Posted by: matheuscmss | September 27, 2015

## Third Bourbaki seminar of 2015: Sophie Morel

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 ${\mathfrak{h}=\{z\in\mathbb{C}: \textrm{Im}(z)>0\}}$ be the upper-half plane, and consider the action of ${SL_2(\mathbb{R})}$ on ${\mathfrak{h}}$ by homographies:

$\displaystyle \left(\begin{array}{cc} a & b \\ c & d \end{array}\right)z := \frac{az+b}{cz+d}$

A modular form of weight ${k\in\mathbb{N}-\{0\}}$ and level ${SL_2(\mathbb{Z})}$ is a holomorphic function ${F:\mathfrak{h}\rightarrow\mathbb{C}}$ such that

• (i) ${F(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)z) = (cz+d)^{k}F(z)}$ for all ${z\in\mathfrak{h}}$ and ${\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\in SL_2(\mathbb{Z})}$;
• (ii) ${F}$ is holomorphic as ${z\rightarrow i\infty}$.

A modular form ${F}$ is called cuspidal if ${F}$ satisfies (i), (ii) and

• (iii) ${F}$ vanishes at infinity.

Note that the condition (i) implies that any modular form ${F}$ is ${1}$-periodic: ${F(z+1)=F(z)}$ for all ${z\in\mathfrak{h}}$. In particular, any modular form ${F}$ has a Fourier series

$\displaystyle F(z) = \sum\limits_{n\in\mathbb{Z}} a_n q^n \quad \textrm{ where } q:=e^{2\pi i z} \ \ \ \ \ (1)$

Since ${q\rightarrow 0}$ as ${z\rightarrow i\infty}$, we see that the condition (ii) is equivalent to

• (ii)’ ${a_n=0}$ if ${n<0}$

Also, the condition (iii) (${F}$ is cuspidal) is equivalent to ${a_0=0}$.

1.2. Hecke operators

The modular forms of weight ${k}$ and level ${SL_2(\mathbb{Z})}$ form a finite-dimensional vector space. For each ${n\in\mathbb{N}-\{0\}}$, the Hecke operator

$\displaystyle T_n F(z) = n^{k-1}\sum\limits_{\substack{a, d >0 \\ ad=n}} \frac{1}{d^k} \sum\limits_{b \textrm{ mod } d} F(\frac{az+b}{d})$

acts on this vector space.

Remark 2 This formula gets significantly simpler when ${n}$ is a prime number.

Among the basic properties of Hecke operators, it is worth to mention that ${T_n}$ and ${T_m}$ commute, and ${T_nF}$ is cuspidal if the modular form ${F}$ is cuspidal.

These properties suggest the study of the following objects:

Definition 1 ${F}$ is called a proper normalized cuspidal modular form if

• ${F}$ is proper (i.e., an eigenvector) for all Hecke operators ${T_n}$;
• the Fourier coefficient ${a_1}$ in (1) is normalized: ${a_1=1}$.

Remark 3 One can show that a proper normalized cuspidal modular form satisfies ${T_p F = a_p F}$ for all ${p}$ 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 ${F}$ of weight ${k}$ satisfy the following bound:

$\displaystyle |a_p|\leq 2p^{(k-1)/2}$

for all ${p}$ prime.

As it is well-known, this conjecture was settled by Eichler-Shimura in the case ${k=2}$, 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 ${F}$, we can divide Deligne’s strategy into two steps:

• (1) for each prime ${\ell}$, there exists a continuous irreductible representation ${\rho_F=\rho_{F,\ell}: Gal(\overline{\mathbb{Q}}|\mathbb{Q})\rightarrow GL_2(\overline{\mathbb{Q}}_\ell)}$ such that
• ${\rho_F}$ is non-ramified at all ${p\neq\ell}$ prime;
• if ${p\neq\ell}$, the characteristic polynomial of ${\rho_F(Frob_p)}$ is ${X^2-a_pX+p^{k-1}}$;
• ${\rho_F}$ appears in the (Betti) cohomology of a (proper, smooth) algebraic variety over ${\mathbb{Q}}$.
• (2) the Riemann hypothesis part of Weil’s conjectures (also established by Deligne) provides useful information on the eigenvalues of ${\rho_F(Frob_p)}$ acting on the cohomology of algebraic varieties, and this can be exploited to establish the Ramanujan-Petersson conjecture because ${a_p}$ is the sum of eigenvalues of ${\rho_F(Frob_p)}$ (cf. the expression for the characteristic polynomial of ${\rho_F(Frob_p)}$).

