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>> Kristin Lauter: Okay. So we're very pleased to have Ken Ribet here visiting us today
from UC Berkeley, and he'll speak to us about -- uh-oh, password required -- modularity
of two-dimensional reducible Galois representations. Thank you.
>> Kenneth Ribet: Thank you very much for the introduction and thank you very much
for the invitation. I've known Kristin for a long time and I've known Microsoft Research
for a very long time, and I've sent graduate students here as interns and corresponded
with some number of you over the years. This is my very first visit to Microsoft, and it's
wonderful. So I really like it.
I wondered what to talk about here, and I realized there's a very interesting subject that
grew out of a question that I was asked by a mathematician in Barcelona. And this really
has to do with some very classical objects that are probably familiar to many of you in
the audience and have just served as a wonderful source of very profound questions for -I wrote at least a hundred years. It's kind of a benchmark. But a lot of the arithmetic of
these objects has been going on roughly for 40 years. I think that's another benchmark,
as I'll explain.
So I'm talking about modular forms in a very classical sense, so these are functions, F of
Z, and the number Z is just a complex number with positive imaginary part. These are
holomorphic functions. And these functions, the way that they are usually discussed
arithmetically is through their Fourier series representations. These functions have some
large group of symmetries, some group of functional equations that they satisfy.
And one of the simplest functional equations is just in variance under integer translation.
And so it's very natural to regard these as functions of the exponential, either the 2 pi IZ
and then represent them as some power series, and in addition to the holomorphicity on
the upper half plane, there are a number of growth conditions that I don't discuss
specifically or explicitly that are imposed on these functions that imply that these
functions can be written as Fourier series beginning with a constant term and then
positive powers of either the 2 pi I times the natural variable.
And these Fourier series for us are basically the objects that we study, so even though
these are holomorphic functions, from another perspective, all they are is just formal
power series.
And the coefficients that occur a priori can be any complex numbers, but the types of
functions that we're looking at -- I used the word arithmetical a moment ago -- really
have the property that the numbers that occur are in the most interesting cases ordinary
integers and slightly more generally they're algebraic integers.
Not only that, but if they're algebraic integers as opposed to rational integers, all of them
as N varies will lie inside of some fixed number field. So it's not as though as you
generate more and more of these coefficients you increase the size of the field that you
need to contain them; they're all contained in some finite extension of the rational number
field, they're all going to be algebraic integers. And typically the field in which they'll be
contained in the examples I discussed will be totally real. That means that all of the
conjugates of all of the numbers in this field will be real numbers as opposed to complex
numbers. So it's very restrictive and kind of a very good place to work.
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Here are some of the equations that may satisfy some of these functional equations, as I
call them. Basically aside from some automorphy factor, what's going on is that the
functions have to be invariant under fractional linear transformations coming from some
large subgroup. It will be a subgroup of finite index in the group that I write, SL(2, Z),
it's just the group of integer matrices with determinant 1.
In addition to finite index, they actually lie in a congruent subgroup, which in this case is
a stronger condition. It just means that they will -- a congruent subgroup is a subgroup
that contains all of the matrices congruent to the identity modulo some number, usually
called positive n -- positive capital N, rather. That number is kind of the level of the
group.
The number K that appears in automorphy factor CZ plus D to the K is called the weight
of the modular form, and because I am going to be keeping things very simple, the weight
will be a positive even integer in all examples. And aside from one example that I give,
basically the weight is going to be 2.
So you have a very simple equation: F of AZ plus B over CZ plus D is CZ plus D
squared, F of Z. And that means, equivalently, if you think about calculus, that if you
write F of Z and multiply it by a formal differential, DZ, that thing's going to be actually
invariant under the group.
There are different standard congruent subgroups. The deepest one that people use
usually after some conjugation consists of matrices ABCD, where C is divisible by some
integer capital N, that's the level, as I called it, and A and D are congruent to 1 modulo
that same N. If you do that you get a slightly smaller group than the one that I've written
down. It's called gamma one of N. And then in addition to the types of considerations
that we have, there is a Dirichlet character, modulo N, that plays a role. And here I'm just
taking the character to be the identity for simplicity. And we get a bigger group called
gamma zero of N. It's one of the standard subgroups of SL(2, Z).
And not only will I take N to be a positive integer, I'm really chickening out here and
letting N be square free, so that means it's just a product of some number of primes; for
example, it could be 1, it could be a prime. And in my examples, the worst that it will
actually be is a product of two different primes. So it's not a very complicated situation.
>>: [inaudible].
>> Kenneth Ribet: Yeah. Did I write something wrong? I'm sorry. That's -- have my
laser pointer here. Thank you for the misprint. That's C is zero, not N. Okay. Thank
you, Pierre. Good. Okay.
Let's see what I have on this slide. In addition to being modular forms, which I said
implies satisfying some growth conditions, we actually want these forms to go to zero
along the boundary. And that means, for example, that when you go back -- I don't want
to make you too dizzy -- and look at the Fourier series in particular this condition will
mean that the series begin with a constant multiple of Q or some higher power of Q, and
the constant term is actually zero.
And in favorable situations, there is a theory of Eisenstein series that shows that any
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modular form can be written as a sum of some fairly standard series that you write down
a priori and a cusp form. The cusp forms are much wilder. You don't know them kind of
a priori. And so by looking at cusp forms, in a sense I am looking at the interesting part,
F to the D composition.
