p-adic interpolation of half-integral weight
modular forms
Adriana Sofer
asofer@math.princeton.edu
1
Introduction
The p-adic interpolation of modular forms on congruence subgroups of SL2 (Z)
has been succesfully used in the past to interpolate values of L-series.
In [12], Serre interpolated the values at negative integers of the ζ-series
of a totally real number field (in fact of L-series of powers of the Teichmuller
character) by interpolating Eisenstein series, which are holomorphic modular forms, and looking afterwards at the resulting constant terms of the
q-expansions. This can be interpreted as evaluating at q = 0, the degenerate
elliptic curve.
In [8], Katz interpolated the values of the Hecke L-series L(ψk , r) by
interpolating first “derivatives” of Eisenstein series, which are in general nonholomorphic, and then evaluating at elliptic curves.
In [18], we interpolated square roots of values of Hecke L-series by interpolating “derivatives” of theta functions, which are non-holomorphic halfintegral weight modular forms, and evaluating at elliptic curves. This last
step required appealing to the theory of integral weight p-adic modular forms,
since there is not a complete theory for half-integral weight yet. There is
ongoing research towards this (see the recent work of Jochnowitz [2], and
Stevens [19]).
There is renewed interest in half-integral weight forms, after the work of
Hida, Kohnen, Rodriguez-Villegas, Waldspurger, Zagier.
1
We here present a p-adic interpolation of the sequence
½³
d
N q dq
´k ¾
f
k≥0
where f is a given p-adic modular form of half-integral weight and of level
N ≥ 1, 4|N , p 6 |N . The integral weight version of this appears in [8, 1].
For integral weight, we use Serre’s and Katz’s theories of p-adic modular
forms ([12, 8, 6, 5, 7, 1]). We use Koblitz’s ([10]) and Jochnowitz’s ([2, 3, 4]
definitions for half-integral weight p-adic modular forms ([10]).
I would like to thank N. Jochnowitz for very helpful conversations, and
the referee for many suggestions and corrections.
2
2.1
Definitions and properties
p-adic modular forms
Let p ≥ 3 be a rational prime. Fix Cp the completion of an algebraic closure
of Qp . Let B ⊂ Cp be the ring of integers of a finite extension of Qp . We fix
an isomorphism C ' Cp and use it to consider B as included in C.
The reader is referred to [16] or [9, Ch.4] for the following definitions.
Let j be the multiplier system on Γ0 (N ), 4|N , corresponding to the stanP
2
dard theta series
θ = 1!+ 2 n≥1 e2πin z , and χ : (Z/N Z)∗ → C be a character
Ã
a b
∈ Γ0 (N ) via χ(d). H is the upper-half plane in C.
acting on γ =
c d
For k ∈ Z, we denote by Mk (Γ0 (N ), χ) the set ofÃholomorphic
f :H→C
!
a b
such that f (γz) = χ(d)(cz + d)k f (z) for every γ =
∈ Γ0 (N ) and f
c d
is holomorphic at all cusps.
For k ∈ Z + 21 , we denote by Mk (Γ0 (N ), χ) the set of
holomorphic
f :
Ã
!
a b
H → C such that f (γz) = χ(d)j(γ)2k+1 f (z) for every γ =
∈ Γ0 (N )
c d
and f is holomorphic at all cusps.
In both cases, and for every cusp t for Γ0 (N ), we associate to f ∈
Mk (Γ0 (N ), χ) a q-expansion ft ∈ C[[qN ]], (qN = e2πiz/N ) in a consistent
manner. We choose f∞ (qN ) = f (z). Given a ring A ⊂ C, we denote
Mk (Γ0 (N ), χ, A) the set of f ∈ Mk (Γ0 (N ), χ) satisfying ft ∈ A[[qN ]] for
L
every t. We extend these definitions to Mk (Γ1 (N )) ' Mk (Γ0 (N ), χ).
