I'm starting my PhD. My advisor's papers are not exactly the most beautiful works I've ever seen (it's a matter of taste), so I'm searching by my own some topics that could fit better with me. It's not that easy.
I've gave a glimpse to the modular forms, they seems attractive. My advisor likes algebraic geometry, complex geometry, if I talk about algebraic curves he's happy; so
1) I was asking my self: what is the connection between elliptic curves and modular forms? Can someone suggest me some reference?
2)Are there open problems in the intersection of modular forms/elliptic curves?
I like complex analysis in one variable, integral transforms, special functions, that kind of stuff. I like also algebraic arguments like groups, Galois and representation theory. I know it's a vague question, but it couldn't be different.
3)Is there any direction coming into your mind? Any suggestion?
You've probably already found in researching these topics online that the problem is not finding something to read, but knowing how to narrow it down. I recommend finding the most recent PhD theses you can that discuss one or more of these topics; many are readable online as PDFs. You needn't read entire theses, but their abstracts and literature reviews will reveal what if anything they can tell you about these topics and their connections. And if you find a thesis (or other piece of research, which may lack a literature review) that's actually contributed to such areas, they're likely to conclude with an acknowledgement of further-study-is-needed areas. Theoretically, that could give you ideas.