Another way to say this in weight two is that when you have a weight-two modular form
and you multiply it by a formal differential, DZ, it's not the same D that appears in the
matrix, that formal differential is invariant by the fractional linear transformations. So it's
really a differential on the quotient on the upper-half plane by gamma zero of N requiring
that the form be a cusp form is the requirement that the differential extends to the
boundary and doesn't blow up so it becomes a differential on the natural compact
identified quotient, which is usually called X zero of N. It's a standard modular curve.
This modular curve is I'm going to be say in a little bit begins life in this description as a
Riemann surface. It's just an object over the complex numbers. But there is a theory that
is very often associated with Shimura of canonical models, which shows that the modular
curve actually has a standard description as an algebraic curve over the rational number
field.
And not only that, but there is a well-defined way that you can take this curve and you
can start looking at it over finite fields, reduce it modulo of prime, P, for prime's P, not
dividing N at least. That theory began with Aguzza [phonetic] in the 1950s.
And perhaps the final thing that I'm going to ask about these -- well, not really final, but I
want to constrain things to really get at the arithmetic by requiring some recursion
conditions among the coefficients of the Fourier series, and these conditions are really
simply encapsulated by a single formula that are these standard Hecke operators in
certain cases were initially defined by Mordell that operate on spaces of modular forms.
And we're going to ask that these modular forms by eigenvectors for all of these Hecke
operators.
So that means that if you hit F with teese [phonetic] of N and hitting it with teese of N is
traditionally written with a right action, you get a constant multiple of the form. And if
you read, for example, Shimura's book on automorphic representations -- I think that's
what it's called -- you find that things can be normalized so that the modular form is
really wearing its eigenvectors -- eigenvalues, rather, on its sleeve. What will happen is
that the first coefficient, the coefficient of Q in the Fourier series is necessarily going to
be nonzero in -- if you have this formula, otherwise all of the coefficients will be zero.
You want a nonzero eigenvector. And once the first coefficient is nonzero, you can scale
the form by it, just divide by that nonzero number, make the first coefficient be 1.
And then what happens is that the Nth coefficient becomes equal to the Nth eigenvalue.
And so you have this very beautiful situation where hitting the modular form by the Nth
Hecke operator just means that the Nth coefficient comes out as a factor.
And this implies the Hecke operators are set up so that they're multiplicative in the sense
of functions in number theory. That means that the coefficient ace [phonetic] of NM, if
N and M are two integers that are relatively prime, is just the product of a respective
coefficients for N and for M. And not only that, being an eigenvector for the teese of P
means that you can express by some simple formulas A sub-P to the K, for P being a
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prime and K some higher power. That's not the weight. I'm sorry. That's just a random
power of P. You can express all the P power index coefficients simply in terms of the
Pth coefficient.
So, in other words, when you have a situation where the form satisfies this condition of
being an eigenvector for the Hecke operators, all of the information about its Fourier
coefficients is already carried by the prime-indexed coefficients, ace of P.
And therefore what you're interested in basically is the packet of eigenvalues associated
to this form. And what we're interested in is packets that are occurring for the first time
as you look at the different possible levels. So you can imagine that there's a packet that
occurs for level 11 that will occur for level 22 and level 33 and for all multiples of 11 if
you get some packet that occurs at level 77, that hadn't occurred at level 7 or level 11,
then you're very interested in it and you say that you have a new form; otherwise, you
kind of ignore it except when you're really at the first level where it occurs for the first
time.
There's a little technical point to mention which is that as you go from level 11 to level
77, say, what happens is that you perturb the eigenvalue for the prime 7, in my example.
And so what we want to do is really look at these packets for all but finitely many primes
and require that the packet is new in that sense, that they're -- that something is old if it's
occurred before even where you take out a couple of exceptions.
And that's the definition. This comes from an article in the late 1960s I think by Atkin
and Lehner. And they call forms that are normalized in this way new forms. And new
forms are really the object of study in the subject. Nowadays, if you do Langlands'
theory, you say that the new forms are really the same as the automorphic repetitions of
the adelic group, GL(2), with some condition involving the component infinity. That
condition is exactly what determines the weight. And here we're taking a very classical
perspective. We're really looking at these things as functions on the upper-half plane or
as Fourier series.
And now I said already that the case that mostly interests me is the case K equal 2. On
the other hand, there's another case where the level is 1, but you let the weight grow.
And that's a case that many of you have heard of before. If you have the full modular
group SL(2, Z), you start looking for cusp forms of different weights for weight two and
weight four and weight six. I said the weights were even. There's nothing doing, there
are no nonzero forms. And the first time you get an interesting new form is case K
equals 12.
And what happens when you have K equal 12 is you get the beautiful Ramanujan
Tau-function. And this really was studied by Ramanujan in 1916 or so where he
discovered empirically a number of amazing facts about the coefficients of a certain
formal power series that I've written out there. It's the product of 1 minus Q to the N
raised to the 24th power. You shift it by multiplying the whole thing by Q. The
coefficients of that power series satisfies, as Ramanujan saw, these amazing
multiplicativity properties. For example, tau of a product of relatively prime integers is
the product of a tau values. And tau of a prime power can be traced back to tau of the
relevant prime.
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Ramanujan saw that this was true and I think those facts were proof for the first time by
Mordell in this article where Mordell prediscovered the Hecke operators that Hecke
discovered later in Humborg [phonetic], but there are also kind of amazing congruences.
I guess they occur in the next slide. Here are the first few values of the tau function.
