Talk:Half-period ratio

The article states, toward the end, that:

"Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of the nome."

But it is false to say that the half periods (or periods) "can be given in terms of the elliptic modulus or in terms of the nome". It is precisely the other way around -- only. The elliptic modulus or the nome can be given in terms of the half-periods (or periods). But the actual periods cannot be recovered from the elliptic modulus or nome.Daqu (talk) 18:30, 8 February 2012 (UTC)Reply

Umm, I think the point is that Klein's j-invariant gives exactly that inversion: it is one-to-one and onto. It's not "one-way", it really is "two-way". That's an elementary theorem from analytic number theory and modular forms. That said, yes, of course, there's an infinite number of different but equivalent periods, and the modular group bounces them around; but in the end you end up in the fundamental domain where all of it is one-to-one. The wording in this article could be tightened.

Also I think that the j that is mentioned cryptically in this article really is supposed to be the j-invariant. 67.198.37.16 (talk) 08:24, 24 January 2019 (UTC)Reply