What is the .sigma() function for an elliptic curve's formal group?

asked 2021-02-03 22:13:27 +0200

peter.xu gravatar image

The SageMath documentation page for "formal groups of elliptic curves" (I can't link it because I don't have enough karma) lists among the methods for formal groups of elliptic curves a .sigma(), with essentially no explanation of what it is. This makes me suspect that it's the $p$-adic $\sigma$-function of the elliptic curve, as defined by Mazur and Tate in their paper of that name, but I'd like some security in whether the implementation is accurate, as well as what the "c" variable is - I guess probably it somehow corresponds to a choice of invariant differential on the elliptic curve/formal group since that's the only other thing the $p$-adic $\sigma$-function depends on, but how this works isn't transparent to me.

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here's the link i couldn't add: https://doc.sagemath.org/html/en/refe...

peter.xu gravatar imagepeter.xu ( 2021-02-03 22:27:50 +0200 )edit

I don't know the answer, but you can view the implementation by entering the following:

sage: E = EllipticCurve('14a')
sage: F = E.formal_group()
sage: F.sigma??

Perhaps something will look familiar.

rburing gravatar imagerburing ( 2021-02-03 23:21:30 +0200 )edit