They grow quite rapidly, as you can see experimentally, and they've been tabulated -- it's
a great project to check your computer algebra program to see how fast and how well you
can tabulate these numbers.
There's an open conjecture that none of these values is ever zero, for example. That's
something that people have tried unsuccessfully to attack. And as you'll see in a moment,
there are a group of congruences that are known for the tau values that were noticed and
in many cases proved by Ramanujan himself.
A word that describes this form and the one that's at the bottom is eta products. This is
related to the Dedekind eta function. Here is a very closely related example which has
weight two. But if you want the weight to be two, since for SL(2, Z) there are no nonzero
things for weight two, you want the capital N to be something nontrivial. And if you take
N to be 11, that's going to be the first example of a nonzero cusp form on gamma zero of
N and weight two.
And I've written down the cusp form here. It's a related project, a product. And this was
studied by Eichler and Shimura and Serre and lots of other people.
If you take this product and you write it out, I just called these things ace of N. They
don't have a special name like tau of N. And then as far as the congruences go, a very
famous congruence that was proved by Ramanujan is modulo to the prime number of
691, tau of P is congruent to 1 plus P to the 11th. And an analogous congruence for the
other modular form, the one of weight two and level 11, is simply that ace of piece
congruent to 1 plus P to the 1st power, modulo 5. If you want to compare them, you'll
probably notice that 11 is one less than the weight and this P could probably be written P
to the first power, 1 is again one less than the weight, so they really are very parallel.
And I ask the question what is the meaning of those congruences. And I'll give an
answer only for the second of the two, the one that has to do with weight two, and there
the answer involves elliptic curves, and the answer has been known for a very long time.
And this is really roughly the 40th anniversary of a seminar that Jean-Pierre Serre gave
on the Ramanujan tau-function in a Paris number theory seminar where he asked himself
what is the meaning of the first congruence. And he posited the existence of some analog
of elliptic curves for forms of higher weight.
And that was an incredibly productive remark. It gave rise to the whole theory of
motives attached to modular forms, which are analogs of the elliptic curves that are
attached to certain modular forms of weight two. For more complicated examples, you
need abelian varieties, if the coefficients are algebraic integers but not rational integers.
And basically there's an analog for 691 of what I'm about to say for the ace of P. But
historically it took a while for people to really understand the importance of having
motives attached to modular forms of higher weight. And that was really tremendous
advance in our understanding of the arithmetic of these Fourier series in the general case.
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Well, now, as kind of an aside, I mostly want to focus in this talk about weight two. And
there there's an elliptic curve. So if you have a new form of weight two, the fact of the
matter is that it gives rise to something in relatively simple algebraic geometry. If the ace
of N attached to a new form of weight two for gamma zero of N generate over the
rational field some algebraic extension of Q of dimension D, D might be 17, let's say,
then you get a 17 dimensional abelian variety with a big field of endomorphisms.
Actually, what acts on it is the ring of integers of the field generated by the coefficients.
When the coefficients are ordinary integers, the abelian variety is an elliptic curve. The
elliptic curve in this case is the Jacobian of the modular curve that I wrote down before,
X zero of 11. That modular curve is of genus 1. The Jacobian is there for essentially the
modular curve. And you can write down the simple equations for the elliptic curve. I
should have done that, and I'll say, you know, it's kind of Y squared plus Y equals X
cubed minus X squared minus two further coefficients. Very easy to find them on online
tables, for example, those of William Stein and John Cremona.
This is an elliptic curve that we know very, very well. And the point about the
relationship between the cusp form and the object and algebraic geometry, even in the
general case, is that the coefficients can be seen by taking traces of Frobenius. So in the
elliptic curve case, the simple thing to say is you have a Weierstrass equation, you have a
cubic equation defining the elliptic curve. Take a prime number P different from 11.
Remember I said before that Aguzza proved that X zero of N was kind of okay. Modulo
prime's not dividing N. That's I think from around 1957.
And when you reduce an elliptic curve, mod P, there's some Frobenius endomorphism
that's widely used in cryptography. Lots of you know about it. But its main raison d'être
in the whole subject is that it counts points. You want to know how many points your
elliptic curve has mod P. If you know the coefficients of the modular form, the answer is
very easy: it's 1 plus P modulo, the ace of P.
And so that's one connection between the elliptic curve and the modular form. Another
connection which is strongly related is that if you take a random prime number, let's call
it L, and you look at the points of order L on the elliptic curve and then what you'll get is
a two-dimensional vector space over the field with L elements. L is a prime number.
And on that two dimensional vector space you have an action of the Frobenius coming
from the field with P elements. Take the trace of the action on a two-dimensional vector
space. You get a number, mod L, and that number is ace of P, modulo L. And if you
have some knowledge about ace of P modulo 5, L is going to be equal to 5, then you say
to yourself, oh, what's going on. Well, what's going on is that there's something special
happening with the group of L division points, 5 division points for this elliptic curve.
So what I can do is I can take the elliptic curve. You know this is a complex torus. If
you take the kernel of multiplication by some integer, call that integer 5, do this over the
complex numbers, what you'll get is the integers mod 5, a cyclic group of order of 5 taken
twice.
You can say that that's because you're elliptic curve is the complex plane modulo of
lattice. The lattice is a free abelian group of rank 2. And you ask what it means for 5
times a point in that quotient to be zero, to be the identity element of the group, well, it
means that the point comes from 1 over 5 times the lattice, modulo the lattice. And that's
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something of rank 2 over the integers mod 5.
