It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures.
The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation.
For instance, the canonical Néron–Tate height is a quadratic form with remarkable properties that appear in the statement of the Birch and Swinnerton-Dyer conjecture.
Here a refined theory of (in effect) a right adjoint to reduction mod p — the Néron model — cannot always be avoided.
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.
Since the time of Carl Friedrich Gauss (who knew of the lemniscate function case) the special role has been known of those abelian varieties
This is seen in their L-functions in rather favourable terms – the harmonic analysis required is all of the Pontryagin duality type, rather than needing more general automorphic representations.