Before proceeding further, let us give some explanations about (1).

The absolute Galois group ${Gal(\overline{\mathbb{Q}}|\mathbb{Q})}$ is the group ${Aut(\overline{\mathbb{Q}})}$ of automorphisms of the algebraic closure ${\overline{\mathbb{Q}}}$ of ${\mathbb{Q}}$.

Given a prime number ${p}$, we can choose an algebraic closure ${\overline{\mathbb{Q}_p}}$ of the field ${\mathbb{Q}_p}$ of ${p}$adic numbers and a morphism ${\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_p}}$ such that the diagram associated to the arrows ${\mathbb{Q}\hookrightarrow\mathbb{Q}_p}$, ${\mathbb{Q}\hookrightarrow\overline{\mathbb{Q}}}$, ${\mathbb{Q}_p\hookrightarrow\overline{\mathbb{Q}_p}}$ and ${\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_p}}$ commute.

In this way, we obtain an embedding ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)\hookrightarrow Gal(\overline{Q}|\mathbb{Q})}$ where ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)}$ is the group of continuous automorphisms of ${\overline{\mathbb{Q}}_p}$.

Remark 4 The embedding ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)\hookrightarrow Gal(\overline{Q}|\mathbb{Q})}$ depends on the choice of ${\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}_p}}$. In particular, this embedding is only well-defined modulo conjugation (but, as it turns out, this is sufficient for our purposes).

The group ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)}$ is part of an exact sequence

$\displaystyle 1\rightarrow I_p\rightarrow Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)\rightarrow Gal(\overline{\mathbb{F}_p}|\mathbb{F}_p)\rightarrow 1$

where the arrow ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)\rightarrow Gal(\overline{\mathbb{F}_p}|\mathbb{F}_p)}$ is defined by reduction modulo ${p}$, or, more precisely, by considering a commutative diagram whose arrows are:

• ${\mathbb{Q}_p\hookleftarrow\mathbb{Z}_p\rightarrow \mathbb{F}_p:=\mathbb{Z}_p/p\mathbb{Z}_p}$,
• ${\overline{\mathbb{Q}_p}\hookleftarrow\overline{\mathbb{Z}_p}\rightarrow \overline{\mathbb{F}_p}}$,
• ${\mathbb{Q}_p\hookleftarrow\overline{\mathbb{Q}_p}}$, ${\mathbb{Z}_p\hookleftarrow\overline{\mathbb{Z}_p}}$, ${\mathbb{F}_p\hookleftarrow\overline{\mathbb{F}_p}}$.

Here, ${\mathbb{Z}_p}$, resp. ${\mathbb{Z}_p}$ is the ring of integers of ${\mathbb{Q}_p}$, resp. ${\overline{\mathbb{Q}_p}}$, ${\mathbb{F}_p}$ is the finite field of ${p}$ elements, and ${\overline{\mathbb{F}_p}}$ is the algebraic closure of ${\mathbb{F}_p}$.

The group ${Gal(\overline{\mathbb{F}_p}|\mathbb{F}_p)}$ is topologically generated by a single element, namely, Frobenius endomorphism ${Frob_p:x\mapsto x^p}$.

The kernel ${I_p}$ of the arrow ${Gal(\overline{\mathbb{Q}_p}|\mathbb{Q}_p)\rightarrow Gal(\overline{\mathbb{F}_p}|\mathbb{F}_p)}$ is called inertia group (but its structure is not need for the sake of this post).

