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Results tagged with arithmetic-geometry
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user 35416
Diophantine equations, rational points, abelian varieties, Arakelov theory, Iwasawa theory.
17
votes
Accepted
Order of Ш (Sha)
No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish fi …
- 13k
13
votes
Fermat's last theorem over larger fields
There might well be an elementary construction of infinitely many points (which I cannot think of right now), but in any case, I think that there are experts out there who expect there to be infinitel …
- 13k
9
votes
Does a curve have infinitely many $K$-rational points under these hypotheses?
Mazur in his article "Rational points of abelian varieties with values in towers of number fields", Invent. Math. 18 has produced lots of curves over number fields that have a finite number of points …
- 13k
6
votes
Accepted
How locally ubiquitous are totally real fields?
The answer to the first question is "yes". See this paper of the Dokchitser brothers, Lemma 3.1 for the case where $K/\mathbb{Q}_p$ is Galois. In the general case, apply the result to the Galois closu …
- 13k
2
votes
Rank of jacobians of twists of hyperelliptic curves of genus one
If you are willing to assume finiteness of Sha, then the answer is "yes".
The first claim, that for infinitely many $d$ (square-free, presumably?) this has a rational point, is easy and does not need …
- 13k