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:


1. 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?
2. 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.
3. 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.


Thanks



The arithmetic of higher genus curves
There is a conjecture (or variants of one) due to Scharaschkin, Skorobogatov and Stoll predicting that Brauer-Manin or finite descent obstructions (these are fancy forms of a local-global principle) determine all rational points. This gives an algorithm to determine if the set of points is empty or not. See Stoll, Algebra and Number Theory 2 (2008), 595–611. The algorithm is actually practical, as demonstrated by Bruin and Stoll (for genus two over the rationals). This answers your 3 and part of 1.


For the rest of 1, Vojta's conjecture or ABC (with some choice of constant which in the original conjectures is not specified) would give an upper bound for the height of rational points on a curve and would allow the determination of all such points by a finite search.


The answer for 2 is (likely) "no".