2
Recall that θ does not vanish on H, and that its q-expansions are:



 θ∞
θ
0


 θ1
2
2
= 1 +³2 n≥1 q n
´
P
2
= √12 1 + 2 n≥1 q4n
= θ( z4 ) − θ(z)
P
(1)
[Serre-Koblitz] We call a p-adic modular form over B for Γ1 (N ) a collection of q-expansions {ft (qN )}{t cusp} ⊂ B[[qN ]] such that there is a sequence
of complex modular forms fj ∈ Mkj (Γ1 (N ), B), kj ∈ Z (or kj ∈ Z + 21 ) with
p
(fj )t (qN ) −→ ft (qN )
uniformly on the coefficients, for every cusp t.
[Serre-Jochnowitz] We call a p-adic modular form over B for Γ1 (N ) a
power series f (qN ) ∈ B[[qN ]] such that there is a sequence of complex modular
forms fj ∈ Mkj (Γ1 (N ), Q̄) with p-adically integer coefficients , kj ∈ Z (or
kj ∈ Z + 21 ) with
p
fj (qN ) −→ f (qN )
uniformly on the coefficients, for every cusp t.
The Corollary in the next section shows that these two definitions are
equivalent. Until that is established, we will work with definition 2.1.
Any p-adic modular form has a weight k ∈ X (or k + 21 ∈ X + 12 ) (see
[2, Theorem 1.2], [12, 10]), where X = Zp × Z/(p − 1)Z. Moreover, if p 6 |N ,
k is the p-adic limit of (kj , kj mod p − 1). We denote Mpk (Γ1 (N ), B) (or
Mpk+ 1 (Γ1 (N ), B)) the set of such f ’s.
2
We immediately obtain that any classical holomorphic modular form with
coefficients in B can be interpreted as a p-adic form.
Proposition 2.1 The Ramanujan function P (z) = 1 − 24 n≥1 σ(n)e2πinz
is not modular in the classical sense, but P ∈ Mp2 (SL2 (Z), Zp ).
P
Proof: [12, Example p.210].
d
1
Let “q dq
” be the operator D + 12
P H, where D is the Halphen-Fricke
operator, P is the Ramanujan
and H(g) = kg if g ∈ Mk (Γ0 (N ), χ)
³
´function,
P
P
d
n
n
(see [8, 18]). We have N q dq f
= nan qN
if f∞ = an qN
.
∞
3
Proposition 2.2 Let p ≥ 5, k ∈ X ∪ X + 21 , and f ∈ Mpk (Γ1 (N ), B). Then
d
f ∈ Mpk+2 (Γ1 (N ), B), where the weight k + 2 means k + (2, 2).
q dq
N
d
= N D + 12
P H. For k ∈ Z ∪ Z + 12 , N D is a derivation
Proof: N q dq
1
Mk (Γ0 (N ), χ, A) → Mk+2 (Γ0 (N ), χ, A[ 12
]) that acts continuously. Hence,
1
for k ∈ X ∪ X + 2 ,
D : Mpk (Γ1 (N ), B)
−→ Mpk+2 (Γ1 (N ), B)
p
H : Mk (Γ1 (N ), B)
−→ Mpk (Γ1 (N ), B)
P ∈ Mp2 (SL2 (Z), Zp )
and we deduce the result.
Note: Also see an alternate proof of proposition 2.2 in [2].
2.2
The q-expansion principle
Recall the q-expansion principle for (integral weight) p-adic modular forms
([8, 5.2], [1, Theorem I.3.1, Prop. I.3.2]). We will prove a similar result for
half-integral weight.
