Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
A Gentle Introduction to Overconvergent
Modular Forms
L. J. P. Kilford
University of Bristol
27 July 2009
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Congruence subgroups
We recall the definition of the modular group SL2 (Z):
SL2 (Z) := ca db : a, b, c, d ∈ Z such that ad − bc = 1 .
A congruence subgroup Γ of SL2 (Z) is a subgroup which contains
Γ(N) := ca db ∈ SL2 (Z) such that ca db ≡ ( 10 01 ) mod N
for some integer N ≥ 2.
We say that Γ has level N.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Modular forms
A classical modular form f of weight k for the congruence
subgroup Γ is classically described as a holomorphic function from
the Poincaré upper half plane to C which satisfies
az + b
a b
k
= (cz + d) f (z) for
∈ Γ.
f
c d
cz + d
A famous example of a modular form for the full modular group is
the ∆-function, given by
∆(q) = q
∞
Y
n 24
(1 − q )
n=1
=
∞
X
τ (n)q n ,
n=1
where q := exp(2πiz) and τ (n) is the Ramanujan τ function.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
The Eisenstein series
We can define modular forms of even weight k ≥ 4 for SL2 (Z) by
Ek (z) :=
X
c,d∈Z
(c,d)=1
1
.
(cz + d)k
These can be shown to transform
correctly under the two
0
1
generators of SL2 (Z) ( −1 0 and ( 10 11 )) and to have no poles on
the upper half-plane.
We can show that the Fourier expansion at ∞ of the Eisenstein
series is
∞
ζ(1 − 2k) X
+
σk−1 (n)q n .
Ek (q) :=
2
n=1
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
What about k = 2?
If we make the same definition for k = 2, then we find that we
can’t rearrange the terms of the series defining E2 , so we fail to
obtain a modular form (this is called a quasi-modular form in the
literature).
We can obtain a Fourier series, though, which is
E2 (q) := 1 − 24
∞
X
σ1 (n)q n .
n=1
We will encounter E2 again later on.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
A complete answer?
We can in fact write every modular form for SL2 (Z) as a graded
polynomial in E4 and E6 ; it can be shown that
M = C[E4 , E6 ] and S = ∆ · M,
where
E43 − E62
,
1728
and a similar result holds for other congruence subgroups.
∆=
However, this does not tell the whole story.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
The j-invariant
The modular j-invariant is an important modular function (as
opposed to modular form). It has weight 0 and is defined by
j(q) :=
1
E4 (q)3
= +744+196884q+21493760q 2 +864299970q 3 +· · ·
∆(q)
q
It has a single simple pole at ∞, and satisfies
several interesting
√
properties; for instance, because τ = 1+ 2−163 is an element of a
field with class number 1, j(τ ) is an integer; this means that
eπ
√
163
= 262, 537, 412, 640, 768, 743.999 999 999 999 25 . . .
≈ 640, 3203 + 744.
It is also closely connected to the Monster group, the largest of the
sporadic finite simple groups (which has an irreducible
representation of dimension 196883).
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Theta series
Another standard example of modular forms are provided by theta
series. We can take a quadratic form and use it to create a
modular form; for instance, we have
X
2
2
θ(q) :=
q x +y = 1 + 4q + 4q 2 + 4q 4 + 8q 5 + · · ·
m,n∈Z
This is a modular form of weight 1 of level 4.
One motivation for studying modular forms is to give bounds on
the number of ways an integer can be represented by a quadratic
form (if one is very lucky, one can derive formulae for them a la
Glaisher).
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Finite-dimensionality
We can prove that given a fixed level and weight, the space of
modular forms of that level and weight is finite-dimensional as a
complex vector space; this allows us to perform computations
effectively.
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1
L. J. P. Kilford (University −
of2 Bristol)
1
2
A Gentle Introduction
to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Number-theoretic identities
Using the finite-dimensionality of spaces of modular forms, we can