So you have a module of five division points and because you know how to write
multiplication or addition on the elliptic curve in terms of polynomials with integer
coefficients or rational coefficients, you see that the solutions to the five division
equation have coordinates that are really algebraic numbers. And these coordinates get
permuted around by a Galois action, action of the group of all -- Galois group of all
algebraic numbers. And you get a representation of that Galois group on this group of
five division points.
And the point is that what you really get is for every element of the Galois group, you get
an automorphism of a vector space. You get a kind of two -- invertible two-by-two
matrix over the field with five elements.
And what you're saying when you look at this congruence and you look at the equation
above, is that you're getting understanding about the trace of that action, the trace of that
representation, because you know what happens when you take Frobenius elements in the
Galois group, and there's a theorem of Chebotaryov
of these Frobenius elements kind of determine everything.
So what you could discern even kind of abstractly just from this congruence is the fact
that this two-dimensional vector space over the field of five -- field with five elements
has something very special going for it, and you could even pin down what that is.
But from another point of view what you can do is just kind of examine that thing as an
object for which you're amassing information and you can see that it contains a very nice
subgroup on which the Galois action is trivial. This is called this cuspidal group. And it
also contains another subgroup on which the Galois action is cyclotomic. That's called
the Shimura subgroup, really discovered by Shimura.
What is mu [phonetic] 5? Mu 5 is the group of roots of unity of order 5, consists of 1 in
the -- 4 nontrivial 5th roots of unity. The action of the Galois group of Q on that cyclic
group defines a little character from the Galois group of Q over to the automorphisms of
this mu 5. Automorphisms of mu 5 is just the invertible integer is mod 5. It's a cyclic
group of order 4.
And there's a wonderful paper which I recommend to all comers. It's from 1977. It's
Barry Mazur's paper called Modular Curves and the Eisenstein Ideal. And he kind of
explains in general for J zero of N where N is a prime that there are special prime
numbers, if N is 11 the special prime number is 5. And in general it's the divisors of N
minus 1 except sometimes you have to ignore 2 and 3. And for those divisors what's
happening is that you have a kernel analogous to the E of 5 and you have a cyclic group
of order L. L here is 5. And also another cyclic group of order L, one of them with
trivial action and one of them with cyclotomic action. And this is kind of the very first
instance of this whole thing.
Shimura, whom I've mentioned quite a bit here, has a beautiful article in Crella's
[phonetic] journal from around also 1967, I think, where he talks about the action of the
Galois group of Q on the division points of this very special elliptic curve. And he says
to himself, well, this is really amazing. You know, class field theory is a study where
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you try to explain what are the abelian extensions of number fields or local fields, and
you do that in terms of the internal structure of the number fields or local fields.
And there was always this question starting with Artin in the 1950s, you know, what
happens if you try to look at nonabelian extensions of arithmetic fields. How can you
describe them. And, you know, Artin is famous for saying make a conjecture and I'll
prove it.
And nowadays I think that the best answer to that conundrum is to say that we can't really
describe very well the absolute Galois group of all extensions of Q or a number field or a
local field, but the best we can do is try to explain what the representations of that group
on various objects explain what those representations are related to. And Shimura said,
well, you know, here's an example where I'm producing in general nonabelian extensions
of Q and they're related to this specific modular curve, X zero of 11. This requires
further study and we should try to understand what we can about these extensions and try
to understand the arithmetic.
And that was a kind of challenge thrown out in 1967, and there were two very prominent
echoes of that. One was a beautiful article that Serre published in 1972 in French called
Properties of Torsion Points of Elliptic Curves, where he said that if you take a random
elliptic curve over the rational number field, one without complex multiplication, the
action of the Galois group of Q on the division points is incredibly rich. The Galois
representations that you get, aside from some finite fooling around, will have images that
are as large as you can contemplate, they'll kind of fill up all of the GL(2)s that are
receiving them.
And then there's this 1977 article five years later by Mazur where he talks specifically
about the arithmetic of the abelian varieties, not only elliptic curves, coming from
Jacobians of modular curves.
So there's something very deep and varied that kind of explains this very simple
congruence; namely, something geometric -- I'm talking so much that my laptop is falling
asleep; I hope you're not -- something geometric is going on that explains these simple
congruences.
So the way I want to write this in kind of capsule form is that you started with a modular
form. The modular form was the one of level 11 and weight 12. That's F. And if you
have the number 5, then there's an associated Galois representation. So the Galois
representation as a module is just this E of 5. And if I think about the homomorphism
that describes the action of the Galois group of Q on this module, well, I call that row F5.
And all of this equation is simply to say that row F5 is a direct sum of two
representations: the trivial representation, that's the Z mod 5; and the cyclotomic
representation, I call it cy [phonetic]. Some people might call it epsilon or omega . Cy
sub-5, the mod 5 cyclotomic character.
And actually, more generally, what might happen is that this row F5 five instead of being
a direct sum like that might be an extension; namely, it might be upper triangular
[inaudible], not irreducible. And in one of the corners you had the trivial representation
and the other you had the cyclotomic representation.
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For me, for this purpose, I'd say that's just as good. Kind of look at situations like that
and say that the mod 5 representation attached to the modular form is this direct sum. So
we just kind of semisimplify. If you have something upper triangular, just look at the
diagonal.
Okay. So there are the words that repeat what I just said. This is a semisimple
representation of the direct sum of the trivial one and the cyclotomic character, it comes
from this form of level 11. And the question is why 11? You know, why is it important
that the number be 11?
