Cup products and L-values of cusp forms

0.07 MB·

by  Romyar T. Sharifi

Cup products and L-values of cusp forms
Romyar T. Sharifi
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Abstract

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In this note, we describe aconjecture that, for an odd prime p, relates special values

]
T of a cup product pairing on cyclotomic p-units in Q(µ ) to the L-values of newforms
p
N
satisfying modulo p congruences withEisenstein seriesforSL (Z).
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h
t
a
m 1 Introduction
[

1 Let p denote an odd prime. In [McS], McCallum and the author began a study of a cup
v
6 product pairingon p-units in the pth cyclotomicfield F Q(µ )with values in aTate twist
p
5 =
3 of the ideal class group of F modulo p. We studied its numerous applications, but the values
1
of this pairing on special cyclotomic p-units were quite mysterious to us. Using the fact that
.
1
0 p-units of the form x and 1 x pair trivially with each other, we were able to narrow down
−
8
the possibilities for these values for p less than 10,000 (currently, 25,000). In fact, we were
0
: able to find a unique nontrivial possibility for the pairing up to a scalar in the F -group ring
v p
i
X of Gal(F/Q). The problem was that the scalar might not be a unit. Still, we conjectured that
r itisaunitandthat,ingeneral,thepairinghadimagespanningatleasttheentireminuspartof
a
theclass group(see Conjecture1).
Verysoonafter,theauthorcametorealizeaconnectionwiththeperiodsofmodularforms
indirectly through the work of Ihara and Takao [Ih]. We were led to check the conjectural
pairing values on special cyclotomic p-units against an old table of Manin’s of algebraic pe-
riods of cusp forms of small weight [Ma], and we saw that the ratios of the pairing values
agreed with the ratios of the periods modulo p. Thanks to an observation of Goncharov, the
fullcorrespondencequicklybecameclear: thevaluesofthecupproductpairingcorrespondin
adirect mannerto algebraicperiods of anewform satisfyingacongruence with an Eisenstein
series modulo p. This, in turn, led to the author formulate the conjecture (Conjecture 3) that
islaidout inthisarticle.
1
Sincethattime,theauthorhasgeneralizedthisconjecturetocompareaparticularvalueof
areciprocity map constructed out ofan inverselimitofcup productsto a two-variable p-adic
L-function of Mazur and Kitagawa. This and several variants may be found in [Sh2]. Still,
we feel that the original conjecture at level 1 and weight k is well worth stating, in that its
down-to-earthstatementservesasmotivationforthatwork. Forcomparisonwithmorerecent
work,wealso stateone, cleaner variant(Conjecture6).
We remark that recent independent but similarprogress has been made on this conjecture
byBusuioc[Bu]andtheauthor[Sh2,Section5]. Wereferthereaderespeciallytotheworkof
Busuioc in terms of the relation with the conjecture as stated in this article. She constructs a
modularsymbolofweightk onSL (Z)with L-valuescorrespondingto(again,possiblyzero)
2
projectionsofourpairingvalues.
The author would like thank everyone who gave their advice to him on his way to for-
mulating this conjecture, including, among others, Jordan Ellenberg, Sasha Goncharov, Dick
Gross,YasutakaIhara, BarryMazur,BillMcCallum,andWilliamStein. HealsothanksCelia
Busuioc and Glenn Stevens for motivating him to write this article. This material is based
upon work supported by the National Science Foundation under Grant No. DMS-0102016
and the Natural Sciences and Engineering Research Council of Canada. This research was
undertaken,in part,thanksto fundingfromtheCanada Research Chairs Program.
2 Cup product values
Let p be an odd prime, and set F Q(µ ), where µ denotes the group of pth roots of
p p
=
unity in an algebraic closure Q of Q. Let 1 Gal(F/Q). Let S denote the set consisting of
=
the unique prime above p in F, let G denote the Galois group of the maximal unramified
F,S
outside p extensionof F, and let O denotethe ringof p-integersin F. McCallum and the
F,S
authorconsidered thecup product
H1(GF,S,µp) H1(GF,S,µp) ∪ H2(GF,S,µ⊗p2)
⊗ −→
in Galois cohomology. (All tensor products in this article are taken over Z.) We recall that,
byKummertheory, H1(G ,µ )iscanonicallyidentifiedwithasubgroupof F /F p con-
F,S p × ×
taining O×F,S/O×F,pS. Furthermore, H2(GF,S,µ⊗p2) is canonically identified with AF ⊗ µp as
an F [1]-module, where A denotes the p-part of the class group of F. We obtain a Galois
p F
equivariantpairing
O O A µ .
