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

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This is a subject of an old (from 1970s) theorem by Chudnovsky. You have to distinguish the CM and non-CM cases. BTW, this is "transcendental" rather than "algebraic" number theory. –  Wadim Zudilin Jun 16 '10 at 13:29
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2 Answers 2

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.

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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 http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183540631 of Masser's Elliptic Functions and Transcendence.)

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You are right: David Masser had several transcendence results for elliptic functions, periods and quasi-periods. (Schneider proved that the values of the modular invariant at non-CM points are transcendental.) Don't remember anything quantitative in this respect from Baker and Coates (the non-Archimedean case?). Chudnovsky proved that among the two periods and two corresponding quasi-periods there are at least 2 algebraically independent numbers. –  Wadim Zudilin Jun 16 '10 at 13:50
    
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. –  Charles Matthews Jun 16 '10 at 14:07
    
Michel is definitely in this business for many years; I guess one can dig more from his webpage. –  Wadim Zudilin Jun 16 '10 at 14:11
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