CM reciprocity theorem

For a complex CM torus $\mathbb{C}/\mathfrak{a}$ and an idele $s$ of its CM field, the automorphism corresponding to $s$ by Artin reciprocity takes its lattice to $s^{-1}\mathfrak{a}$ and acts on torsion through the adelic scalar $s^{-1}$. When the curve is defined over a larger field, use the idelic norm to the CM field. Comparison with the original curve gives a scalar in the CM field, producing the Hecke character of a CM elliptic curve.