Definition 2 We say that a representation ${\theta}$ of ${Gal(\overline{\mathbb{Q}}|\mathbb{Q})}$ is non-ramified at ${p}$ if and only if ${\theta(I_p)=\{1\}}$. In particular, ${\rho(Frob_p)}$ is well-defined modulo conjugation whenever ${\theta}$ is non-ramified at ${p}$ (cf. Remark 4).

Remark 5 The fact that ${\theta(Frob_p)}$ 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 ${\rho_F(Frob_p)}$ (and this polynomial depends only on the conjugacy class of ${\rho_F(Frob_p)}$).

Closing this subsection, let us outline the construction of the representation ${\rho_F}$ as in the first step of Deligne’s strategy in the particular case of proper normalized cuspidal modular forms ${F}$ of weight ${k=2}$.

We begin by introducing the modular surface ${Y=SL_2(\mathbb{Z})\backslash \mathfrak{h}}$. The modular form ${F}$ is not a function on the non-compact Riemann surface ${Y}$ (because it is not ${SL_2(\mathbb{Z})}$-invariant): in fact, the modularity condition (i) (with ${k=2}$) implies that ${F}$ is a section of the line bundle ${H^0(Y,\Omega^1)}$ of holomorphic differentials on ${Y}$.

Note that the one-point compactification ${X}$ of ${Y}$ is an orbifold topologically isomorphic to a sphere. The fact that ${F}$ is holomorphic at infinity (by condition (ii)) implies that ${F}$ can be extended to ${X}$ (i.e., ${F\in H^0(X,\Omega^1)}$).

Next, we consider the Hodge decomposition

$\displaystyle H^1(X,\mathbb{Q})\otimes_{\mathbb{Q}} \mathbb{C} = H^0(X,\Omega^1)\oplus H^1(X,\mathcal{O}_X)$

of the first (Betti) cohomology group of ${X}$.

In these terms, the Hecke operators ${T_p}$ admit the following geometrical interpretation. For each ${p}$, we can choose a finite-index subgroup ${\Gamma_p\subset SL_2(\mathbb{Z})}$ and an element ${\gamma_p\in SL_2(\mathbb{Z})}$ such that the action of ${T_p}$ on ${F\in H^0(Y,\Omega^1)}$ (or ${H^0(X,\Omega^1)}$ or ${H^1(X,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}}$) consists into taking the pullback under the natural arrows

$\displaystyle X\gets Y=SL_2(\mathbb{Z})\backslash\mathfrak{h}\gets \Gamma_p\backslash\mathfrak{h}$

and then taking the “trace of the operator” induced by the arrows

$\displaystyle \Gamma_p\backslash\mathfrak{h}\stackrel{\gamma_p}{\rightarrow} Y\rightarrow X$

where ${\stackrel{\gamma_p}{\rightarrow}}$ is the multiplication by ${\gamma_p=\left(\begin{array}{cc} p & 0 \\ 0 & 1 \end{array}\right)}$. (See, e.g., the subsection 2.3 of this PhD thesis here for more details.)

In particular, a normalized proper cuspidal modular form

$\displaystyle F= \sum\limits_{n>0} a_n q^n$

gives rise to a non-trivial simultaneous eigenspace of the operators ${T_p}$ on ${H^0(X,\Omega^1)}$ and, a fortiori, ${H^1(X,\overline{\mathbb{Q}})}$, with eigenvalues ${a_p}$.

This geometrical interpretation of ${T_p}$ permits to construct ${\rho_F=\rho_{F,\ell}}$ along the following lines.

First, we observe that ${X}$ and ${Y}$ are algebraic varieties over ${\mathbb{Q}}$ in a “canonical way” (because ${Y}$, resp., ${\Gamma_p\backslash\mathfrak{h}}$ are moduli spaces of elliptic curves, resp. elliptic curves with extra (“level”) structure).

