Two open research problems for you

With the kind permission of Elie Studnia, I want to share here two open research problems presented by him at the ANTS "rump session" (i.e. the ultra-short talks). The first one might be for you if you are looking for a concrete computational unsolved problem where you can start experimenting immediately. The second connects directly to the recent Chapter 3 where we discussed the complexity of an L-function, since the problem asks about a very special case of the general principle that if two elliptic curve L-functions share the same coefficients up to a bound which is small in relation to the conductor(s), then they must be equal. The question is if you can prove this special case without assuming GRH or the abc conjecture!

Here are the slides from the brief talk. If some piece of notation is unclear I'd be happy to answer questions in the comments below, and if you have ideas or code you'd like to share with others, feel free to post these as well!