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This used to be a comment, but as Kevin pointed out you might never have found out that I left one. So just in case this is still of any relevance, I will repeat it here. I know, this thread is old s …

As Xandi Tuni said, most of the answers to your questions can be found in standard references. Silverman, Knapp's book on elliptic curves, Milne's book, many more (just google for elliptic curves). …

In general, for any integer $N$ and any fixed elliptic curve $E$, the elliptic curves $E'$ for which $E[N]\cong E'[N]$ as Galois modules (and such that the isomorphism respects the Weil pairing) are p …

You would usually want the principal Arakelov divisors, i.e. those of the form $(\sum_{\mathfrak{p}}{\rm ord}_{\mathfrak{p}}(a), \sum_\sigma -\log|\sigma(a)|)$ for $a\in F^\times$, to be cocompact in …

To put this one to rest, I will answer the more precise question that, after much prodding, we got Adam to formulate in the comments. I am merely paraphrasing a comment of Qiaochu. If you are given t …

$l$-adic cohomology has been used in a crucial way to investigate the representation theory of finite groups of Lie type. Basically, all irreducible characters of such a group are summands of characte …

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 …

In addition to Felipe's reference, you can also have a look at Tom Fisher's papers https://www.dpmms.cam.ac.uk/~taf1000/papers/congr7and11.html and https://www.dpmms.cam.ac.uk/~taf1000/papers/congr9.h …