When k ∈ Z, the weight k means (k, k), and k + 12 , (k, k) + 12 . Let us
denote Mp•+ 1 (Γ1 (N ), B) (and Mp• (Γ1 (N ), B)) the set of Z-linear combina2
S
tions of forms in k∈Zk≥0 (Mpk+ 1 (Γ1 (N ), B) ⊗ Qp ) (and, for integral weight,
2
S
p
k∈Zk≥0 (Mk (Γ1 (N ), B) ⊗ Qp )) such that the q-expansions coefficients (of the
Z-linear combinations) at all cusps belong to B (the q-expansions are defined by linearity). Notice that any sequence approximating an element of
Mp•+ 1 (Γ1 (N ), B) (or Mp• (Γ1 (N ), B)) will eventually have coefficients in B,
2
rather than in B ⊗ Qp , due to the uniform convergence. Also, for k ∈ Z,
k ≥ 0,
(Mpk+ 1 (Γ1 (N ), B) ⊗ Qp ) ∩ Mp•+ 1 (Γ1 (N ), B) = Mpk+ 1 (Γ1 (N ), B),
2
2
2
d
on Mp•+ 1 (Γ1 (N ), B) and Mp• (Γ1 (N ), B)
and similarly for weight k. We define q dq
2
by linearity.
Jochnowitz independently proved results similar to the next proposition.
4
√
Proposition 2.3 (q-expansion principle) Let µN ∪ { 2} ⊂ B 0 ⊂ B ⊂
Cp , B and B 0 the ring of integers of K, K 0 ⊂ Cp , finite extensions of Qp . Let
p ≥ 3, p 6 |N , and assume f ∈ Mp•+ 1 (Γ1 (N ), B).
2
0
1. If f∞ ∈ B [[qN ]], then f ∈
Mp•+ 1 (Γ1 (N ), B 0 ).
2
2. Let π be a prime element in B. The map
Mp•+ 1 (Γ1 (N ), B)
2
q-exp at ∞
−→
B[[qN ]]
is injective, and the cokernel has trivial π-torsion.
Proof:
1. Since p ≥ 3, θ ∈ M 1 (Γ0 (4), Z[ √12 ]) ⊂ Mp (Γ1 (N ), B 0 ). We know
2
that f θ ∈ Mp• (Γ1 (N ), B), and (f θ)∞ ∈ B 0 [[qN ]]. By the integral-weight
q-expansion principle , f θ ∈ Mp• (Γ1 (N ), B 0 ), i.e. (f θ)t ∈ B 0 [[qN ]] for every cusp t. Since θt |(f θ)t , and θt ∈ B 0 [[qN ]] with invertible first coefficient
(see equations (1)), (f )t = (f(θ)θ)tt ∈ B 0 [[qN ]] for every cusp t. Take a sequence of classical modular forms, with coefficients in B approaching f in
P
the following way: if f = i hi , hi ∈ Mpki + 1 (Γ1 (N ), B) ⊗ Qp , take sequences
2
fij ∈ Mkij (Γ1 (N ), B) ⊗ Qp with
(fji )t −→ (hi )t .
We have
(fj )t =
X
i
(fji )t −→ ft .
Since f has integral coefficients, the uniform convergence guarantees that fj
has coefficients in B, for every j >> 0.
Let K̄ be the Galois closure of K in Cp . Let d = [K̄, K 0 ]. Define
X1
1
trK̄,K 0 fji ,
gj = trK̄,K 0 fj =
d
i d
a sequence such that dgj has coefficients in B 0 ∩ C for j >> 0. The forms gji =
1
tr
f belong to Mki + 1 (Γ1 (N ), B 0 ) ⊗ Qp . Call si ∈ Mpki + 1 (Γ1 (N ), B 0 ) ⊗
d K̄,K 0 ji
j
2
2
Qp the limit of gji . So
(gj )t =
X
i
X
(gji )t −→ (
5
i
si )t .
Also
(gj )t −→ ft .
p
M•+ 1 (Γ1 (N ), B 0 ).
2
We deduce that f = i si ∈
2. Assume f, g ∈ Mp•+ 1 (Γ1 (N ), B), f∞ = g∞ . Then (f θ)∞ = (gθ)∞ , and
2
f θ, gθ ∈ Mp• (Γ1 (N ), B). Again by the integral-weight q-expansion principle,
(f θ)t = (gθ)t for every t, hence ft = gt for every t, and f = g.
