In Mathematics, often a theory becomes popular because it tells us something new or gives different proof for already established facts. For example, I have read that algebraic number theory is helpful because it helped us to solve some diaphontine equations (most notably, Fermat’s last theorem). I want to know what kind of questions are […]

Let $\rho_{\ell}$ be the “mod $\ell$” Galois representation associated to an elliptic curve $E/K$ (i.e., corresponding to the action of Galois on the $\ell$-torsion points). Serre proved that in the case where the image of Galois is the normalizer of a nonsplit Cartan subgroup, this defines a quadratic extension of $K$ which is actually unramified. […]

I am not very familiar with the theory of Galois representations, but I do know a bit about both Galois theory and representation theory. Recently I learned about the notion of a normal basis for a Galois extension which led me to consider following straight-forward Galois representation: Let $L/K$ be a Galois extension and choose […]

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