prove theorems like
σ7 (n) = σ3 (n) + 120
n−1
X
σ3 (i) · σ3 (n − i);
i=1
this follows because M4 (SL2 (Z)) = CE4 and M8 (SL2 (Z)) = CE8 ,
so we can prove it by noting that E8 = E42 and comparing their
Fourier coefficients.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Hecke operators
We can define linear operators called Hecke operators Tp
(called Up if p | N) on modular forms in many intrinsic ways. The
effect on Fourier expansions is given by
(Tp f )(q) =
∞ X
X
apn + χ(p)p k−1 an/p q n =
bn q n ,
n=0
n=0
where χ is a Dirichlet character and an/p = 0 if n/p is non-integral.
We can find examples of forms which are eigenforms for all of these
Hecke operators; examples are ∆, θ, and the Eisenstein series Ek .
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Slopes of modular forms
If f is an eigenform for Tp , then we say that the p-slope of f is
ap
vp
.
a1
For instance, ∆(q) = q − 24q 2 + 252q 3 + O(q 4 ), so the 2-slope
of ∆ is 3.
The slopes will play an important role in the p-adic theory we will
be introducing later, but they can be understood purely in terms of
the classical forms.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Example of slopes
The T2 operator acting on the space S42 (SL2 (Z)) has
characteristic polynomial
x 3 + 344688x 2 − 6374982426624x − 520435526440845312;
the Newton polygon of this is:
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the slopes here are [4, 9, 12].
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Two Conjectures
Maeda’s Conjecture says that the characteristic polynomials of the
Hecke operators Tp acting on Sk (SL2 (Z)) are irreducible. This is
still open (it has been checked for many choices of p and k, and
density results are also known).
It is also not known whether the positive Fourier coefficients of
∆(q) = q
∞
Y
(1 − q n )24 =
n=1
∞
X
τ (m)q m
m=1
are all nonzero!
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
p-adic modular forms
If p is a prime number, then we can define the q-expansion of a
p-adic modular form of weight κ to be the p-adic limit of
q-expansions of classical modular forms of weights k which tend
p-adically to κ.
For instance, it is well-known that
Epr (p−1) ≡ 1
mod p r +1 · (p − 1) for r ≥ 1,
and using this formulation means therefore that we can think of
these Eisenstein series as being p-adic approximations to the
constant form 1 in weight 0. They form a “p-adic family”.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Fitting E2 into the picture
We recall the quasi-modular form E2 that we saw earlier. This is
not a modular form, but given any p we can write it as a p-adically
convergent sum of p-adic modular forms, so it is a p-adic modular
form for any p. If we define
Ek∗ (z) := Ek (z) − p k−1 Ek (pz),
for any Eisenstein series Ek , then we can expand Ek as
Ek = Ek∗ (z)+p k−1 Ek∗ (z)+p 2(k−1) Ek∗ (pz)+· · ·+p m(k−1) Ek∗ (p m z)+· · · ,
and each of these terms is a classical modular form. If we put in
k = 2 then we have our result.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Caveats about the p-adics
The previous definition is known as “p-adic forms in the sense of
Serre”; this definition is fairly clear but fails to encompass
everything. A more comprehensive but rather more abstract
definition was given by Katz; this is in terms of a rule sending
p-adic test objects to elements of a p-adic ring; the rule must
satisfy conditions similar to those defining classical modular forms.
The definition also uses the theory of canonical subgroups of
elliptic curves.
Also, the definition in terms of Serre does not give the whole
picture for very small primes (such as 2 and 3).
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Problems with the p-adics
Unfortunately the p-adic modular forms as defined above are not
optimal for computations, because although we can define
analogues of the Hecke operators for them, these extended Hecke
operators are not compact.
Here is an example of the sort of

