Three months ago, Sophie Morel gave her Bourbaki seminar talk “Construction de représentations galoisiennes [d’après Scholze]”.
As it turns out, Sophie Morel was kind towards the non-experts in this subject (like myself): indeed, a large part of the talk (see the video here) was introductory, while the more advanced material was delegated to the lecture notes (available here).
In the remainder of this post, I’ll try to summarize some of the topics discussed by Sophie Morel’s talk (using the video in the link above as the main source).
Disclaimer: Since I’m not an expert on this subject, all mistakes in this post are my responsibility.
Remark 1 If you are in the Oxford area this week, then you will have the opportunity to learn about the main results in Morel’s talk directly from the authors: indeed, Boxer and Scholze (resp.) will give talks tomorrow, resp. Wednesday on their works (see this schedule here of the corresponding 2015 Clay research conference.)
1. Motivations
The basic references for this section are Zagier’s text on modular forms and the lecture notes from Deligne’s Bourbaki seminar talk in 1968–1969.
1.1. Modular forms
Let be the upper-half plane, and consider the action of
on
by homographies:
A modular form of weight and level
is a holomorphic function
such that
- (i)
for all
and
;
- (ii)
is holomorphic as
.
A modular form is called cuspidal if
satisfies (i), (ii) and
- (iii)
vanishes at infinity.
Note that the condition (i) implies that any modular form is
-periodic:
for all
. In particular, any modular form
has a Fourier series
Since as
, we see that the condition (ii) is equivalent to
- (ii)’
if
Also, the condition (iii) ( is cuspidal) is equivalent to
.
1.2. Hecke operators
The modular forms of weight and level
form a finite-dimensional vector space. For each
, the Hecke operator
acts on this vector space.
Remark 2 This formula gets significantly simpler when
is a prime number.
Among the basic properties of Hecke operators, it is worth to mention that and
commute, and
is cuspidal if the modular form
is cuspidal.
These properties suggest the study of the following objects:
Definition 1
is called a proper normalized cuspidal modular form if
is proper (i.e., an eigenvector) for all Hecke operators
;
- the Fourier coefficient
in (1) is normalized:
.
Remark 3 One can show that a proper normalized cuspidal modular form satisfies
for all
is prime. In other terms, the eigenvalues of the Hecke operators can be read through Fourier coefficients of proper normalized cuspidal modular forms.
1.3. Deligne’s proof of Ramanujan-Petersson conjecture
In this context, the Ramanujan-Petersson conjecture states that the Fourier coefficients (1) of a proper normalized cuspidal modular form of weight
satisfy the following bound:
for all prime.
As it is well-known, this conjecture was settled by Eichler-Shimura in the case , and in full generality by Deligne.
The strategy of Deligne was to “reduce” the Ramanujan-Petersson conjecture to Weil conjectures. Very roughly speaking, given a proper normalized cuspidal modular form , we can divide Deligne’s strategy into two steps:
- (1) for each prime
, there exists a continuous irreductible representation
such that
is non-ramified at all
prime;
- if
, the characteristic polynomial of
is
;
appears in the (Betti) cohomology of a (proper, smooth) algebraic variety over
.
- (2) the Riemann hypothesis part of Weil’s conjectures (also established by Deligne) provides useful information on the eigenvalues of
acting on the cohomology of algebraic varieties, and this can be exploited to establish the Ramanujan-Petersson conjecture because
is the sum of eigenvalues of
(cf. the expression for the characteristic polynomial of
).
Before proceeding further, let us give some explanations about (1).
The absolute Galois group is the group
of automorphisms of the algebraic closure
of
.
Given a prime number , we can choose an algebraic closure
of the field
of
–adic numbers and a morphism
such that the diagram associated to the arrows
,
,
and
commute.
In this way, we obtain an embedding where
is the group of continuous automorphisms of
.
Remark 4 The embedding
depends on the choice of
. In particular, this embedding is only well-defined modulo conjugation (but, as it turns out, this is sufficient for our purposes).
The group is part of an exact sequence
where the arrow is defined by reduction modulo
, or, more precisely, by considering a commutative diagram whose arrows are:
,
,
,
,
.
