A Integral Points of an Elliptic Curve over a Cyclotomic Tower

[lugita15]

$\mathbb{Q}(\zeta_{p^\infty})$, also written as $\mathbb{Q}(\mu_{p^\infty})$ or $\mathbb{Q}(p^\infty)$, denotes $\mathbb{Q}$ adjoined with the $p^{n}$th roots of unity for all $n$. It's the union of a cylotomic tower, and it's studied in subjects like Iwosawa theory and class field theory.

Let $R$ be the ring of integers of this field, and let $\lambda$ be a nonzero element of $R$.
My question is, are there always only finitely many ordered pars $(x,y)$of elements of $R$ such that $y^2-x^3+8\lambda x=0$? Or does there exist a $\lambda$ for which there are infinitely many such ordered pairs?

 

[WWGD]

Seems to be along @mathwonk 's alley.

 

[mathwonk]

I am not a number theorist, and the author of this post is clearly more knowledgable than I am. However, I have noticed a post on math overflow, also noticed by the author, from which the answer seems to be "yes", provided the prime p is "good", due to the theorems of Rohrlich and Siegel. I.e. according to that post, Rohrlich proves this elliptic curve is equal to one defined over a number field, and Siegel proves such curves over number fields have finitely many integral points. Perhaps this does not work if the notion of integral points does not carry over from the tower to the number field? Or perhaps the problem is that the constant lambda can be chosen so that the prime is no longer "good"?
https://mathoverflow.net/questions/...ank-growth-in-iwasawa-tower?noredirect=1&lq=1

http://matwbn.icm.edu.pl/ksiazki/aa/aa68/aa6827.pdf

Well, a little reading reveals that a prime p is "bad" for an elliptic curve E, iff E becomes singular mod p. Thus in our case p becomes bad if and only if p divides 8.lambda. So I guess a basic question here is whether the curve y^2 = x^3 - 8px, has only a finite number of integral points in R. Notice the author has required lambda ≠ 0, since otherwise the pairs (a^2, a^3) all solve, for any a ≠ 0 in R.

 

[lugita15]

I am not a number theorist, and the author of this post is clearly more knowledgable than I am. However, I have noticed a post on math overflow, also noticed by the author, from which the answer seems to be "yes", provided the prime p is "good", due to the theorems of Rohrlich and Siegel. I.e. according to that post, Rohrlich proves this elliptic curve is equal to one defined over a number field, and Siegel proves such curves over number fields have finitely many integral points. Perhaps this does not work if the notion of integral points does not carry over from the tower to the number field? Or perhaps the problem is that the constant lambda can be chosen so that the prime is no longer "good"?
https://mathoverflow.net/questions/...ank-growth-in-iwasawa-tower?noredirect=1&lq=1

http://matwbn.icm.edu.pl/ksiazki/aa/aa68/aa6827.pdf

Well, a little reading reveals that a prime p is "bad" for an elliptic curve E, iff E becomes singular mod p. Thus in our case p becomes bad if and only if p divides 8.lambda. So I guess a basic question here is whether the curve y^2 = x^3 - 8px, has only a finite number of integral points in R. Notice the author has required lambda ≠ 0, since otherwise the pairs (a^2, a^3) all solve, for any a ≠ 0 in R.

“I am not a number theorist, and the author of this post is clearly more knowledgable than I am.” Actually I know nothing at all about elliptic curves. It’s just that this question arose in my research as a logician.

 

[mathwonk]

I believe you, but my comment is based on the fact that, until I did some research, I didn't even know what a cyclotomic tower was. (I did know how to spell Iwasawa though, which maybe was a bit of a tip- off.)

 

[WWGD]

I believe you, but my comment is based on the fact that, until I did some research, I didn't even know what a cyclotomic tower was. (I did know how to spell Iwasawa though, which maybe was a bit of a tip- off.)

Not that it's any consolation, but I didn't know what a Cyclotomic polynomial was. I had heard the term, but didn't know what it was, let alone a Cyclotomic Tower.