to zero with the characters z 7! z jhðzÞ (extended to

functions on Zp zero on pZp). Since M�1

ð1þ pÞj t

p

� �k�1!

does not vanish at these char-acters, Lp;1 must vanish. r

This proposition, together with the preceding discussion, imply that the distribution

Lp;1M�1

ð1þ pÞj t

p

� �k�1!

vanishes at every locally algebraic character of degreee k � 2

that is not algebraic, and at every locally algebraic character in the split case. In the splitcase, this distribution is simply Lp;b, and one deduces the well-known fact that Lp;b ‘‘pre-tends rather convincingly to be 0’’ (see [5], Remarque 4.12).

Remark 6.6. From the formula a ¼ tk�1axk þ p1�kmjðaÞ, we deduce that

aðzp j � 1Þ ¼ p1�kmaðzp

p j � 1Þ

for any j f 1, where zp j is a p j-th root of unity. Since p1�km ¼ a=b, this gives

aðzp j � 1Þ ¼ a

b

� �j

að0Þ:

Since a� að0Þ vanishes to orderf k � 1 at 0, a� a

b

� �j

að0Þ vanishes to order k � 1 at

zp j � 1. This gives a purely analytic proof that for any F A HðGÞ satisfying the interpola-tion property of (1.1) for the critical-slope root b, the distribution

G ¼ aLp;aM�1�ð1þ pÞjðaÞ

�� að0ÞbF

vanishes at all special characters of G of conductor > 1. Hence G factorises as

M�1

ð1þ pÞj t

p

� �k�1!

H

for some distribution H, and if F has order k � 1, then H must be in LEðGÞ.




In particular, taking F to be the analytic critical-slope L-function LPSp;b, we obtain a

decomposition of LPSp;b analogous to Proposition 6.4. However, without the above interpre-

tation of H via Wach modules, it is not clear how one could determine whether or not H

was integral.

In the non-split case, we also obtain a formula for Lp;1ðw iÞ, for i ¼ 0; . . . ; k � 2, whichallows us to show that it is non-vanishing in some cases:

Proposition 6.7. If V �f is non-split and at least one of the L-values Lð f ; jÞj¼1;...;k�1 is

non-zero, then Lp;1 3 0.

Proof. For 0e j e k � 2, we have

M�1�ð1þ pÞjðaÞ

�ðw jÞ ¼ að0Þ;

since a� að0Þ vanishes to degreef k � 1 at the origin. Thus

Lp;1ðw jÞM�1

ð1þ pÞj t

p

� �k�1!ðw jÞ ¼ að0ÞðaLp;a � bLp;bÞ:

On the right-hand side,

að0Þ ¼ ðk � 2Þ!a1� p1�km

¼ ðk � 2Þ!a1� a=b

¼ ðk � 2Þ!abb � a

;

and substituting the values of Lp;a and Lp;b at w j from equation (1.1) and simplifiying, weobtain (eventually)

Lp;1ðw jÞ �M�1

ð1þ pÞj t

p

� �k�1!ðw jÞ ¼ �ðk � 2Þ!ðp� 1Þpk�2e~LLð f ; 1; 1þ jÞ: r

If k f 3, this is su‰cient to show that Lp;1 3 0, since the complex L-function Lð f ; jÞ

does not vanish for j >k

2. If k ¼ 2, the only character at which we can relate the value of

Lp;1 to the complex L-function is the trivial character, so when Lð f ; 1Þ ¼ 0 (which can ofcourse happen) we cannot show that Lp;1 3 0.

7. Newton polygons and Mellin transforms

In this section, we take k ¼ 2, and present some explicit consequences of the aboveanalysis for the algebraic critical-slope L-function Lp;b. For s > 0 let Cs denote theclosed a‰noid disc fX : jX je p�sg. (For our purposes it will su‰ce to take s rational;if s is irrational, this space is not defined as an a‰noid space, but it can be interpretedas a Berkovich space.) For a rigid-analytic function f on the open unit disc, we writevsð f Þ ¼ inf

x ACs

ordp

�f ðxÞ

�; note that vsð f Þ is clearly an increasing function of s.




Proposition 7.1. The function s 7! vsð f Þ is continuous, piecewise-linear, and concave.

For any sf 0, the left-hand derivative of vsð f Þ at s is the number of zeros of f on Cs (counted

with multiplicity), and the right-hand derivative is the number of zeros of f on the open disc

fX : jX j < p�sg.

Proof. This is simply a restatement of the standard theory of the Newton polygon.r

As above, let g1 be a generator of G1, and let x ¼ g1 � 1, so HðG1Þ is the ring ofpower series in x converging on the unit disc.

