The CTO is not responsible for your application, except at very small companies (for example at my current company the CTO is also the hiring manager for all dev positions so he does know these things, but this is the exception not the rule). So I wouldn't ask him directly about your application in particular, because he does not know, nor does he care (he has other more important things to do).
What you can do is:
1) Introduce yourself. It's likely at least that there is a hiring committee, of which the CTO may be a part of. He may see your application eventually, and building a personal relationship is worthwhile.
2) Express interest in his company, in a general sense. Explain that you have applied there, and you are interested in his products/work/research/whatever else it is that interests you about the company.
3) Ask if he can suggest ways to make your application more attractive (if you haven't yet applied), or if there is a person you can reach out to directly for recruitment matters (if you have, or even if you haven't). It's always good to have a direct contact with a first-party company recruiter for any company you want to apply for, rather than sending your application into the void.
4) If after performing the above steps, you feel like you have a good rapport with the CTO of the company, you can ask him for his business card or direct contact information to follow up and keep in touch. There's nothing wrong with trying to build a professional network with a company CTO. However he may decline, and that's ok; he's a busy dude and probably doesn't have time to field random emails from random university students. Don't take it personally. If you do receive his personal contact info, don't treat him like a recruiter or like HR; don't follow up with him just to ask about the status of your application or whatnot, that's neither his job nor his interest. If you want to follow up with him, you can ask him about things like his research, developments in the company, software engineering ideas/strategies, and so on. But remember, he's not HR and he's not going to help you with your application.