Assume f∞ ∈ B[[qN ]] and π r f ∈ Mp•+ 1 (Γ1 (N ), B) for some r ≥ 0. Equiv2
alently, f ∈ Mp•+ 1 (Γ1 (N ), B) ⊗ Qp . Again we multiply by θ and use the
2
integral-weight q-expansion principle for π r f θ ∈ Mp• (Γ1 (N ), B), and deduce
that f θ ∈ Mp• (Γ1 (N ), B). Therefore, since θt is π-integral (see equations (1)),
π r ft is divisible by π r for every t. Hence ft has coefficients in B for every
t. Take a sequence (fj )j with coefficients in B, approaching π r f . Then the
sequence (π −r fj )j will approach f . We deduce that f ∈ Mp•+ 1 (Γ1 (N ), B).
2
This ends the proof.
P
√
Corollary (Jochnowitz). If p ≥ 3, p 6 |N , µN ∪ { 2} ⊂ B, definitions 2.1
and 2.1 are equivalent.
Proof: The reader is referred to [2, 3, 4].
Therefore, we identify f to f∞ in what follows.
2.3
Measures
A measure on Zp with values in a p-adic ring A (i.e. A = lim A/pn A) is an
←
additive function
µ : {compact open subsets of Zp } −→ A.
We call M eas(Zp , A) the ring of measures (with convolution as product).
The Fourier transform
Ã
!
!
Ã
Z
X Z
x
x
dµ X n
(X + 1) dµ(x) =
µ̂(X) =
n
Zp
Zp
n≥0
gives a ring isomorphism between M eas(Zp , A) and A[[X]]. For example,
Ã
! the
P
a
X n.
Dirac measure at a ∈ Zp has Fourier transform (X + 1)a = n≥0
n
6
By abuse of notation, we will often use µ̂(T ) instead of µ̂(X), where T =
X + 1. Also, M eas(Zp , A) = lim
M eas(Z/pn Z, A), and
←
M eas(Z/pn Z, A) ' A[Z/pn Z] = {
X
a mod
pn
ma T a , ma ∈ A}.
If µ ∈ M eas(Zp , A) and C is a compact open subset of Zp , µ|C , the restriction
of µ to C, extended by 0, is in M eas(Zp , A) too.
The following properties can easily be checked by proving them first for
the dense subring of finite linear combinations of Dirac measures.
• If µ̂(T ) =
P
n≥0
an T n , then µ̂|pZp (T ) =
P
n≥0
apn T pn .
• For any a ∈ Zp , and any measure µ ∈ M eas(Zp , A), define µa ∈
M
eas(Zp , A) Rto be the unique measure with µ̂a (T ) = µ̂(T a ). Then
R
h(x)dµa = h(ax)dµ for every continuous function h : Zp −→ Zp .
References for measures are [8, 17, 20].
3
Interpolation
3.1
The measure associated to a modular form
Let p ≥ 5, p 6 |N . Define M = Mp (Γ1 (N ), B) to be the smallest p-adic ring
containing Mp•+ 1 (Γ1 (N ), B). We define the q-expansions on M by continu2
ity. Given f ∈ Mp•+ 1 (Γ1 (N ), B), we define µf by
2
µ̂f (X) =
X
m≥0
where cm,r ∈ Q are such that
P
n≥0
n
an q N
,
m
X
Ã
r=0
Ã
x
m
µ̂f (T ) =
cm,r
!
n≥0
7
d
Nq
dq
=
X
Proposition 3.1 µf ∈ M eas(Zp , M).
Ã
!r !
Pm
f Xm
r=0 cm,r x
n n
T .
an q N
r
. Notice that, if f =
Proof: To see that µf is a measure, it is enough to prove that µ̂f ∈ M[[X]]
N
(rather that in M Qp as the definition seems to suggest). An easy computation shows that
µ̂f (X) =
X
m≥0


X
n≥0
Ã
n
m
!

