L-functions
For abelian varieties such as Ap, there is a definition of local zeta-function available. To get an L-function for A itself, one takes a suitable Euler product of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the Tate module of A, which is (dual to) the étale cohomology group H1(A), and the Galois group action on it. In this way one gets a respectable definition of Hasse–Weil L-function for A. In general its properties, such as functional equation, are still conjectural – the Taniyama–Shimura conjecture (which was proven in 2001) was just a special case, so that's hardly surprising.
It is in terms of this L-function that the conjecture of Birch and Swinnerton-Dyer is posed. It is just one particularly interesting aspect of the general theory about values of L-functions L(s) at integer values of s, and there is much empirical evidence supporting it.
Read more about this topic: Arithmetic Of Abelian Varieties