You might say what is the kind of true level of this representation, the direct sum of the
mod 5 cyclotomic representation and the trivial representation?
And one answer -- well, I'll just stick by my slide so I don't do too much digression
anymore -- is that one of the things that Mazur shows is that if you start with a level
capital N, then aside from 2 and 3, which I said were special, the primes that play the role
of 5 are the divisors of N minus 1. So the point about 11 is just simply that it's congruent
to 1 mod 5.
So if you have another such prime, for example, 101, then you'll find by taking one of the
modular forms, one of the new forms of that level and one of the primes dividing five in
the ring of integers of the field containing the coefficients, you will find the congruence
like the one that we had a couple of slides ago, ace of P congruent to 1 plus P mod 5.
So to say it in a different way, this trivial-looking direct sum, the trivial character and the
cyclotomic character, will occur as -- from a new form of weight two and level N
anytime N is congruent to 1 mod 5.
Okay. And this is just some slightly technical point, which I already said; namely, if you
had a form whose coefficients were, let's say, you know, in the ring of integers of Q to
the square root of 101, then what would happen perhaps, and in the case it does, is that 5
would split into different prime ideals. And what you do is you take the different
reductions of your modular form, one of them might satisfy the congruence, another one
doesn't.
A perfectly good example is instead of 5 you could take 11 and then you take L to be 23.
In that case there is a unique new form up to Galois conjugation of weight two and level
N, and the coefficients are not rational integers, they're integers in I believe the ring of
integers of Q to the square root of 5. And in that ring the prime 11 splits as a product of
two different maximal ideals. These maximal ideals live their own independent lives.
Modulo 1 of them, the ace of P, will be congruent to 1 plus P. Modulo the other, there's
no such congruence whatsoever.
So when I say that there's a new form that is doing this thing, mod L, that just is
shorthand for saying that it's doing it modulo 1 of the primes dividing L, but probably not
more than 1.
And here's the thing that I've now said twice, which is that the key congruence in this
example is N congruent to 1 mod L.
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And earlier in discussing this with you, I mentioned Eisenstein series. If you have an
arbitrary weight-two modular form on gamma zero of N, you can write it as a multiple of
this very well-mannered series, which I'll discuss in a moment, the sum of a multiple of
that and a cusp form, the kind of thing that goes to zero at infinity.
And this Eisenstein series, it has a 1 minus N in it floating around, and then aside from
that the major term in this N minus 1 over 24 is the second Bernoulli number, which is
minus the 6, I think. And then there's also a 4 in the denominator that's twice the weight.
There's all this numerology of the thing.
And the coefficients of the Fourier series, the Nth coefficient is the sum of the positive
divisors of N, except you throw out divisors that are divisible by the prime capital N.
So, for example, if you take N to be a random prime number P, the coefficient is 1 plus P.
And if you take it to be N, the coefficient is 1.
And the point what's really behind -- kind of drives Mazur's paper -- remember, Mazur's
paper has the word "Eisenstein" as part of the title -- is that if you take this cusp form,
this Eisenstein series modulo L, L is a prime different from 2 and 3, if L happens to
divide N minus 1, then when you look at this Fourier series, it kind of starts with zero.
So it looks as though it is a cusp form. And what happens is that in that case you can
really produce a cusp form that's congruent to it. That's how you get the kind of
congruence that we've been discussing.
And the converse is that if you have a cusp form which is congruent to the nonconstant
part of the Fourier series attached to the Eisenstein series, then there's a theory, a
so-called Q expansion principle that forces the constant term to be zero module L as well.
And that's how you get that L divides N minus 1.
So you really have this very tight theory. You know exactly when it's true that the
representation 1 plus cy 5 comes from a modular form of level N; namely, N has to be
congruent to 1 mod 5. And in that description 5 can be replaced by any prime number
greater or equal to 5 in fact.
And although I'm sure -- I haven't looked at Ramanujan's paper, at least lately, one way in
which you can interpret what happens modulo 691 is some similar circumstance; namely,
you can start writing down Eisenstein series. In the case of level 1 and weight 12, the
Eisenstein series begins with a constant -- trivial constant multiple of the 12th Bernoulli
number, which has a 691 in the numerator. And it's exactly that numerator of the
Bernoulli number that drives the congruence that was discovered by Ramanujan.
And there are articles by Serre and Swinnerton-Dyer, again, from around 1972, where
they kind of used the Galois representations attached to modular forms. Remember, I
mentioned something about motives before. Well, even though motives took a while to
get going, almost immediately after Serre suggested the Galois representations for these
higher-weight modular forms might exist, Deligne constructed them.
And using Deligne's Galois representations, Serre and Swinnerton-Dyer really kind of
ordered in a sense of bringing order to. You know, they kind of -- they cleaned up the
whole subject. They explained why Ramanujan and his successors could prove the
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congruences that they did. And they also show that aside from one case where there was
a new congruence that hadn't been found by the classical people, there were no more
congruences to be found, had a way of showing that the Galois representations in their
case had large images. So, for example, they couldn't be upper triangular. They did for
modular forms the same kind of thing that Serre had been doing for elliptic curves, if that
makes sense.
And now there's a little bit of larger context, which I guess I won't go -- well, I should say
a little bit about it. So I already have in some sense; namely, in general, if you have a
new form, which for this discussion we can think of as just having weight two, all the
coefficients rely on some interger ring. For example, for that 23 example it would be the
ring of integers of Q to the square root of 5. And you can start reducing the Fourier
coefficients modulo different maximal ideals.