n
X m.
an q N
So the mth -coefficient is in Mp•+ 1 (Γ1 (N ), B) ⊗ Qp , and its q-expansion at ∞
Ã
2
!
n
n
∈ B[[qN ]].
an q N
is n≥0
m
Therefore, the mth -coefficient belongs to the p-torsion of B[[qN ]]/Mp•+ 1 (Γ1 (N ), B),
2
which is trivial by the q-expansion principle. We get that all coefficients are
in Mp•+ 1 (Γ1 (N ), B).
P
2
By density, we extend the definition to f ∈ M.
The expression Zp xr dµf is called the rth -moment of the measure (r ∈ Z).
The rth -moment of µf satisfies
R
Z
r
x dµf =
Zp
X
n
an nr q N
n≥0
³
Ã
d
= Nq
dq
!r
f.
´r
d
Therefore, the rth -moment is N q dq
f.
Any measure µ ∈ M eas(Zp , A) “interpolates” its moments, via the Gamma
transform
Γµ : Zp −→ A
R
s
∗
s 7→
Zp < x > dµ|Zp
p
This
can be seen as follows: if r → s, |r| → ∞, r ≡ 0 mod p − 1, then
R
r
Zp x dµ → Γµ (s). For r ≡ 0 mod p − 1,
Γµ (r) =
Z
Zp
xr dµ|Z∗p =
Z
Zp
xr dµ −
Z
Zp
xr dµ|pZp .
This is the (rth -moment) − (something related to it). We next pay a closer
look at Γµ (r).
8
3.2
Restriction of the measure to pZp
If µ̂f (T ) =
P
n≥0
n n
an q N
T , then
µ̂f |pZp (T ) =
X
pn pn
apn qN
T = µ̂Rf (T )
n≥0
pm
where Rf = Zp dµ|pZp ∈ M, with q-expansion at ∞:
m≥0 apm qN . The
P
P
n
n
,
if f = an qN
operator R equals V U , where V f = f (q p ), and U f = anp qN
operators that respect the level. (See [4]).
Let p ≥ 5, p 6 |N , and f ∈ Mp•+ 1 (Γ1 (N ), B), and let µf be the measure on
2
Zp with values in M associated to it. Then its Gamma transform Γf = Γµf
´r
³
d
f . More precisely, for r ≡ 0 mod p − 1,
“interpolates” the sequence N q dq
R
P
!r
Ã
!r !
ÃÃ
d
d
f − R Nq
f
Γf (r) = N q
dq
dq
Ã
!r
ÃÃ
!r
!
d
d
= Nq
f−
Nq
Rf
dq
dq
Ã
!r
ÃÃ
!r
!
d
d
r
= Nq
f −p V
Nq
Uf
dq
dq
Proof: We prove the first equality. The other two are left to the reader.
Γf (r) =
Z
Zp
=
Ã
=
Ã
=
Ã
xr dµf −
d
Nq
dq
!r
d
dq
!r
d
Nq
dq
!r
Nq
Z
xr dµRf
Zp
f−
Z
xr d(
Zp
n≥0
f−
X
pn
apn (pn)r qN
X
pn pn
apn qN
T )
n≥0
f −R
ÃÃ
d
Nq
dq
!r !
f
We include the following particular case.