×F,S × ×F,S → F ⊗ p
Now, suppose that p divides the numerator of B /k for some positive even k, where B
k k
2
denotesthekthBernoullinumber. Wesaythat(p,k)isanirregularpair. Thentheeigenspace
(1 k)
AF− of AF upon which 1 acts by the (1 − k)th power of the Teichmu¨ller character is
nontrivial. Wehavean induced pairing
h , ik: O×F,S ×O×F,S → A(F1−k) ⊗ µp.
McCallumandtheauthormadethefollowingconjecture(see[McS,Conjecture5.3],butnote
(q)
thatVandiver’sconjecturethat A 0forall evenq isnotassumedhere):
F =
(1 k)
Conjecture 1(McCallum-Sharifi). The imageofh , ik spans AF− ⊗ µp.
In particular, theauthorprovedthefollowing[Sh1, Corollary 5.9].
Theorem 2. Conjecture1 is truefor p < 1000.
Of particular interest are the values of this pairing on cyclotomic p-units. Fix a primitive
pth rootofunityζ . Forpositiveoddi, let
p
p 1
ηi = − (1−ζpu)ui−1 ∈ O×F,S.
u 1
Y=
For instance, η p. Note that the image of η modulo pth powers lies in the (1 i)th
1 i
= −
eigenspace ofO×F,S/O×F,pS, and it generates it if and only if A(F1−i) = 0. Since h , ik is Galois
equivariant,we have η ,η 0 ifi j k mod p 1. When i j k mod p 1, we
i j k
h i = + 6≡ − + ≡ −
set
(1 k)
ei,k = hηi,ηjik ∈ AF− ⊗ µp.
Notethat if j i mod p 1,then e e .
j,k i,k
≡ − =
Under Vandiver’s conjecture, which holds for p < 12,000,000 [BCE], there exists an
isomorphism
(1 k)
φ: AF− ⊗ µp −→∼ Fp
ofgroups. Choosingsuch an isomorphism,we may viewthe e as elementsof F . Foreach
i,k p
irregularpair(p,k)with p <25,000,McCallumandtheauthorhavefoundauniquenonzero
possibilityfortheresultingvector
(φ(e ),φ(e ),...,φ(e )) F(p 1)/2
1,k 3,k p 2,k p −
− ∈
upto scalar.
3
3 The conjecture in terms of L-values
AsinSection2,wesupposethat(p,k)isanirregularpair,sok isevenwithk 12. Consider
≥
theEisensteinseries ofweightk and level1:
G Bk ∞ σ (n)qn, σ (n) dk 1.
k k k −
= −2k + =
n 1 d n
X= Xd|1
≥
Let us fix an embedding of Q into Q , which in particular fixes a prime above p in every
p
algebraicextensionofQ. Then thereexistsnewform f ofweightk and level1 suchthat
f G mod p ,
k f
≡
where p is the chosen maximal ideal above p in the ring O generated over Z by the co-
f f
efficients of f. In fact, it is well-known that the space of weight k cusp forms for SL (Z)
2
congruentto G moduloourfixedprimeabove p inQ hasabasis ofeigenforms.
k
Wewilluse K todenotethequotientfield ofO . Let uswritetheq-expansionof f as
f f
∞
f a qn.
n
=
n 1
X=
Considerthe L-values L(f, j)of f for j with1 j k 1, and set
≤ ≤ −
(j 1)!
3(f, j) − L(f, j)
= ( 2π√ 1)j
− −
forsuch j. Foroddi with3 i k 3, let
≤ ≤ −
ρ(f,i) 3(f,i)/3(f,1),
=
whichby Eichler-Shimurais anelement of K .
f
Let H betheO -submoduleof K spannedbytheρ(f,i). TheEisensteinideal I ofO
f f f f f
generated by the a σ (n) for all n 1 is contained in H , as follows from [Ma, Theorem
n k f
− ≥
(k 1)
1.3]. In fact, thecompletionsof Hf and If at pf agree whenever AF− = 0, as we shallsee
inProposition8.
Let r denote the image of ρ(f,i) in H /p H for odd i with 3 i k 3. We are
i,f f f f
≤ ≤ −
nowready tostatethefirstform ofourconjecture:
(1 k)
Conjecture 3. Let ψ: Hf/pHf → Fp. Then there exists φ: AK− ⊗ µp → Fp such that
thetwo vectors
v (φ(e ),φ(e ),...,φ(e ))
φ 3,k 5,k k 3,k
= −
4
and
w (f) (ψ(r ),ψ(r ),...,ψ(r ))
ψ 3,f 5,f k 3,f
= −
k/2 2
inFp − areequal.
