*This proof was made possible by a couple helpful and fabulous conversations with Yifeng Liu. All errors are mine and mine alone*.

This is toward my understanding of the phrase “Why is is height so important as an invariant? Because the height of a formal group law comes from the symmetry of the underlying variety.”

One method of getting a lower dimensional formal group laws from a abelian varieties of higher dimension is via using the theory of complex multiplication — splitting the abelian variety by splitting the prime (as I exposited in my paper here).

**I show that for abelian varieties with CM, the height of the formal group law pieces are expressible as a formula in terms of the degree of some field extensions of $Q_p$ ****(one corresponding to each prime living over $p$) and the dimension of the rational endomorphism ring of the variety as a $Q$-vector space.** Continue reading The Height of a Formal Group Law in terms of the Symmetry of the Underlying CM Abelian Variety