This implies that, for each ${\ell}$ prime, we have an (algebraically defined) isomorphism

$\displaystyle H^1(X,\mathbb{Q})\otimes\overline{\mathbb{Q}}_{\ell}\rightarrow H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})$

from ${H^1(X,\mathbb{Q})\times\overline{\mathbb{Q}}_{\ell}}$ to the étale cohomology group ${H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})}$ of the algebraic variety ${X}$ over ${\mathbb{Q}}$. Moreover, this isomorphim behaves “equivariantly” with respect to the Hecke operators ${T_p}$.

In particular, the non-trivial simultaneous eigenspace of ${T_p}$ associated to ${F}$ can be transferred to ${H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})}$.

Secondly, we note that this isomorphism is interesting because ${Gal(\overline{\mathbb{Q}}|\mathbb{Q})}$ acts on ${H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})}$. Furthermore, since ${T_p}$ is defined over ${\mathbb{Q}}$ (and ${T_p}$ behaves “equivariantly” with respect to this isomorphism), the transferred non-trivial simultaneuous eigenspace of ${T_p}$ in ${H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})}$ is invariant under this action of ${Gal(\overline{\mathbb{Q}}|\mathbb{Q})}$.

In summary, the normalized proper cuspidal modular form ${F}$ induces a representation of ${Gal(\overline{\mathbb{Q}}|\mathbb{Q})}$ in a transferred non-trivial simultaneous eigenspace of ${T_p}$ in ${H^1_{et}(X,\overline{\mathbb{Q}}_{\ell})}$, and, as it turns out, this is (more or less) the representation ${\rho_{F,\ell}:Gal(\overline{\mathbb{Q}}|\mathbb{Q})\rightarrow GL_2(\overline{\mathbb{Q}}_\ell)}$ 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 ${Y_{\Gamma}:=\Gamma\backslash\mathfrak{h}}$ (for appropriate choices of ${\Gamma\subset SL_2(\mathbb{Z})}$).

Nevertheless, for the sake of generalizing this discussion, it is helpful to replace ${Y_{\Gamma}}$ by

$\displaystyle S_{K(n)} = GL_2(\mathbb{Q})\backslash GL_2(\mathbb{A})/K_{\infty} K(n)$

where ${\mathbb{A} = \mathbb{R}\times\mathbb{A}_f}$ is the ring of adeles (with ${\mathbb{A}_f=\widehat{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{Q}:=\prod\limits_{p \textrm{ prime}}\mathbb{Z}_p}$ denoting the “finite adeles”),

$\displaystyle GL_2(\mathbb{A}) = GL_2(\mathbb{R})\times GL_2(\mathbb{A}_f)$

(with ${GL_2(\mathbb{A}_f) = \{(g_p)_p\in\prod\limits_p GL_2(\mathbb{Q}_p): g_p\in GL_2(\mathbb{Z}_p) \textrm{ for all but finitely many } p\}}$),

$\displaystyle K_{\infty} = \mathbb{R}_+\cdot SO(2)\subset GL_2(\mathbb{R})$

and

$\displaystyle K(n)=\{g\in GL_2(\widehat{\mathbb{Z}}): g\equiv \textrm{id} \textrm{ mod } n\}\subset GL_2(\mathbb{A}_f)$

The fact that ${S_{K(n)}}$ is closely related to ${Y_{\Gamma}}$ comes from the equality

$\displaystyle S_{K(n)} \simeq \Gamma(n)\backslash\mathfrak{h}^{\pm}$

where ${\Gamma(n)=K(n)\cap GL_2(\mathbb{Z})}$ and ${\mathfrak{h}^{\pm}=\mathbb{C}-\mathbb{R}}$.

The key points of this complicated definition of ${S_{K(n)}}$ are:

• each operator ${T_p}$ acts only on the ${p}$-th component of ${GL_2(\mathbb{A})}$;
• 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 ${G=GL_2}$ can be generalized to ${G=GL_d}$, ${d\geq 3}$.

We start by introducing

$\displaystyle S_{K(n)} = GL_d(\mathbb{Q})\backslash GL_d(\mathbb{A})/K_{\infty} K(n)$

for ${n\geq 3}$ (where all implied groups are defined by their natural counterparts in ${GL_d}$).