So these maximal ideals, they can't be called L, L is a prime number, so we call them
lambda. And then associated by Deligne, or in this case, for weight two it's especially
easy, one could say it's really kind of Shimura, in fact, for every lambda there's a
two-dimensional Galois representation. It's just what happens is you take an abelian
variety attached to the modular form, it has an action of this interger ring O. That's
explained in chapter 7 of Shimura's book that was published in the early 1970s. And you
take the lambda division points of this abelian variety, instead of taking the five division
points of the initial elliptic curve, Galois acts on it and you get these automorphisms of
the vector space, which are really two-by-two invertible matrices.
And if you look at the literature in the subject, some of which is by me, the main point is
that when you take these representations, you fix F and you start looking at the different
lambda, the main point is that for most of them the Galois action is rich. As I just said, in
1972 Serre and Swinnerton-Dyer were showing kind of representations, filled up the
group of matrices, and that would impede congruences.
What we show is that this happens for all by finitely many cases. So there just aren't that
many congruences to be found. But, on the other hand, it does occur that you can have a
congruence every now and then. And so maybe we should be spending more of our time
thinking about the reducible representations that come from these modular forms.
And as I've already said, if you have a reducible representation that means an upper
triangular matrix, just look at the diagonal. So you semisimplify. And in the simple case
that I've described -- namely, square free level -- it's quite easy to show that the two
characters that occur on the diagonal of the matrix up to permutation will be the trivial
character and this mod L cyclotomic character. L is the prime that lambda divides. And
since we're semisimplifying up to permutation doesn't matter.
So it's just the -- reducible case means that you're getting this 1 plus cy cybele [phonetic].
And now what's the question? The question, if we back up a little bit, is to take any
two-dimensional Galois representation and ask ourselves does it come from a modular
form. And, if so, what kind of modular form's given. So the simplest case would be take
the two-dimensional representation here and you say does it come from cusp forms.
Well, the answer is it does. And we know, for example, that if you take a weight-two
cusp form, you look at the weight-two cusp forms of level N where N is congruent to 1
mod L, you'll get these representations. But you can ask what else? You know, what if
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the level is not prime, what if it's a product of several primes and so on. What's the
condition that guarantees that you get this from some new form.
And if we go back now and ask the question in generality where the representation might
be irreducible as well as reducible, then we're really in the landscape of other conjectures
that Serre made around 1987. Serre published an article in the Duke journal in which he
made what are called "Serre's Conjectures," but they kind of started with correspondence
with Tate as early as 1972 that kind of took off from this -- these articles by Serre and
Swinnerton-Dyer that I mentioned; namely, they try to ask is there a converse to
Deligne's construction. That was kind of the question. Given a modular form, Deligne
makes a Galois representation, can you go backwards. You know, if you start with a
Galois representation, does it come from a modular form.
And the irreducible case seemed to be the essential one. It's a very hard question. And I
have been going around in the last couple of years explaining a method by Khare and
Wintenberger in which they basically have resolved the conjectures; namely, they use a
lot of technical algebraic geometry, much of which is due to Mark Kisin from the
University of Chicago. And they kind of throw all kinds of different things into the pot,
everything in the kitchen. And they really come to the conclusion that if you have an
irreducible two-dimensional Galois representation that satisfies some very tiny necessary
conditions that come from modular forms, that would be satisfied if the determinant is an
odd power of cy L. Odd is essential. They show that the thing actually comes from
modular forms, some modular forms.
And then there's a lot of literature. A buzz phrase for that is level lowering, that shows
that if the thing comes from any modular form whatsoever, you kind of understand the
different weights and the different levels of different modular forms that give that same
representation.
And what I'm asking for is really some recipes to whether or not you can do that sort of
thing in the reducible case. In the reducible case, the questions are a little bit different
because you know going in that the thing does come from some modular form, it
comes -- you can kind of make these Eisenstein series and then make cusp forms if you
like that are congruent to them. But then you ask, you know, can you describe all levels
and all weights that give that given representation. And all of a sudden the question
becomes interesting.
If somebody just gave you this 1 plus cy L, you might say that the natural level is 1. I
kind of mentioned that here and might have said that before. Because there's kind of -there's no prime that naturally comes into the picture. If you have 5 being L, why 11?
Why 101? Well, okay, there's a congruence. But those primes don't seem to be
essentially connected to the initial representation.
But we do have Mazur's answer. If you look at prime level, if things are congruent to 1
mod N, mod L, and we know there are lots of primes congruent to 1 mod L, but as I was
trying to say just a moment ago, these different primes have nothing to do with each
other. 11 and 101, they don't all come down to some root that would give rise to them.
In the irreducible case, there's kind of like the greatest common divisor. There's kind
of -- is that what they're called? No. Least common multiple, or something, you know,
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take the least common multiple -- there's a smallest one, and then the others are all
multiples of it. For the levels in the irreducible case, it's exactly like that. There's a
smallest level. It's the minimal level, and all the levels are multiples of it. But in the
reducible case, that doesn't happen.
And the question is what does? Now, you might ask why do I care? And the answer is
that there are lots of mathematicians in Barcelona who are interested in abelian varieties
and modular forms and cryptography and the relations among all these things. And one
of them is someone named Luis Dieulefait, who's been at Harvard this past spring. I
don't know if he's back in Barcelona. And he asked me this very question I'm discussing
here. And the question is why did he do that?
