# Wiman's method for bounding the rank of an elliptic curve

In 1945 Wiman [W] showed that certain elliptic curves $$E$$ over $$\mathbf Q$$ have rank* at least 4. (It seems this was the highest known rank of an elliptic curve over $$\mathbf Q$$ until 1974, when Penney--Pomerance found a curve of rank at least 6.)

The method of his proof appears to be rather elementary, defining a map from $$E(Q)$$ to a certain abelian 2-group $$A$$ by using the $$p$$-valuations of the $$x$$-coordinate of a point $$(x,y) \in E(\mathbf Q)$$ for various primes $$p$$.

However, due to some combination of my inadequate German and Wiman's somewhat archaic mathematical style, I cannot decipher the exact definition of the group $$A$$ and the map $$E(\mathbf Q) \rightarrow A$$. So I ask

Question: What is the precise definition, in modern terms, of Wiman's group $$A$$ and map $$E(\mathbf Q) \rightarrow A$$?

[W] Wiman, A., Über den Rang von Kurven $$y^2=x(x+a)(x+b)$$, Acta Math. 76, 225-251 (1945). ZBL0061.07109.

*: A bit confusingly, "rank" is used in this paper to mean "minimal number of generators of $$E(\mathbf Q)$$" rather than "minimal number of generators of $$E(\mathbf Q)/E(\mathbf Q)_{\operatorname{tors}}$$", so to get the "correct" rank one has to subtract 2 from each of the values reported by Wiman.

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I have not read all the details of the article, but most of what I see is just descent by the isogeny $$[2]$$. The map is the Kummer map $$E(\mathbb{Q})\to \mathbb{Q}^{\times}/\square \times \mathbb{Q}^{\times}/\square \times \mathbb{Q}^{\times}/\square$$ which sends $$(x,y)$$ to $$(x-e_1,x-e_2,x-e_3)$$. In modern terms this is the connecting homomorphism $$E(\mathbb{Q})\to H^1\bigl(\mathbb{Q}, E[2]\bigr)$$. The kernel of the map is $$2E(\mathbb{Q})$$ and the image in each of the component lies in the group generated by the prime divisors of $$\Delta$$. Looking at valuations at those primes corresponds to studying the image of the local map $$E(\mathbb{Q}_p) \to \mathbb{Q}_p^{\times}/\square\times\dots$$. So the article actually studies the $$2$$-Selmer group. Similar to Selmer's work around the same time using $$3$$-descent. The idea to define Selmer groups and relate it to Galois cohomology only appears (if I am not mistaken) later in the work of Cassels, Lang, Tate, ...