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.)
1.1. 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.
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
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 .
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”),
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 .
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.