# Galois representations attached to a cusp form for different primes

If I have a cusp form $$f$$, I can consider the associated Galois representation $$\rho_l(f)$$ for any prime $$l$$. For two distinct primes $$p$$ and $$q$$, what is the relationship between $$\rho_p(f)$$ and $$\rho_q(f)$$? Are they basically the same? How can I make the relationship precise?

• They have the same $L$-function (locally and globally), which is both pretty impressive and quite far from being "basically the same". Sep 28, 2021 at 9:48

## 1 Answer

At the most basic level, $$\rho_p$$ and $$\rho_q$$ are "nothing to do with each other". E.g. the kernels of $$\rho_p \bmod p$$ and of $$\rho_q \bmod q$$ cut out two finite Galois extensions of $$\mathbf{Q}$$ which are essentially unrelated.

On the other hand, there is quite an industry devoted to formulating properties of $$\rho_p$$ which are independent of $$p$$ (with the independence sometimes being easy, and sometimes very hard indeed); these results go under the name of "local-global compatibility in the Langlands programme".

Here is an example at a rather deeper level of how the different $$\rho_p$$'s complement each other. If $$f$$ corresponds to an elliptic curve of analytic rank 0, then $$L(f, 1) / \Omega_E$$ is a nonzero rational number; and the BSD conjecture gives a formula for what that number should be. There is a rather substantial body of theory aiming to prove, for a given prime $$p$$, that the power of $$p$$ dividing $$L(f, 1)/\Omega_E$$ is the "correct" one for the BSD formula (i.e. BSD holds up to a $$p$$-adic unit); and this strategy involves a deep study of the arithmetic of $$\rho_p$$ -- but you really need $$\rho_p$$ here, there's no way that $$\rho_q$$ for $$q \ne p$$ will tell you anything about the powers of $$p$$ dividing $$L(f, 1)$$. If you can do this for all $$p$$, then you can nail down $$L(f, 1)$$ exactly, but you need all the $$\rho_p$$'s available; each one is giving you different information.