(k) (1 k)
Recall from the theory of cyclotomic fields that if A 0, then A is cyclic. In
−
F = F
particular, the hypotheses of the following proposition, which will be proved in Section 5,
holdunderVandiver’sconjecture.
(2 k) (1 k)
Proposition 4. If AF− = 0 and AF− is cyclic, then Hf/pf Hf is one-dimensional over
F .
p
UnderthehypothesesofProposition4, wethenhave:
(2 k) (1 k)
Corollary 5. If A − 0 and A − is cyclic, then Conjecture 3 asserts the existence of
F = F
isomorphisms
(1 k)
φ: AF− ⊗ µp −→∼ Fp and ψ: Hf/pf Hf −→∼ Fp
suchthatφ(e ) ψ(r )foralloddi with 3 i k 3.
i,k i,f
= ≤ ≤ −
Clearly, Corollary 5impliesConjecture1underitshypotheses.
4 The conjecture in terms of modular symbols
We continue with our earlier notation. Let V denote the subgroup of Z[X,Y] of homoge-
k 2
−
neouspolynomialsofdegreek 2. Thishas aleft actionof M (Z) via
2 +
−
a b
P(X,Y) P(dX bY, cX aY)
c d · = − − +
!
for P V . Let 1 denote the group of divisors on P1(Q) and 1 the subgroup of divisors
k 2 0
∈ −
of degree 0, equipped with the usual action of M (Z) . The Z-module M (Z) of “modular
2 + k
symbols”(see[St,Chapter 8])ofweightk onSL (Z) isthen defined as
2
M (Z) (V 1 ) /torsion,
k = k−2 ⊗ 0 SL2(Z)
i.e.,asthemaximaltorsion-freeSL (Z)-invariantquotientofV 1 . (Weremarkherethat
2 k 2 0
− ⊗
this is dual to another, perhaps more common definition of the module of Z-valued modular
symbols.) We represent the image of P ([b] [a]) in this quotient, for P V and
k 2
⊗ − ∈ −
5
a,b P1(Q), by P a,b . In our setting, any modular symbol may be written as an Z-linear
∈ { }
combinationofsymbolsoftheform Xi 1Yk 1 i 0, with1 i k 1.
− − −
{ ∞} ≤ ≤ −
Foranycommutativering R, weset
M (R) M (Z) R.
k k
= ⊗
If we replace 1 by 1 in the definition of M (R), we obtain an R-module of boundary
0 k
symbols B (R). We define the space of cuspidal modular symbols S (R) to be the kernel of
k k
theobviousmapM (R) B (R). ThenS (R)isgeneratedasan R-modulebythesymbols
k k k
→
of the form Xi 1Yk 1 i 0, with 2 i k 2. Furthermore, the image of Xk 2 0,
− − − −
{ ∞} ≤ ≤ − { ∞}
generates B (R). We may also consider the maximal quotients M (R) and S (R) upon
k k ± k ±
whichtheinvolution
P(X,Y) 0, P( X,Y) 0,
{ ∞} 7→ − { ∞}
acts by 1.
±
LetT denotethecuspidalweightk HeckealgebraforSL (Z)overZ ,andletH denote
k 2 p k
the full modular Hecke algebra. Then H acts on M (Z ) and B (Z ) using the above-
k k p k p
described M (Z) -action, and T acts on S (Z ). Let I denote the Eisenstein ideal in T ,
2 + k k p k k
generated by T 1 lk 1 for all primes l, and let m pT I be the maximal ideal in
l − k k
− − = +
whichitis contained.
We now present a rather sleeker variant of Conjecture 3, which matches better with the
discussionin[Sh2].
Conjecture 6. There existsanisomorphism
4k: A(F1−k) ⊗ µp −→∼ Sk(Zp)+/mSk(Zp)+
satisfying
4 (e ) Xi 1Yk i 1 0,
k i,k − − −
= { ∞}
foroddi with3 i k 3.
≤ ≤ −
Conjecture1 isan immediatecorollaryofConjecture6.
(2 k)
Remark. If A 0,thentheworkofKuriharaandHarder-Pinkshowsthatthetwospaces
−
F =
inConjecture6areindeedisomorphic(seeRemark 3.8,Proposition3.10,andthefootnoteon
p.292 of[Ku]).
