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Given an elliptic curve $y^2 = x(x^2 + bx + c)$ is a non-singular curve, say $c > 0$ and $b^2 - 4c > 0$. Can we show the bound on the rank $r$ in terms of $\nu(c)$ and $\nu(b^2 - 4c)$ without quoting the Selmer groups? Well, a classical approach on $2$-descent says $$ r \leqslant \nu(c) + \nu(b^2 - 4c) - 1. $$ I suspect that we can just show $$ r \leqslant \nu(c) + \nu(b^2 - 4c) $$ by using a simpler argument, however I stuck here. Could you please give any hints?

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This is done in Silverman and Tate's "Rational points on elliptic curves" in Section 5 of Chapter III (see in particular the discussion at the top of page 85). Notice however that even though the proof does not mention Selmer groups explicitly, they are lurking in the background. The authors try hard to keep the proof elementary for undergraduates, but once you learn about Selmer groups you would quickly realize that the argument they discuss is essentially using Selmer groups without defining them as such.

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