I am reading the following article on isomorphism classes of elliptic curves over a finite field https://pdfs.semanticscholar.org/7280/b7c66cf02f1d43cbe042d7a6f6b4b7de269c.pdf and noticed some open problems at the end of the paper, which I might be interested in working on.
I was wondering if these had perhaps already been solved by others or if some significant progress had been made, as I do not wish to embark on a study of an open problem if it has essentially already been done by someone else.
For example, one of the open problems is to take an elliptic curve $E$ in Weierstrass form:
$y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6, \:\:\:a_i \in \mathbb{Z}$
such that all the coefficients are $M^{o(1)}$. Is it true that the number of integer points $(x,y) \in [0,M] \times [0,M]$ on $E$ is $M^{o(1)}$?
There are also two other problems mentioned in the paper, but you will have to read the paper for those as it would take a while to explain the notation. If anyone has more information or suggestions, that would be greatly appreciated.
