This is more of what I’m currently looking at, if you have any suggestions for any of them do contact me and we can discuss them!
The Smart attack solves the discrete log problem of points on an elliptic curve over \(\mb F_q\) by lifting to \(\mb Q_q\). However some lifts fail. So far I have a script to find every lift that fails and a mostly complete (I hope) proof that it is indeed all lifts. The goal is to find curves in \(\mb Q_p\) of the form \(y^2=x^3+ax+b\) such that the attack fails and \(0\leq a,b<p\). However the only case I know off is when \(a=0\).
The maskit slice is a moduli space of punctured torus subgroups groups of \(\PSL[2]{\mb C}\). It turns out that when we take a limit inside this Teichmüller space, the algebraic and geometric limits may be different. So far I have some rough computations on what the geometric limit may be but nothing really proven so far.
We know that \(\left(\frac{\mc O_K}{I}\right)^*\) can be written in terms of product of cyclic groups as it is finite. Can we find an explicit representation of this product? My current progress can be found in the blogs section.