show · ec.galois_rep all knowls · up · search:

If $E$ is an elliptic curve defined over a field $K$ and $m$ is a positive integer, then the mod-$m$ Galois representation attached to $E$ is the map $\overline\rho_{E,m}: \Gal(\overline{K}/K) \to \Aut(E[m])$ describing the action of the absolute Galois group of $K$ on the $m$-torsion subgroup $E[m]$.

When the characteristic of $K$ does not divide $m\gt 1$, we may identify the finite abelian group $E[m]$ with $(\Z/m\Z)^2$ and hence view the representation as a map $\overline\rho_{E,m}: \Gal(\overline{K}/K) \to \GL(2,\Z/m\Z)$ defined up to conjugation. In particular, when $m=p$ is prime different form the characteristic of $K$, we have $\overline\rho_{E,p}: \Gal(\overline{K}/K) \to \GL(2,\Z/p\Z).$ Taking the inverse limit over prime powers $m=p^n$ yields the $p$-adic Galois representation attached to $E$, $\rho_{E,p}: \Gal(\overline{K}/K) \to \Aut(T_p(E)) \cong \GL(2,\Z_p),$ which describes the action of the absolute Galois group of $K$ on $T_p(E)$, the $p$-adic Tate module of $E$.

When $K$ has characteristic zero one can take the inverse limit over all positive integer $m$ (ordered by divisibility) to obtain a Galois representation $\rho_{E}: \Gal(\overline{K}/K) \to \GL(2,\hat \Z).$

Authors:
Knowl status:
• Review status: reviewed
• Last edited by John Jones on 2018-06-19 22:22:36
Referred to by:
History: