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Browsing the wealth of functions available in Mathematica, one encounters two not-so-common functions: the "elliptic logarithm", which is an elliptic integral in another garb, and the "elliptic exponential", its inverse which is an elliptic function. The only substantial thing I know about this pair is that they can be expressed in terms of the Weierstrass ℘ function and its inverse (and in fact Mathematica internally uses the elliptic logarithm/exponential to compute the Weierstrass functions).

My questions on this pair are the following:

  1. A bit of searching around seems to indicate that these are used in the theory of elliptic curves. Considering that the other elliptic integrals and the elliptic functions of Jacobi and Weierstrass have much wider applicability (e.g. in physics), why are these two seemingly marginalized? (Or, are there references where using these instead of the classical functions made for better exposition?)

  2. Why exactly would one use an elliptic logarithm and/or an elliptic exponential? More precisely, why would not the other more common elliptic integrals/functions suffice that one has to resort to these? Do they satisfy more convenient identities than the classical one?

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The usual logarithm can be thought of as a function on the multiplicative group $\mathbb{C}^*$. The multiplicative group is a degeneration of an elliptic curve, so a natural idea is to "deform" the logarithm (or other given function) to one that makes sense on any given elliptic curve. Usually this is done by averaging over the lattice L for which $\mathbb{C}/L$ is the given curve. (It's not obvious or trivial that an elliptic analogue of any given construct exists. But one looks for it.)

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Sounds great. If so, is the only purpose of the elliptic exponential one of it being used to parametrize elliptic curves (but then again the Weierstrass ℘ can do that as well)? Or is there something more "convenient" about the elliptic exponential? –  J. M. Aug 6 '10 at 3:04
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The elliptic objects don't parametrize elliptic curves, they are parameterized by elliptic curves. For each lattice L in the complex plane one gets a logarithm function Log(L,z). When the elliptic curve (genus 1) is varied so as to tend to a singular curve (genus 0) the elliptic object is supposed to degenerate to the usual, non-elliptic object. In this paradigm, the "object" can mean: logarithm, exponential, gamma function, quantum group, integrable model, etc. These all have elliptic analogues. –  T.. Aug 6 '10 at 4:39
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