Similarly to the case ${d=2}$, we have that ${S_{K(n)}}$ is a real-analytic variety (given by the disjoint union of two symmetric spaces for ${GL_d(\mathbb{R})}$).

In this setting, the Hecke operators are obtained by taking a prime number ${p}$ and an adequate element ${\gamma_p\in GL_d(\mathbb{Q}_p)}$, and considering the composition ${T_p}$ of the pullback and trace operators induced on ${H^*(S_{K(n)},\overline{\mathbb{Q}}_{\ell})}$ by the natural arrows (inclusion and multiplication by ${\gamma_p}$):

$\displaystyle S_{K(n)}\leftarrow S_{K(n)\cap \gamma_p K(n)\gamma_p^{-1}}\stackrel{\gamma_p}{\rightarrow} S_{K(n)}$

This construction can be slightly generalized by noticing that, if ${p\not| \,\, n}$, then we have an action on ${H^*(S_{K(n)},\overline{\mathbb{Q}}_{\ell})}$ of the Hecke algebra

$\displaystyle \mathcal{H}_p = C_c(GL_d(\mathbb{Z}_p)\backslash GL_d(\mathbb{Q}_p)/GL_d(\mathbb{Z}_p),\overline{\mathbb{Q}}_{\ell})$

equipped with the product given by the convolution with respect to the Haar measure with volume one.

Remark 6 For ${d=2}$, we studied exclusively the action of certain elements ${T_p\in\mathcal{H}_p}$, but from now on we will consider the whole action of ${\mathcal{H}_p}$.

After the works of Satake, we know that ${\mathcal{H}_p}$ is a commutative algebra whose characters ${\mathcal{H}_p\rightarrow \overline{\mathbb{Q}}_{\ell}}$ correspond to the conjugations classes of ${GL_d(\overline{\mathbb{Q}}_{\ell})}$ under a canonical bijection called Satake isomorphism.

For the sake of convenience, we will put “together” the Hecke algebras ${\mathcal{H}_p}$ for ${p\not\vert \,\, n}$ by defining the Hecke algebra unramified off ${n}$ as:

$\displaystyle \mathcal{H}^{(n)} := \bigotimes\limits_{p \,\not\vert\,\, n} \mathcal{H}_p$

2.2. Characters of Hecke algebras and Galois representations

The following two conjectures are part of the so-called Langlands program.

\noindentConjecture. Let ${\varphi:\mathcal{H}^{(n)}\rightarrow\overline{\mathbb{Q}}_{\ell}}$ be a character such that the corresponding eigenspace in ${H^*(S_{K(n)}, \overline{\mathbb{Q}}_{\ell})}$ is non-trivial (i.e., ${\varphi}$ is a sort of “normalized proper cuspidal modular form”). Then, there exists a (continuous) Galois representation ${\rho_{\varphi}: Gal(\overline{\mathbb{Q}}|\mathbb{Q})\rightarrow GL_d(\overline{\mathbb{Q}}_{\ell})}$ such that

• ${\rho_{\varphi}}$ is not ramified on ${p}$ (i.e., ${\rho_{\varphi}}$ is trivial on the inertia group ${I_p}$);
• ${\rho_{\varphi}(Frob_p)\in GL_d(\overline{\mathbb{Q}}_p)/\textrm{conjugation}}$ corresponds to ${\varphi|_{\mathcal{H}_p}}$ 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 ${S_{K(n)}}$ is not algebraic when ${d\geq 3}$.

\noindentConjecture’. The same conjecture as above is true when ${H^*(S_{K(n)}, \overline{\mathbb{Q}}_{\ell})}$ is replaced by the cohomology with torsion ${H^*(S_{K(n)}, \overline{\mathbb{F}}_{\ell})}$.

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 ${\mathbb{Z}/\ell^m\mathbb{Z}}$) implies Conjecture (by taking the limit ${m\rightarrow\infty}$). 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 ${G=GL_d}$, ${d\geq 3}$, but we can also study them for other reductive connected groups ${G}$ such as ${Sp_{2g}, U(p,q)}$, etc.

