I have an ignorant question about elliptic curves which I'll be slightly imprecise about. If I have an elliptic curve $X$ defined over $\mathbb Z$, I can base change to $\mathbb C$, and then $X(\mathbb C)$ is isomorphic to the quotient of $\mathbb C$ by the lattice generated by $1, \tau$ for some $\tau \in \mathbb C$.

I can also base change to a finite field $\mathbb F_q$, and then the Frobenius operator on $H^1 (X( \bar {\mathbb{F} }_q), \bar {\mathbb{Q} }_l)$ has eigenvalues $\sigma, \bar \sigma$.

Q: Is there any relationship between $\sigma, \bar \sigma$, and $\tau$?

(I may have glossed over some things, like fixing an ~~identification~~ embedding of $\bar {\mathbb Q}$ into $\mathbb C$, or requiring $X(\mathbb F_q)$ to be smooth, or something else.)

3more comments