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It's well known that an elliptic curve of the form $y^3 = x^3 +ax +b$ admits a group structure, as long as a point at infinity $O$ is added to serve as identity. If we look at a real elliptic curve as a subset of the real projective plane, then it's easy to see that this point at infinity will be [0: 1: 0], which makes sense since real elliptic curves intersect the line at infinity in the direction of (0, 1). It also seems reasonable to assume that the multiplication operation would be continuous under the subspace topology. A real elliptic curve is also a closed subset of the real projective plane, since it is the locus of a polynomial plus a single point. Since the real projective plane is compact, the curve will be compact as well with respect to this topology. As such, it seems to make sense to me to say that it forms a compact topological abelian group that is also Hausdorff (since the real projective plane is), which would imply that it has a (non-discrete) finite Haar measure.

With all that in mind, my question is: Is there any interest in analyzing Pontryagin Duality and Harmonic Analysis restricted to elliptic curves? So far I haven't been able to find anyone referencing this, which I found odd.

Pedro Lourenço
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    I may be mistaken, but as topological groups complex elliptic curves are $S^1\times S^1$. So the Pontryagin dual is simply $\Bbb{Z}^2$. I think that's exactly what the complex lattice way of looking at elliptic curves is all about (i.e. the periods of elliptic curves, see chapter VI in Silvermann). – Jyrki Lahtonen Jul 10 '23 at 17:42
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    OP asks about the situation over $\mathbb R$, not $\mathbb C$. One should still start with the situation over $\mathbb C$ of course. Cf. https://math.stackexchange.com/q/40577/96384, https://math.stackexchange.com/q/425176/96384. – Torsten Schoeneberg Jul 10 '23 at 17:42

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All complex elliptic curves are isomorphic as topological groups (or even real Lie groups) to $S^1 \times S^1$, so Pontryagin duality or harmonic analysis for complex elliptic curves is a special case of working with any finite product of copies of a circle.

A real elliptic curve will be isomorphic to $S^1$ (one loop on a torus, or negative discriminant) or $S^1 \times \mathbf Z/(2)$ (two loops on a torus, or positive discriminant). So doing Pontryagin duality and harmonic analysis on a real elliptic curve amounts to working on a circle or a product of a circle with a group of order $2$.

KCd
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    I'm very ill informed on this subject so apologies if this question is silly, but does that imply that real elliptic curves can then be seen as subgroups of the torus? – Pedro Lourenço Jul 10 '23 at 17:55
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    I added a paragraph to my answer about the topological appearance of real elliptic curves. They are a subgroup of the complex elliptic curve (a suitable torus) defined by the same equation. All accounts of elliptic curves have the same kinds of graphs in the real plane with one or two components: those are the real points except one point at infinity is missing from the picture. – KCd Jul 10 '23 at 18:21