An interesting feature of this generalization of Conjecture/Conjecture’ is that sometimes ${S_{K(n)}}$ becomes an algebraic (Shimura)variety (e.g., for ${G=Sp_{2g}, U(p,q)}$, etc.), and this gives us a hope of mimicking the arguments from the first section of this post.

Remark 9 The fact that ${S_{K(n)}}$ is algebraic is not sufficient in general to reproduce the strategy employed for ${GL_2}$ in the first section of this post.For example, if ${E}$ is an imaginary quadratic extension of ${\mathbb{Q}}$ and ${G=U(2,2)}$ act on ${E^4}$, then the strategy used for ${GL_2}$ does not produce a “good” representation ${\rho_{\varphi}}$ attached to a character ${\varphi: \mathcal{H}_G^{(n)}\rightarrow\overline{\mathbb{Q}}_{\ell}}$.

In fact, one gets a good representation ${\rho_{\varphi}}$ only after applying Langlands functoriality principle to “transfer” the problem from ${U(2,2)}$ to ${U(1,3)}$, and then using the strategy for ${GL_2}$ in this new setting.

In summary, even if ${S_{K(n)}}$ is algebraic, one can’t apply the strategy for ${GL_2}$ in a simple-minded way in order to deduce Conjecture/Conjecture’.

On the other hand, the fact that ${S_{K(n)}}$ is algebraic for certain choices of ${G}$ does not seem to help us in the context of the results of Boxer and Scholze because we know that ${S_{K(n)}}$ is not algebraic in their setting.

Nevertheless, Clozel noticed that the non-algebraic varieties ${S_{K(n)}}$ 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 ${\varphi:\mathcal{H}^{(n)}\rightarrow \overline{\mathbb{Q}}_{\ell}}$ appearing in ${H^*(S_{K(n)}, \overline{\mathbb{Q}}_{\ell})}$.

Let us illustrate this idea of Clozel in the first non-trivial case of the generalization of the Conjecture, i.e., ${G=GL_{2,E}}$, ${E|\mathbb{Q}}$ imaginary quadratic extension (where ${S_{K(n)}}$ is a non-algebraic variety of real dimension 3).

We take ${H=U(2,2)}$, so that ${S_H}$ is algebraic (but not compact). Its Borel-Serre compactification ${S_H^{BS}}$ is not algebraic but its boundary has some components associated to parabolic subgroups ${P}$ of ${H}$ such as Levi’s parabolic subgroup

$\displaystyle P=\left\{\left(\begin{array}{cccc} \ast & \ast & \ast & \ast \\ \ast & \ast & \ast & \ast \\ 0 & 0 & \ast & \ast \\ 0 & 0 & \ast & \ast \end{array}\right)\in H\right\}$

which turns out to be isomorphic to ${GL_{2,E}}$.

In this way, we get two arrows ${S_{H}^{BS}\leftarrow S_P\rightarrow S_G}$ allowing to relate ${H^*(S_G)}$ to ${H^*(S_H)}$ with the advantage that ${S_H}$ 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 ${H^*(S_G)}$ to ${H^*(S_H)}$ as above, an important ingredient in Boxer and Scholze proof of Conjecture’ is the following theorem:

Theorem 4 (Boxer; Scholze) If ${\varphi:\mathcal{H}_H^{(n)}\rightarrow\mathbb{F}_{\ell}}$ appears in ${H^*(S_{K(n)},\overline{\mathbb{F}}_{\ell})}$, then there exists a (cuspidal) ${\psi: \mathcal{H}_H^{(n)} \rightarrow \mathbb{F}_{\ell}}$ in characteristic zero appearing in ${H^*(S_H,\overline{\mathbb{Z}}_{\ell})}$ such that ${\varphi\equiv \psi}$ (mod ${\ell}$).

Then, she told that a “one-sentence proof” of this result is the following: one uses a comparison theorem to relate ${H^*(S_{K(n)},\overline{\mathbb{F}}_{\ell})}$ to cohomology groups of affinoid spaces.