And the answer is that every time you have a number field -- every time you have a new
form, there's an associated number field, field generated by its coefficients, and you
might ask whether these number fields can be arbitrarily large in their degree. And this
question -- once again, there's some article by
Serre in [speaking french] where he's
applying analytic number theory to modular forms. And he has some very weak result
that the degrees grow in certain situations.
And there's -- in Mazur's 1977 article, he has a proposition based on the remark that I'm
about to make that the degrees grow if you look at odd prime level. But Dieulefait is
interested in levels that are product of two primes or several primes. And he actually sent
me this argument where for three or more primes using some variant of Goldbach's
conjecture, the result proved by Chen in the 1970s, you could actually treat the problem.
And he said, well, we could do it for a prime level and for products of three or more
primes, how about products of two primes. So that was kind of interesting to me. I
started looking at the product of two primes.
And the point is -- I kind of don't want to speak for much longer than an hour, so I'll
really slough over this, but if you have a congruence, AP congruent to 1 plus P mod
lambda, and it applies for P equal 2, which will be the case if N is odd, then that
congruence for very idiotic reasons forces the degree of the field to be large. And the
reason is that A2 has to be smaller than 3.
For example, you could say, well, if 3 is congruent to 3 mod 5, so -- and Q is not a large
field, but there's something called the Riemann hypothesis for algebraic curves proved by
Weil in the late 1940s that implies that an A2 is some algebraic integer and all of its
absolute values, kind of stick it into the real numbers with all of its conjugates, all of
those absolute values are at most 2.83. So they can't be 3. And if you use the triangle
equality in some idiotic way, you'll see that the degree of the field has to be bounded
from below by essentially the logarithm of the number L. So if you can make
congruences modulo larger and larger L, then you can make the degree of the field go to
infinity.
So that's kind of Mazur's observation, and also Dieulefait's in the case when N is prime.
The question is can you make arguments like that for more complex-looking levels, not
just simply for prime level.
And so, for example, N could be a product PQ, just -- well, here. Product of two distinct
primes, suppose L is an arbitrary prime, you make L as large as you like to force the
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degree up. Can you make the mod L representation 1 plus cy L come from some new
form of level N. N can vary. But just a product of two primes. A new form of
weight-two in level N.
And the answer is you can do that, and I actually can tell you very quickly what the
answer is. But when I do that and you have level that's a product of two primes, there are
some signs that you can keep track of; namely, if you hit your new form with a Hecke
operator TP, you'll get plus or minus the new form, and if you hit it with TQ, you'll also
get plus or minus the new form, so there are two signs: a sign at P and a sign at Q.
And there are really two essential cases: one is where the sign is plus-minus and the
other is where it's plus-plus. And the reason for that is that the minus-minus case never
occurs for these congruences. You can rule that out very quickly. And the minus-plus by
permuting P and Q can be reduced to the case of plus-minus. So, you know, there are
only two essential cases.
And in the two cases, the answers are really different. The first case has a kind of very
boring answer, and the second case has a much more interesting answer that you can
prove theoretically and also verify empirically by looking at different tables. So it's kind
of very interesting.
In the case of the first situation, you want the eigenvalue for TQ to be minus 1 and the
eigenvalue for TP to be 1. All you need is for the second prime to be congruent to 1 mod
L. So you want to make a mod L congruence for a product P and Q and some new form
of level PQ and weight two. All you have to do is choose a prime. In Mazur's thing you
choose primes that are 1 mod 5. Here you choose a prime that's minus 1 mod 5. And
then you multiply it by any other prime you like. And you will end up with a
congruence. And you can see that empirically.
The necessity of this condition is very easy to see by comparing Galois representations,
and the sufficiency really requires some argument. You have to look at the way that
some variance of modular curves, called Shimura curves, behave when you reduce
modulo, the prime Q.
One thing I can say on the closing moments of this talk, I mentioned Aguzza a couple of
times by saying that you have these modular curves, they have level N, you reduce
modulo of prime, not dividing N, you can kind of understand what happens. It took a
very long time for mathematics to understand what should happen at primes that do
divide N; namely, there was a beautiful article again from 1972 by Deligne and
Rappaport where they talk about, for example, the reductions of X zero of N where N is
square free at primes dividing N.
And there is a -- that's already kind of the length of a small book. And a follow-up article
is the length of a very thick book by Katz and Mazur about the arithmetic of modular
curves. And when you look at these Jacobians, there's kind of an interesting story of
Shimura curves. Mauneed [phonetic] had two very good students, one of them
Cherednik, who I think is now at the University of North Carolina in Chapel Hill, and
Drinfeld, who's at the University of Chicago. And they worked on their thesis. And
Drinfeld -- Cherednik's, rather, was on the reductions of Shimura curves. And Mauneed
said, well, you guys are both very good students, I'd like you to give seminars in which
15
you explain each other's work. And Shimura and -- sorry, Drinfeld took Cherednik's
work and re-explained it in this way that kind of began a lot of the modern fine theory of
the reductions of Shimura curves. And using Drinfeld's work, which was initially
Cherednik's, you get a proof that Q congruent to minus 1 mod L is sufficient to make this
kind of congruence.
The other case is much more interesting, and I'll just tell you the answer. It's when the
two signs are plus 1. In that case by comparing with Eisenstein series, just as in Mazur's
case, you know that L divides N minus 1, here you know that L divides P minus 1 times
Q minus 1, which means it divides either P minus 1 or Q minus 1 or both. So let's
assume again permuting if necessary that it divides P minus 1.
