# Comparison of two $GL_N(\mathbb{Z}_\ell)$ Galois representations

I have a question about comparing two $$\ell$$-adic Galois representations. Suppose we have two irreducibible Galois representaions $$\rho_1,\rho_2: Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_N(\mathbb{Z}_{\ell}).$$ If $$\rho_1$$ is not isomorphic to $$\rho_2$$, then there must be a largest integer $$a$$ such that $$tr(\rho_1)\equiv tr(\rho_2)\ (mod\ \ell^a).$$ Similarly, there is a largest integer $$b$$ such that up to a conjugation by an element in $$GL_N(\mathbb{Z}_{\ell})$$, we have $$\rho_1 =\rho_2 \ (mod\ \ell^b).$$ It is obvious that $$b\leq a$$. I am wondering is there any known conditions to ensure that $$b=a$$?

• $N=1$ for instance. You want all other coefficients of the characterstic polynomials, like $\det(\rho_i)$ to be congruent , too. – Chris Wuthrich Oct 12 '18 at 8:43

This is not true if the representations are residually reducible. Suppose that $$N=2$$, then $$b$$ can be strictly less than $$a$$. Let $$\bar{\rho}_i$$ for $$i=1,2$$ denote the mod $$l$$ representations. Then if $$\bar{\rho}_1$$ is diagonal and $$\bar{\rho}_2$$ is upper-triangular with the same diagonal entries as $$\bar{\rho}_1$$ but not isomorphic to $$\bar{\rho}_1$$ it would follow that $$a\geq 1$$ and $$b=0$$.