16 votes

What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?

Here is a rough idea of the proof of 1. I'll highlight (with italics) concepts that you may not know now, but will be useful to learn in number theory, which hopefully will be motivated by this ...
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  • 115k
5 votes
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Lang's proof concerning ray class fields of imaginary quadratic number fields

I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $K$ is the ray class field of $k$ modulo $N$ - but ...
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  • 179
3 votes

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to the conjecture of Schinzel and Tijdeman (Schinzel, Α., Tijdeman, R., On the equation $y^n = P(x)$, Acta Arith. 31 (1976), 199-204), if a polynomial $P(x)$ with rational coefficients has ...
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2 votes

A constructive proof of the theorem of the cube

This is not really an answer, but a rephrasing together with some comments on why this is difficult. In summary, the question reduces to the case where the three restrictions of $D$ are trivial as ...
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2 votes
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Tri-homogenous polynomials of tridegree $(3,3,3)$ to add three points on an elliptic curve

The divisor $D$ is rationally equivalent to an effective divisor, hence $H^0(E^3,D) \neq 0$. To see this, let $p_0,p_1,p_2 \colon E^3 \to E$ be the canonical projections. For $i,j \in \{0,1,2\}$, ...
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