Proposition 7.2. Let f A Bþrig;Qpand let g ¼M�1

�ð1þ pÞjð f Þ

�A HðG1Þ. Then for

any s with 0 < s < 1, we have vsð f Þ ¼ vsðgÞ.

Proof. Let us suppose f ¼P

anpn. Then

g ¼Pnf0

anfnðxÞ;

where fnðxÞ ¼M�1�ð1þ pÞjðpÞn

�.

We know that fnðxÞ ¼ tnðxÞxðx� l1Þ � � � ðx� ln�1Þ, where li ¼ wðg1Þi � 1 A pZp and

tn A LQpðG1Þ�. Since ð1þ pÞjðpÞn A AþQp

npAþQp, we must have tn A LZp

ðG1Þ�.

Let us write dn ¼ tnð0Þ. Then we have

vs

�fnðxÞ � dnxn

�¼ vs

�xn�tnðxÞ � dn

�þ tnðxÞ

�xðx� l1Þ � � � ðx� ln�1Þ � xn

��:

We evidently have xn�tnðxÞ � dn

�A xnþ1ZpJxK, so

vs

�xn�tnðxÞ � dn

��f ðnþ 1Þs:

For the second group of terms, the coe‰cient of xn�j in xðx� l1Þ � � � ðx� ln�1Þ � xn isclearly divisible by p j; since tnðxÞ A Z�p , we have

vs

�tnðxÞ

�xðx� l1Þ � � � ðx� ln�1Þ � xn

��f inf

1ejenðn� jÞsþ j

¼ ðn� 1Þsþ 1:

Since 0 < s < 1, both ðnþ 1Þs and ðn� 1Þsþ 1 are strictly bigger than vsðdnxnÞ ¼ ns. Thus,in particular, vs

�fnðxÞ

�¼ ns.

We now write

g ¼P

nf0

anfnðxÞ ¼�P

nf0

andnxn

�þ�P

nf0

an

�fnðxÞ � dnxn

��:




Clearly we have

vs

�Pnf0

andnxn

�¼ inf

nf0ðnsþ ordp anÞ ¼ vsð f Þ:

On the other hand,

vs

�Pnf0

an

�fnðxÞ � dnxn

��f inf

nf0

�inf�ðn� 1Þsþ 1; ðnþ 1Þs

�þ ordp an

�¼ vsð f Þ þ infðs; 1� sÞ:

Hence we must have vsðgÞ ¼ vsð f Þ. r

Combining the two preceding propositions, we see that the zeros of the power series f

and g lying ‘‘near the boundary’’ must have the same valuations; the zeros inside the closeddisc jX je p�1 are equal in number, but can be in very di¤erent places within this disc, asthe examples f ¼ pn show.

Corollary 7.3. Let m A O�E , m3 1, and let f be the unique element of E nBþrig;Qpsuch

that

1� m

pj

� �ð f Þ ¼ t

pþ t

2:

Then f has ðp� 1Þ zeros of valuationf1

p� 1, piðp� 1Þ2 zeros of valuation

1

piðp� 1Þ2for

each integer i f 0, and no other zeros. Moreover, vsð f Þ < vs

t

p

� �for s <

1

ðp� 1Þ2; and we

have

lim infs!0

vs

t

p

� �� vsð f Þ

!¼ 1

ðp� 1Þ2;

lim sup

vs

t

p

� �� vsð f Þ

!¼ 1

ðp� 1Þ :s!0

(We have f ¼ a=a in the notation of the previous sections, in the case k ¼ 2.)

Proof. Let us calculate vsð f Þ. We suppose first that s >1

p� 1. Then the formal

series expansion

f ¼P

nf0n31

Bntn

n!ð1� pn�1mÞ




of Proposition 5.6 is convergent, and the disc jpje p�s corresponds to jtje p�s; hence

vf ðsÞ ¼ infnf0n31

ordp

Bn

n!ð1� pn�1mÞ

� �þ ns:

For nf 2, ð1� pn�1mÞ A O�E , and hence ordp

Bn

n!ð1� pn�1mÞ

� �¼ ordp

Bn

n!

� �. We

have

infnf2

ordp

Bn

n!