To compare Conjecture 6 with Conjecture 3, note that we have a Hecke equivariant inte-
grationpairing
( , ) : M (C) S (C) C, (P 0, ,g) ∞g(z)P(z,1)dz,
k k k k
× → { ∞} =
0
Z
6
wheretheintegralistakenalongtheimaginaryaxisand S (C)denotesthespaceofcuspforms
k
ofweightk onSL (Z). ThispairingisnondegeneratewhenrestrictedtoS (C) . Pairingwith
2 k +
f inducesaperiod mapping
8 : M (Z) K (Xk 2 0, , f) ,
f k + f − k
→ · { ∞}
and weremark that3(f,i) 8 (Xi 1Yk 1 i 0, ).
f − − −
= { ∞}
Now,ifψ: H /p H F , thenthemap
f f f p
→
ψ (3(f,1) 18 ): S (Z) F
− f k + p
◦ →
extendstoamaponS (Z ) thatfactorsthroughtheEisensteincomponent. Thatis,itinduces
k p +
a map ψ : S (Z ) F . If we define φ ψ 4 , then we see that φ satisfies the
′f k p +m → p = ′f ◦ k
statementofConjecture 3. That is,wehave:
Proposition7. Conjecture6impliesConjecture3.
The question of whether every F -valued homomorphismon S (Z ) arises as a sum of
p k p +m
maps of the form ψ (varying f) is rather more subtle, and we will not attempt to address it
′f
here.
5 Complements
Inthissection,webrieflyexploretheconsequencesoftheworkofKurihara,Harder-Pink,and
Ohta for our conjectures. We begin by proving Proposition 4, which claimed that H /p H
f f f
(2 k) (1 k)
isone-dimensionalwhen A − 0 and A − iscyclic.
F = F
Proofof Proposition4. Kurihara[Ku,Theorem0.4]hasshownthattheEisensteincomponent
(T ) ofT isGorenstein. Since
k m k
S (Z ) Hom((T ) ,Z )
k p +m ∼ k m p
=
(this is standard: see, for instance, [Ku, Proposition 2.5]), we have that S (Z ) is free of
k p +m
rank 1over(T ) .
k m
Note that O /p has residue field F , as the completion (T ) itself has residue field F
f f p k m p
and has the p -completion of O as a quotient. We conclude that H /p H is indeed one-
f f f f f
dimensional over F , as it is the homomorphicimage of S (Z ) under the map induced by
p k p +m
3(f,1) 18 .
− f
7
ForanyO -module M,let M denotethecompletionof M atp . LetMbethemaximal
f pf f
idealcontainingtheEisensteinideal ofH .
k
Proposition8. Supposethat(H ) isGorenstein. Then (H ) (I ) .
k M f pf = f pf
Proof. Weneed onlyshowthat H is containedin (I ) . Considertheusualsurjection
f f pf
9 : M (Z ) Z , Xk 2 0, 1
k k p + p −
→ { ∞} 7→
taking S (Z ) to zero. This satisfies 9 (T x) (1 lk 1)9 (x) for all primes l and has
k p + k l − k
= +
kernel S (Z ) . Since
k p +
M (Z ) Hom((H ) ,Z ),
k p +M ∼ k M p
=
ourGorenstein conditionimpliesthat M (Z ) is free of rank 1 over(H ) withgenerator
k p +M k M
Xk 2 0, .
−
{ ∞}
Let usdefine ahomomorphism
8 : M ((O ) ) (K )
′f k f pf + → f pf
by extending (O ) -linearly the map 3(f,1) 18 . By what we have said above, its image
f pf − f
is generated by 8 (Xk 2 0, ) 1 as an (O ) -module, so the image is (O ) . If we
′f − { ∞} = f pf f pf
compose9 withthecanonicalquotientmap
k
Z (O ) /(I ) ,
p → f pf f pf
thentheresultingmaphasHeckeeigenvaluesequaltothea (f)modulo(I ) . Thiscomposi-
n f p
f
tionthereforeagreeswiththecompositionof8 withreductionmodulo(I ) . Inparticular,
′f f pf
since9 (S (Z ) ) 0,wehavethat H iscontainedin (I ) .
k k p + = f f pf
(k 1)
WeremarkthatOhtahasshownthat(Hk)MisGorensteinundertheconditionthat AF− =
0, as follows from [Oh, Theorem 3.3.2]. One might ask if it is Gorenstein under the weaker
(2 k)
conditionthat A − 0, thoughwewillnotexplorethisfurtherhere.
F =
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