Tate module

Summary

Fix a prime $\ell$. The $\ell$-adic Tate module of an abelian group collects its elements of $\ell$-power order into one module over the $\ell$-adic integers. Let $A$ be an abelian variety of dimension $g$ over a field $k$ whose characteristic is not $\ell$. Then the Tate module of $A$ is free of rank $2g$. The absolute Galois group of $k$ acts on it, which gives an $\ell$-adic Galois representation. These representations carry much of the arithmetic of $A$.