� �þ ns ¼ inf

jtjep�sordp

t

et � 1� 1þ t

2

� �

¼ infjpjep�s

ordp

t

p� 1þ t

2

� �

¼ infnf2

ordp

1

nþ 1� 1

2n

� �þ ns:

Clearly, if nf p� 1 and n 0 is the largest integere n of the form p j � 1, or n 0 ¼ 2 ifn < p� 1, then we have

ordp

1

n 0 þ 1� 1

2n 0

� �þ n 0s < ordp

1

nþ 1� 1

2n

� �þ ns:

Thus the infimum is attained either at n ¼ 2 or at n ¼ p j � 1 for some j f 1. We calculatethat the term for n ¼ p j � 1 is ðp j � 1Þs� j, which is a strictly increasing function of j for

any s >1

p� 1. Hence the infimum is

inf�ðp� 1Þs� 1; 2s

�:

(If p ¼ 3, then the 2s term does not appear, but the other term is 2s� 1 which is smalleranyway, so the formula is true as stated.)

If we include also the term in the original sum for n ¼ 0, we deduce that for any s inthis range

vsð f Þ ¼ inf�1; 2s; ðp� 1Þs� 1

�:

One checks that if pf 5, this gives

vsð f Þ ¼

ðp� 1Þs� 1 if1

p� 1e se

1

p� 3;

2s if1

p� 3e se

1

2;

1 if sf1

2;

8>>>>>>><>>>>>>>:




whereas if p ¼ 3, we obtain

vsð f Þ ¼2s� 1 if

1

2e se 1;

1 if sf 1:

8<:

Hence the zeros of f with valuation >1

p� 1are: two zeros of valuation 1 if p ¼ 3; four

zeros of valuation1

2if p ¼ 5; and two of valuation

1

2and p� 3 of valuation

1

p� 3if pf 7.

In each case, the total number of zeros with valuations in this range is p� 1.

We now use this intensive study of vf ðsÞ for relatively large s to describe vf ðsÞ forall smaller s. From the equation f ¼ t=pþ t=2þ p�1mjð f Þ, we deduce that for s in the

interval1

p� 1;

p

p� 1

� �we have

vs=pð f Þf inf�vs=pðt=pÞ;�1þ vsð f Þ

�¼ inf

�vs=pðtÞ � s=p;�1þ vsð f Þ

�

¼ �1þ inf

1� 1

p

� �s; vsð f Þ

!:

We find that if1

p� 1< s <

p

ðp� 1Þ2, the term �1þ vsð f Þ is strictly smaller, whereas for

larger s, the term �1þ 1� 1

p

� �s is strictly smaller; hence this is the exact value of the

left-hand side (by the ultrametric property), and by continuity this is the case at the cross-

over point s ¼ p

ðp� 1Þ2. This determines vsð f Þ in the interval

1

pðp� 1Þ ;1

p� 1

� �: we

have

v 1pð p�1Þð f Þ ¼ �1; v 1

ð p�1Þ2ð f Þ ¼ �1þ 1

p� 1; and v 1

p�1ð f Þ ¼ 0;

and vsð f Þ is linear between these points.

We now consider the interval from1

pnþ1ðp� 1Þ to1

pnðp� 1Þ , for nf 1. By iteratingthe functional equation, we find that

vs=pið f Þf�i þ inf

vf ðsÞ; 1� 1

p

� �s; 1� 1

p2

� �s; . . . ; 1� 1

pi

� �s

!:

The terms 1� 1

p2

� �s; . . . ; 1� 1

pi

� �s are clearly strictly larger than 1� 1

p

� �s, so we de-

duce that for sf1

p� 1, we have vf

s

pi

� �¼ �i þ inf

vf ðsÞ; 1� 1

p

� �s

!¼ 1� i þ vf

s

p

� �.




This gives the locations of the zeros in the statement of the proposition, and shows that

vsð f Þ < vs

t

p

� �for all s <

1

ðp� 1Þ2.

Finally, we establish the formulae for the limits inferior and superior. For1

p� 1e se

p

p� 1and i f 1 we have

vs=pi

t

p

� �� vs=pið f Þ ¼ vs=piðtÞ � s

pi

� �� i þ inf

1� 1

p

� �s; vsð f Þ

!

¼ 1� 1

pi

� �s� inf

1� 1

p

� �s; ðp� 1Þs� 1

!:

One checks that the minimum value of this expression is attained at s ¼ p

ðp� 1Þ2, where it

is equal topi�1 � 1

pi�1ðp� 1Þ2. Hence for any se

1

piðp� 1Þ , we have

vs

t

p

� �� vsð f Þf

pi�1 � 1

pi�1ðp� 1Þ2

and equality occurs for s ¼ 1

piðp� 1Þ2; so the limit inferior as s! 0 is

1

ðp� 1Þ2, as

claimed. On the other hand, the maximum value ispi�1 � 1

pi�1ðp� 1Þ , attained at both of the

endpoints, so the limit superior is1

p� 1. r

We now use this to describe the L-functions. We note that HðGÞ ¼Lh

ehHðGÞ,

where the sum is over the characters of D and eh is the corresponding idempotent. Forg A HðGÞ, we let gh be the unique element of HðG1Þ such that ehg

h ¼ ehg.