Here, , resp.
is the ring of integers of
, resp.
,
is the finite field of
elements, and
is the algebraic closure of
.
The group is topologically generated by a single element, namely, Frobenius endomorphism
.
The kernel of the arrow
is called inertia group (but its structure is not need for the sake of this post).
Definition 2 We say that a representation
of
is non-ramified at
if and only if
. In particular,
is well-defined modulo conjugation whenever
is non-ramified at
(cf. Remark 4).
Remark 5 The fact that
is well-defined modulo conjugation is not a serious issue for the first step of Deligne’s strategy: indeed, we imposed only a condition on the characteristic polynomial of
(and this polynomial depends only on the conjugacy class of
).
Closing this subsection, let us outline the construction of the representation as in the first step of Deligne’s strategy in the particular case of proper normalized cuspidal modular forms
of weight
.
We begin by introducing the modular surface . The modular form
is not a function on the non-compact Riemann surface
(because it is not
-invariant): in fact, the modularity condition (i) (with
) implies that
is a section of the line bundle
of holomorphic differentials on
.
Note that the one-point compactification of
is an orbifold topologically isomorphic to a sphere. The fact that
is holomorphic at infinity (by condition (ii)) implies that
can be extended to
(i.e.,
).
Next, we consider the Hodge decomposition
of the first (Betti) cohomology group of .
In these terms, the Hecke operators admit the following geometrical interpretation. For each
, we can choose a finite-index subgroup
and an element
such that the action of
on
(or
or
) consists into taking the pullback under the natural arrows
and then taking the “trace of the operator” induced by the arrows
where is the multiplication by
. (See, e.g., the subsection 2.3 of this PhD thesis here for more details.)
In particular, a normalized proper cuspidal modular form
gives rise to a non-trivial simultaneous eigenspace of the operators on
and, a fortiori,
, with eigenvalues
.
This geometrical interpretation of permits to construct
along the following lines.
First, we observe that and
are algebraic varieties over
in a “canonical way” (because
, resp.,
are moduli spaces of elliptic curves, resp. elliptic curves with extra (“level”) structure).
This implies that, for each prime, we have an (algebraically defined) isomorphism
from to the étale cohomology group
of the algebraic variety
over
. Moreover, this isomorphim behaves “equivariantly” with respect to the Hecke operators
.
In particular, the non-trivial simultaneous eigenspace of associated to
can be transferred to
.
Secondly, we note that this isomorphism is interesting because acts on
. Furthermore, since
is defined over
(and
behaves “equivariantly” with respect to this isomorphism), the transferred non-trivial simultaneuous eigenspace of
in
is invariant under this action of
.
In summary, the normalized proper cuspidal modular form induces a representation of
in a transferred non-trivial simultaneous eigenspace of
in
, and, as it turns out, this is (more or less) the representation
that we were looking for.
1.4. Hecke operators from the adelic point of view
Before closing this introductory section, let us make the following observation about the geometrical interpretation of the Hecke operators.
Our geometrical construction of Hecke operators involved pullback and “trace” operators related to the varieties (for appropriate choices of
).
Nevertheless, for the sake of generalizing this discussion, it is helpful to replace by
where is the ring of adeles (with
denoting the “finite adeles”),
(with ),
and
The fact that is closely related to
comes from the equality
where and
.
The key points of this complicated definition of are:
- each operator
acts only on the
-th component of
;
- this definition can be generalized to other groups (see below).
2. The works of Boxer and Scholze
2.1. Hecke algebras
Let us quickly indicate how the discussion of the previous section for can be generalized to
,
.
We start by introducing
for (where all implied groups are defined by their natural counterparts in
).
Similarly to the case , we have that
is a real-analytic variety (given by the disjoint union of two symmetric spaces for
).
In this setting, the Hecke operators are obtained by taking a prime number and an adequate element
, and considering the composition
of the pullback and trace operators induced on
by the natural arrows (inclusion and multiplication by
):
This construction can be slightly generalized by noticing that, if , then we have an action on
of the Hecke algebra
equipped with the product given by the convolution with respect to the Haar measure with volume one.
