Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?

$\begingroup$ This is a subject of an old (from 1970s) theorem by Chudnovsky. You have to distinguish the CM and nonCM cases. BTW, this is "transcendental" rather than "algebraic" number theory. $\endgroup$– Wadim ZudilinJun 16 '10 at 13:29
On p. 304 of "Contributions to the theory of transcendental numbers" by Gregory Chudnovsky (avalaible from google books) one finds a consequence of Theorem 1.26 which states (even more than) that if $E$ has a complex multiplication in a number field, then any period is transcendental.
You need to assume E defined over the algebraic numbers, or else "the" period makes no sense; and then there are two basic periods, except in the complex multiplication case where the ratio will be algebraic. The first transcendence results were due to Schneider and Baker. These were developed by Coates and Masser. Probably anything you need is in Masser's thesis, which includes results on the quasiperiods too. There are many further results, but I think no real surprises. (Review by Moreno at http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183540631 of Masser's Elliptic Functions and Transcendence.)

$\begingroup$ You are right: David Masser had several transcendence results for elliptic functions, periods and quasiperiods. (Schneider proved that the values of the modular invariant at nonCM points are transcendental.) Don't remember anything quantitative in this respect from Baker and Coates (the nonArchimedean case?). Chudnovsky proved that among the two periods and two corresponding quasiperiods there are at least 2 algebraically independent numbers. $\endgroup$ Jun 16 '10 at 13:50

$\begingroup$ Much more is in Waldschmidt's slides at modular.math.washington.edu/swc/aws/08/slides/… . The first result seems to have been by Siegel. $\endgroup$ Jun 16 '10 at 14:07

$\begingroup$ Michel is definitely in this business for many years; I guess one can dig more from his webpage. $\endgroup$ Jun 16 '10 at 14:11