n
Corollary. If f ∈ Mp•+ 1 (Γ1 (N ), B), f∞ (qN ) = n≥0 an qN
, satisfies vp (n) 6=
2
1 when an 6= 0 (the standard theta function θ, and θψ,t , the generators of all
P
9
forms of weight 12 given in [13] are examples), then the interpolation can be
further refined:
For r ≡ 0 mod p − 1,
Ã
d
Γf (r) = N q
dq
!r
f −p
2r
"Ã
d
Nq
dq
!r
2
#
2
(U f ) (q p )
Proof:
Ã
!r
Ã
!r
f −p
=
Ã
d
dq
!r
f − p2r
=
Ã
d
Nq
dq
!r
Γf (r) =
=
4
d
Nq
dq
d
Nq
dq
Nq
f−
Z
Zp
f −p

r
2r
x d
Z
r
Zp
2r
X
X
n≥0

x d

np2 np2 
anp2 qN
T
X
n≥0

np2 n 
T
anp2 qN
2
np
anp2 nr qN
n≥0
"Ã
d
Nq
dq
!r
2
#
2
(U f ) (q p )
Applications
As mentioned in the introduction, this kind of interpolation was used in
several instances to obtain p-adic L-functions. For that, we start with a
classical modular form f , we interpret it as a p-adic form, and we compose the
interpolating function Γf : Zp −→ M from Proposition 3.2 with evaluation
at an appropriate modular point. For half-integral weight, this can not be
done directly, since we do not know how to evaluate half-integral weight padic forms yet. In [18], we evaluated, instead, the function Γf /θ, that takes
values on Katz’s generalized p-adic modular forms. This provides a function
Zp −→ B, that interpolates the values obtained by evaluating the sequence
n ³³
´r
³³
´r ´´o
1
d
d
N
q
f
−
R
N
q
f
at the modular point.
θ
dq
dq
Via Shimura Reciprocity,
o evaluations become an Euler Factor times
n ³
´r these
1
d
the evaluation of θ N q dq f .
³
´r
d
Finally, the evaluation of 1θ N q dq
f is proved to equal the evaluation
1
r
of θ W f , where W is the Maass-Weil operator. This puts us in the realm of
the classical theory, where the connection between W r f and special values
10
of L-series is established. The extra θ in the denominator is finally absorbed
by the power of the complex period that naturally appears in the resulting
formula.
References
[1] F. Gouvêa, Arithmetic of p-adic Modular Forms, Springer-Verlag LNM
1304 (1988).
[2] N. Jochnowitz, Congruences between modular forms of half integral
weights, and implications for class numbers and elliptic curves, to appear in Invent. Math.
[3] N. Jochnowitz, An alternate approach to non-p divisibility of coefficients
of half-integral forms with implications for Iwasawa λ-invariants and for
Kolyvagin, Birch and Swinnerton-Dyer results, to appear.
[4] N. Jochnowitz, A p-adic theory for half-integral weight forms, to appear.
[5] N. Katz, p-adic properties of modular schemes and modular forms, in
Modular functions of one variable III, Springer-Verlag LNM 350 (1973),
69-190.
[6] N. Katz, p-adic L-functions via moduli of elliptic curves, in Algebraic
Geometry: Arcata 1974, Proceedings of Symposia in Pure Mathematics
29, AMS (1975).
[7] N. Katz, Higher congruences between modular forms, Ann. Math. 101
(1975), 332-367.
[8] N. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of
Math. 104 (1976), 459-571.
[9] N. Koblitz, Introduction to Elliptic Curves and Modular Forms,
Springer-Verlag (1984).
[10] N. Koblitz, p-adic Congruences and Modular Forms of Half Integral
Weight, Math. Ann. 274 (1986), 199-220.
11
[11] S. Lang, Elliptic Functions, Springer-Verlag (1987).
[12] J.P. Serre, Formes Modulaires et Fonctions Zêta p-adiques, in Modular
functions of one variable III, Springer-Verlag LNM 350 (1973), 191-268.
[13] J.P. Serre and H. Stark, Modular Forms of Weight 12 , in Modular functions of one variable VI, Springer-Verlag LNM 627 (1977), 28-67.
[14] G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press (1971).
[15] G. Shimura, Modular Forms of Half-Integral Weight, in Modular Forms
of One Variable I, Springer-Verlag LNM 320 (1972), 57-74.
[16] G. Shimura, On modular forms of half integral weight, Ann. of Math.
97 (1973), 440-481.
[17] W. Sinnott, On the µ-invariant of the Γ-transform of a rational function,
Inv. Math 75 (1984), 273-282.
[18] A. Sofer, p-adic interpolation of square roots of central values of Hecke
L-series, Ph.D. thesis, The Ohio State University (1993).
[19] G. Stevens, article in this volume.
[20] R. Yager, On two variable p-adic L-functions, Annals of Math. 115
(1982), 411-449.
12