For a character h of D, let lh1 , mh

1 , lh2 , mh

2 be the Iwasawa l- and m-invariants of Lhp;1

and Lhp;2. Note that lh

2 , mh2 are equal to the corresponding invariants of the unit root p-adic

L-function Lp;a, which can be calculated in many cases; we know of no easy way to evalu-ate l

h1 , mh

1 , but Proposition 6.3 gives conditions under which mh2 is forced to be non-negative.

Theorem 7.4. Let h be a character of D and let lh1 , m

h1 , l

h2 , m

h2 be as above. Suppose

that V �f is non-split at p.

(a) If mh2 <

1

ðp� 1Þ2þ m

h1 , then for all su‰ciently small s we have

vsðLhp;bÞ ¼ l

h2 sþ m2 þ vsð f Þ;

where f is as above. In particular, for ng 0, Lhp;b has pnðp� 1Þ2 zeros of valuation

1

pnðp� 1Þ2, and the total number of zeros of valuation > rn is pnðp� 1Þ þ l

h2 .




(b) If mh2 >

1

p� 1þ m

h1 , then for su‰ciently small s the formula becomes

vsðLhp;bÞ ¼ l

h1 sþ m1 þ vs

t

p

� �;

so for ng 0 there are pnðp� 1Þ zeros of valuation1

pnðp� 1Þ2and the number of zeros of

valuation > rn is pn � 1þ lh1 .

Proof. We have a decomposition

�að0ÞbLp;b ¼ Lp;1M�1

ð1þ pÞj t

p

� �!� Lp;2M

�1�ð1þ pÞjðaÞ

�:

For all su‰ciently small s, we have vsðLhp; iÞ ¼ m

hi þ sl

hi . Hence, using the bounds of Corol-

lary 7.3, the hypotheses of case (a) force the second term to dominate, giving the stated for-mula for vsðLh

p;bÞ. Similarly in case (b), the first term must dominate for all su‰ciently small

s, giving the formula stated. r

This gives a (conditional) explanation of the phenomena observed in the examples of[13], §9, for the critical slope 3-adic and 5-adic L-functions of quadratic twists of the ellipticcurves X0ð11Þ and X0ð14Þ, with h the trivial character of D. In all of these cases the residualrepresentations are locally non-split at p.

The 3-adic representation of X0ð11Þ (and hence of any of its twists) is surjective1).Hence the Kato zeta element for this curve is integral, and any twist such that m

h2 ¼ 0

must satisfy the hypotheses of (a) above.

In the case of 3-adic L-functions of twists of X0ð14Þ, the residual representation isglobally reducible (but still locally non-split). Thus we cannot show that Lp;1 is integral;but if we assume that this is the case, then again any twist of X0ð14Þ with m

h2 ¼ 0 satisfies

the hypotheses of (a), which is consistent with the numerical results of op.cit.

In the case of the 5-adic L-functions of twists of X0ð11Þ, let us also assume that Lp;1 isintegral. From this assumption, it follows that for twists by even quadratic characters,where m

h2 ¼ 0, we obtain the pattern of zeros of part (a) of the theorem; but for twists by

odd quadratic characters, where mh2 > 0, we are in the situation of (b), unless mh

1 is also pos-itive.

Acknowledgement. We would like to thank Chris Wuthrich for useful conversationsrelating to the integrality of Kato’s zeta-elements. This paper was written while the secondauthor was visiting the University of Warwick; she thanks the number theory group fortheir hospitality.

1) This follows from the fact that its mod 3 representation is surjective, as can be explicitly seen by calcu-

lating the Galois group of the 3-torsion field; and its j-invariant does not lie in the image of the rational func-

tion f ðxÞ of [7]. So its mod 9 representation is surjective, which implies that its 3-adic representation is also

surjective.




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Bordeaux (Univ. Bordeaux, 1974), Asterisque 24–25 (1975), 119–131.

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248–260, 296.

Warwick Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

e-mail: D.A.Loe¿[email protected]

Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK

e-mail: [email protected]

Eingegangen 6. Dezember 2010, in revidierter Fassung 28. Juli 2011




original citation - university of...

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