1 0
 0 1

 0 0

.. ..
. .
L. J. P. Kilford (University of Bristol)
thing that goes wrong:

0 ...
0 ... 

1 ... 

.. . .
.
.
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Overconvergence
We are therefore motivated to find a large subspace of the p-adic
modular forms where the Up operator is compact.
The formal definition of overconvergence is rather complicated, but
here is an idea. If the space of p-adic modular forms of weight 0 is
given by sums of the form
∞
X
ai j i , where ai ∈ Cp ,
i=0
then the overconvergent modular forms of weight 0 are
∞
X
ai j i , where ai ∈ Cp , and |ai |p → 0 as i → ∞.
i=0
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Handwavy overconvergence
The idea of overconvergence is to consider the definition of p-adic
modular forms as sections of structure sheaves on a modular curve
viewed as a rigid analytic space, and extend it slightly.
The problems one has to deal with is that there are bad areas on
the modular curve that we don’t want to define our forms in, but if
we exclude too much of the curve then we get way too many
functions (these are the p-adic modular forms).
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Details for experts
We also need to be careful about the positions of the poles of the
classical Eisenstein series we are using.
If we (very carefully) extend the range of definition of our forms a
little way into the bad areas, then this excludes a lot of the more
badly-behaved forms and gives us a nicer ring of forms to deal with
that have good arithmetic (the overconvergent modular forms).
The area we are extending into is the locus of
not-too-supersingular curves; these are the supersingular curves for
which there is a defined canonical subgroup.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Motivation for the overconvergent forms
Why do we care about these obscure objects that one needs to
know about structure sheaves and rigid geometry to even define?
1. The classical modular forms inject into the overconvergent
forms; if we can prove results about the latter we can intuit
stuff about the former.
2. We can consider many overconvergent weights at once,
3. This is in some way a more “natural” way to think about
modular forms than the complex approach.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
The different sorts of modular forms
Let us consider the different types of modular forms by considering
the example of the quasimodular form E2 .
E2
E2∗
E2
E2
classical
classical
p-adic
overconvergent
no
yes
yes
no
In fact, E2 is transcendental over the ring of overconvergent
modular forms.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
The Arithmetic Approach
Although modular forms are normally defined (at least to begin
with) as complex functions, and the proof that the spaces are
finite-dimensional relies on their complex definition, most number
theorists care about them because they have an interesting
arithmetic theory.
“. . . the reason that modular forms enter into number
theory at all, in some sense, is exactly because modular
curves can be defined over schemes other than Spec C”
— Wikipedia talk page for Modular form
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Choice of basis
The definition given can be refined to allow us to choose a specific
basis for the overconvergent forms; for instance, when p = 2, if we
define g := ∆(q 2 )/∆(q) then a suitable Banach basis is
6
1, 2 g , 212 g 2 , 218 g 3 , . . . .
We can do this because there are results which guarantee that a
wide range of choices of power of p will work here and give the
same characteristic power series for the Hecke operator Up .
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Using Magma
Magma does not natively support overconvergent modular forms,
but it can work with Fourier expansions, so one can investigate
them using it. It does support classical modular forms well, which
is a good reason to choose it. Sage could also be used for much
of the work (and the newest version of Sage now does support
overconvergent modular forms natively (this is work of Loeffler)).
Although overconvergent modular forms are defined over the
p-adic field, which is an inexact ring (and some instantiations of
which can actually fail to be associative!), a lot of the calculations
can in fact be taken to be over finite extensions of Q, which is
much better for computing.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
An explicit example
We say that a classical modular form f of weight k has character χ
if it satisfies
az + b
a b
k
f
= χ(d) · (cz + d) f (z) for
∈ Γ0 (4).
c d
cz + d
Let τ : (Z/4Z)× → C× be the nontrivial Dirichlet character.
The slopes of the Hecke operator U2 acting on modular cusp forms
of character τ and odd weight are
Weight
5
7
9
11
..
.
L. J. P. Kilford (University of Bristol)
Slopes
2
2,4
2,4,6
2,4,6,8
..
.
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
The theorem
We can in fact prove:
Theorem
Let k be an odd integer. The slopes of the U2 operator acting on
overconvergent modular forms of weight k and conductor τ are
{2i}i∈N .
This can be proved by considering one weight first (weight 1) and
then using a result of Coleman to move to all other odd weights.
Similar results hold for other characters of conductor 2n (for an
integer n ≥ 2); this is joint work with Buzzard.
This result also tells us about the field of definition of the basis of
eigenforms.
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
Extensions to other primes
Similar results have been proved for other (“small”) primes:
Prime
2
3
5
7
K., Buzzard-K.
Roe, Jacobs
K.
K.-McMurdy (in progress)
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms
Classical modular forms
p-adic and overconvergent modular forms
Explicit computations
More complicated cases
When the level is 1, the slopes can be more involved. Here are the
slopes of T2 acting on weight k classical forms:
Weight
12
14
16
18
20
22
24
26
28
30
Slopes
3
3
4
3
5
3,7
4
3,8
6,6
Weight
32
34
36
38
40
42
44
46
48
50
Slopes
3,7
4,8
3,9
5,8
3,7,11
4,9,12
3,8,11
6,6,13
3,7,12,15
4, 8, 13
(Emerton, Clay, Smithline, and Buzzard-Calegari, Loeffler. . . )
L. J. P. Kilford (University of Bristol)
A Gentle Introduction to Overconvergent Modular Forms