# Where does the notation $\operatorname{Tr}(\cdot)\bmod \ell^\alpha$ implies isomorphism come from?

In J-P Serre's article on Faltings-Serre (Resume du Course 1984-1985) he states (without proof) that for two finite-dimensional $\ell$-adic Galois representations of $\operatorname{Gal}(\mathbb{Q})$, $\rho_1$, $\rho_2$ with absolutely irreducible residual representations if $\operatorname{Tr}(\rho_1(s)) = \operatorname{Tr}(\rho_2(s)) \bmod \ell^\alpha$ then they are isomorphic $\mod \ell^\alpha$

There is also a letter of Serre to Tate where he includes many more details (and an additional example beyond the Resume du Course) but not the proof or reference of the above. It is possible I am looking at the wrong letter, and there is another one where the proof appears. Armand Brumer credits H. Carayol with this result, but I have searched in vain for a clear reference (in MR1279611 a generalization of a corollary (in the case $A=\mathbb{Q}_p$) is proved, one proof of which is due to Serre, and this is from 1994).

So where is the original publication of this proof? My best guess is that the theorem is due to J-P Serre and was proved in 1984, but I haven't found the details yet.

• What are $\rho_1$ and $\rho_2$? Galois representations Okay, but over which fields or rings? of What dimensions? What is $\alpha$? A fixed integer? any integer? And when you say "they are isomorphic": what is isomorphic and over what? Thanks – Joël Mar 4 '17 at 4:27
• Based on the comments in MR1279611, I think it is reasonable to infer that Carayol was the first to publish the result but that Serre was aware of it in the mid-1980s. However, I doubt he communicated it explicitly before becoming aware of Carayol's result, as Barry Mazur, for instance, seems to be unaware of it in Deforming Galois Representations, which is from 1989. – Olivier Mar 4 '17 at 14:11
• I think Olivier – Joël Mar 4 '17 at 14:23