This might be a very vague question since I am not very familiar with the theory of automorphic forms. Let $G$ be a connected reductive algebraic group defined over $F$ (a number field). Suppose we have a cuspidal automorphic representation $\pi$ of $G(\mathbb A_{F})$ for which there are Galois representations $\rho_{\pi,p}: \Gamma_{F} \to G^{\vee}(\overline{\mathbb Q}_p)$ attached to it for all primes $p$ satisfying good properties such as for any unramified place $v$ the Langlands parameter of $\pi_v$ matches with the semisimple part of $\rho_{\pi,p}(Fr_v)$, and $\rho_{\pi,p}$ is de Rham at places above $p$. (Here we fix an isomorphism $\iota: \mathbb C \cong \overline{\mathbb Q}_p$.)

Now suppose there are two automorphic representations $\pi_1,\pi_2$ as above such that for some prime $p$, $\rho_{\pi_1,p} \equiv \rho_{\pi_2,p}$ modulo $p$. Are $\pi_1$ and $\pi_2$ then "congruent" in some sense? For example, we have for any unramified place $v$, $\iota\phi_{\pi_1,v}(Fr_v) \equiv \iota\phi_{\pi_2,v}(Fr_v)$ modulo $p$, where $\phi_{\pi,v}$ is the Langlands parameter of $\pi_v$. On the other hand, if one would like to work with integral automorphic forms valued in a finite extension $E$ of $\mathbb Q_p$ with ring of integer $\mathcal O$ and a uniformizer $\lambda$ (as in 3.3 of Clozel-Harris-Taylor), then any integral automorphic representation can be reduced mod $\lambda$. What would be a reasonable definition of congruence between automorphic representations? A naive way would be to define two autormorphic representations $\pi_1, \pi_2$ of $G(\mathbb A_F)$ to be congruent if their mod $\lambda$ reductions are equal. Alternatively, we can define $\pi_1, \pi_2$ to be congruent if for any place $v$ where both $\pi_{1,v}$ and $\pi_{2,v}$ are unramified, the mod $\lambda$ reductions of their Satake parameters (which are valued in $G(\mathcal O)$) are equal.

What is a correct way of understanding the integral theory of automorphic forms and congruences? I know that in order to prove $R=\mathbb T$ theorems, we need an integral version of the Hecke algebra, and hence the integral theory of automorphic form is necessary. Are there other reasons why people study the integral theory of automorphic forms? Does the integral theory carry more information than the classical theory?


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