When is the period of elliptic curve over the rationals transcendental? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:48:06Z http://mathoverflow.net/feeds/question/28385 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28385/when-is-the-period-of-elliptic-curve-over-the-rationals-transcendental When is the period of elliptic curve over the rationals transcendental? norondion 2010-06-16T13:15:35Z 2010-06-16T13:43:10Z <p>Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?</p> http://mathoverflow.net/questions/28385/when-is-the-period-of-elliptic-curve-over-the-rationals-transcendental/28387#28387 Answer by Charles Matthews for When is the period of elliptic curve over the rationals transcendental? Charles Matthews 2010-06-16T13:36:13Z 2010-06-16T13:36:13Z <p>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 quasi-periods too. There are many further results, but I think no real surprises. (Review by Moreno at <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.bams/1183540631" rel="nofollow">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&amp;id=pdf_1&amp;handle=euclid.bams/1183540631</a> of Masser's <em>Elliptic Functions and Transcendence</em>.)</p> http://mathoverflow.net/questions/28385/when-is-the-period-of-elliptic-curve-over-the-rationals-transcendental/28388#28388 Answer by Wadim Zudilin for When is the period of elliptic curve over the rationals transcendental? Wadim Zudilin 2010-06-16T13:43:10Z 2010-06-16T13:43:10Z <p>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.</p>