Genus 0 curves are well understood in number theory. There is also are rich theory a bunch of conjectures about the arithmetic of elliptic curves.
This leads me to the question, what we know about higher genus curves. The only thing that comes to my mind is Falting's theorem saying that any such curve has only finitely many points.
Coming from the theory of elliptic curves one might ask a lot of questions:
Under BSD there is an algorithm computing a basis for Mordell-Weil group of the elliptic curve. Is it expected that such an algorithm will also exist for higher genus curves that determines all the rational points, or even proven to exists assuming some big conjectures such as ABC or Vojta's conjectures. Or does Matiyasevich theorem imply that such an algorithm is not possible?
Do special values of $L$-functions of these curves tell us anything interesting about these curves. I know that there is a generalization of BSD to abelian varieties and hence to Jacobians of higher genus curves, but does the $L$-functions also tells something about the curve itself.
The local-global principle does not hold for elliptic curves. However, (conjecturally
through BSD) local information does indeed give information about global properties. So I regard BSD as a fancier or refined version of local-global principle, since the pure local-global principle does not hold. Is there also a fanicer local-global principle for higher genus curves.
I am not particulary asking answers for these questions, since these are only examples and I could post more. I am rather asking for a place to look, where things like these are treated.