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The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $\Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}-1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $\Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}+1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}-1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $\Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}-1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $Gamma_n$.

Then for $n$ greater than $3$, $H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}-1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)

The answer to this question is probably to be found in the theory of automorphic forms, but (I don't know much about it and consequently) after some tries, I did not catch it. Thus I'd be grateful if somebody could give me a reference.

Let $\Gamma_n$ be the kernel of the natural morphism $\mathrm{SL}_2(\mathbf Z)\to \mathrm{SL}_2(\mathbf Z/n)$.

Let $X_n$ be the quotient of the Poincaré half-plane $\mathcal H$ by the usual action of $Gamma_n$.

Then for $n$ greater than $3$, $\mathrm H_1(X_n,\mathbf Z)$ is a free module of rank $$\frac{\sharp \mathrm{SL}_2(\mathbf Z/n)}{12}-1\ \ \ .$$

Thus we have a sequence of representations $\rho_n$ (of $\mathrm{SL}_2(\mathbf Z/n)$ on $\mathrm H_1(X_n,\mathbf Z)$) together with morphisms $\rho_n\to\rho_m$ for $m\vert n$, and at the limit a representation $\hat{\rho}$ of $\mathrm{SL}_2(\hat{\mathbf Z})$ on $\lim_n\mathrm{H}_1(X_n,\mathbf Z)$.

The question is the following : is there an algebraic construction of this representation $\hat \rho$ ?

(Something along the lines of "the second exterior power of the adjoint representation" would qualify (but it's obviously false ...)).

(Of course the question could be declined in many ways : replace $\mathrm{SL}_2$ by another group, $\mathcal H$ by the associated symmetric space, $\mathrm{H}_1()$ by another homology theory and so on ...)