And then there's a way you can think about the problem that makes it somewhat clear that
the answer I'm going to give might be right; namely, already at level P there's a new
form. That's by this Mazur result that I keep quoting. And what you're trying to do is
kind of promote level P into level PQ.
And just as I mentioned level lowering before, this goes by the term level raising. The
question is can you raise levels. And that was really the way that Dieulefait came to me
with this question. He said, You're an expert on level raising. Can you raise levels also
in the reducible case? And I knew that one could but had to work out the details. And
the answer is the following. When can you raise the level? Well, there are kind of two
cases. These are not in exclusive or if Q divides also Q minus 1 as well as P minus 1,
you can raise the level. You'll get a new form of level PQ.
And in the other case, L divides P minus 1 but not Q minus 1. And what you do is you
look at Q modulo P and ask that it be an Lth power. That's the condition that there be
level raising. And the simplest case that I can think of is when L is 5 and P is 11, and
then Z mod, LZ star is a cyclic group of order 10, so the 5th powers are just plus 1 and
minus 1, the two elements of order 2. And so to say there's a new form of level PQ, 11
times Q in this case, in the case when the second coefficient, when Q is not congruent to
1 mod 5, or even if it is, is to say that Q has to be congruent to plus 1 or minus 1 modulo
11.
So you get a kind of very clean result. And once again what's interesting is you can go to
William Stein's online tables and see empirically that you really get the level raising that
the theory predicts.
Okay. That's my talk.
[applause]
>>: Are there cases in which [inaudible] the case in which [inaudible]?
>> Kenneth Ribet: Well, so this is kind of an interesting question. The question is what
happens when it's not a direct sum. So, you know, you think about different parts of my
brain, I have this theory where I make unramified extensions of cyclotomic fields by
looking at cases where it's not a direct sum.
And basically the theory says that when you have these reducible cases, if it is in a direct
16
sum, you can kind of divide by finite subgroups and make isogonies until you get some
reducible representation, some indecomposable representation, and you work with that.
And the thing that's not very well explored, I believe, is whether or not it ever occurs that
you get a direct sum. So my theory in that direction was, well, you might get a direct
sum and you might not. If you do, that's not interesting for making these extensions, so
let's put ourselves in a situation where it becomes indecomposable.
In Mazur's case, Mazur has this great philosophy in his paper that you just take the
objects as they are, you don't make isogonies, just look at them as they are in nature.
When you look at nature, you really get a direct sum. And you can ask geometrically, in
my case, do you get a direct sum at the higher level. So, in other words, what I'm saying,
or what you're saying, is the following. Let's take a prime -- well, take an example where
P and Q are both congruence 1 mod 5. Okay. So, in other words, you would take kind of
J zero 11 times 101, for example. Okay. And then what you'd want to do is you'd want
to take the kernel of a maximal ideal where you're really looking at the new form, mod 5.
Lambda is going to be dividing 5. And this thing -- you make it so it says it's predicted in
my theory. And you know that after you semisimplify the thing it's going to be a direct
sum of two groups.
And now the question is what happen if you don't semisimplify, will that be a direct sum
of two groups, and I suspect typically not. But I actually thought about that question for
the first time this morning when I was on the plane. And I think it would be interesting to
prove that it's not a direct sum. Or to see what it is.
>>: [inaudible]?
>> Kenneth Ribet: Well, so if L is arbitrary, you start with L. And then the conditions
on P and Q are basically congruence conditions. So it's not -- they're not mysterious. It's
just kind of analytic number theory of Dirichlet.
>>: What was the motivation for producing these large [inaudible]?
>> Kenneth Ribet: Ah. That's an interesting question. All right. Let's take this off. I'm
not exactly sure how he first came to this question, but I think it came from the analytic
number theory of cusp forms. He looked at Serre's paper and was trying to better some
of the results. And he's also interested -- he has a colleague or -- the three of us may end
up writing a joint paper. His coauthor is named Jimenez in Barcelona. And Jimenez is
very strong in analytic number theory and he's also interested in variance of Chen's
results and kind of proving better things in the direction of Goldbach. And it all fits
within his interests.
>> Kristin Lauter: In what sense was it thought that the reducible case was
uninteresting?
>> Kenneth Ribet: Well, that's -- yeah, why is the reducible case uninteresting? I guess
somehow the reducible case, it kind of harkens back to GL(1) and kind of ordinary class
field theory, and we just say, well, we know everything there is to know about that. And
then you're really interested in essentially two-dimensional things, and those should be
attached to modular forms.
17
You know, I mean, for example, in Serre's original conjecture, people would look, for
example, at this 1 plus cy L and they'd say, well, you know, somehow morally the level
of that thing is 1, there should be some kind of missing Eisenstein series of weight two
gamma zero of 1. So it would have as its nonconstant coefficient just the sum of the
divisors of N without a prime, and then the constant coefficient would be something like
minus 1 over 24. And the problem is that this thing is not a modular form, all of this
business about nonholomorphic analytic continuation of it and Hecke. And so people
would say, well, you know, it's just kind of an embarrassment, but the thing doesn't exist,
and morally it should exist and the natural level of this representation is 1 and we should
kind of forget about it and look at the more interesting cases of the irreducible
representations. But I guess I'm arguing that you shouldn't forget about it.
>> Kristin Lauter: More questions? Thank you.
[applause]