Remark 6 For
, we studied exclusively the action of certain elements
, but from now on we will consider the whole action of
.
After the works of Satake, we know that is a commutative algebra whose characters
correspond to the conjugations classes of
under a canonical bijection called Satake isomorphism.
For the sake of convenience, we will put “together” the Hecke algebras for
by defining the Hecke algebra unramified off
as:
2.2. Characters of Hecke algebras and Galois representations
The following two conjectures are part of the so-called Langlands program.
\noindentConjecture. Let be a character such that the corresponding eigenspace in
is non-trivial (i.e.,
is a sort of “normalized proper cuspidal modular form”). Then, there exists a (continuous) Galois representation
such that
is not ramified on
(i.e.,
is trivial on the inertia group
);
corresponds to
under Satake’s isomorphism.
Remark 7 The reader certainly noticed the similarity between the statement of this conjecture and the item (1) of Deligne’s strategy of proof of Ramanujan-Petersson’s conjecture. Of course, this is not a coincidence, but this conjecture is somewhat “surprising” in comparison with Deligne’s setting because
is not algebraic when
.
\noindentConjecture’. The same conjecture as above is true when is replaced by the cohomology with torsion
.
The conjecture above was proved to be true by Harris-Lan-Taylor-Thorne. More recently, George Boxer and Peter Scholze proved (independently) the following result
Theorem 3 (Boxer; Scholze) The Conjecture’ is also true.
Remark 8 It is known that (a version of) Conjecture’ (for
) implies Conjecture (by taking the limit
). In particular, the methods of Boxer and Scholze are able to recover the theorem of Harris-Lan-Taylor-Thorne.
2.3. Some words about the results of Boxer and Scholze
We stated Conjecture and Conjecture’ for ,
, but we can also study them for other reductive connected groups
such as
, etc.
An interesting feature of this generalization of Conjecture/Conjecture’ is that sometimes becomes an algebraic (Shimura)variety (e.g., for
, etc.), and this gives us a hope of mimicking the arguments from the first section of this post.
Remark 9 The fact that
is algebraic is not sufficient in general to reproduce the strategy employed for
in the first section of this post.For example, if
is an imaginary quadratic extension of
and
act on
, then the strategy used for
does not produce a “good” representation
attached to a character
.
In fact, one gets a good representation
only after applying Langlands functoriality principle to “transfer” the problem from
to
, and then using the strategy for
in this new setting.
In summary, even if
is algebraic, one can’t apply the strategy for
in a simple-minded way in order to deduce Conjecture/Conjecture’.
On the other hand, the fact that is algebraic for certain choices of
does not seem to help us in the context of the results of Boxer and Scholze because we know that
is not algebraic in their setting.
Nevertheless, Clozel noticed that the non-algebraic varieties usually are strata in the compactification of an algebraic variety. In particular, one can try to exploit this to build the desired representations from characters
appearing in
.
Let us illustrate this idea of Clozel in the first non-trivial case of the generalization of the Conjecture, i.e., ,
imaginary quadratic extension (where
is a non-algebraic variety of real dimension 3).
We take , so that
is algebraic (but not compact). Its Borel-Serre compactification
is not algebraic but its boundary has some components associated to parabolic subgroups
of
such as Levi’s parabolic subgroup
which turns out to be isomorphic to .
In this way, we get two arrows allowing to relate
to
with the advantage that
is algebraic. In particular, this allows to transfer the problem of showing Conjecture/Conjecture’ from non-algebraic settings to algebraic settings (but this is not the end of the history: cf. Remark 9 above!)
At this point, Sophie Morel runned out of time and she decided to conclude her talk by mentionning that after transferring the problem from to
as above, an important ingredient in Boxer and Scholze proof of Conjecture’ is the following theorem:
Theorem 4 (Boxer; Scholze) If
appears in
, then there exists a (cuspidal)
in characteristic zero appearing in
such that
(mod
).
Then, she told that a “one-sentence proof” of this result is the following: one uses a comparison theorem to relate to cohomology groups of affinoid spaces.
There seems to be some text missing in the beginning of paragraph 1.3. [Corrected. Thanks, Matheus]
By: mathlog on December 1, 2015
at 2:31 pm