In his famous Mordell paper, Faltings proved that two abelian varietes $ A_1, A_2$ defined over a number field $ K$ are isogenous if and only if the local $ L$ -factors of $ A_1, A_2$ are equal at every finite place of $ K$ . Moreover, there is an analogue of the Birch/Swinnerton-Dyer conjecture for abelian varieties over number fields which predicts, among other things, than the $ L$ -funciton of $ A/K$ “knows” the Mordell-Weil rank of $ A(K)$ , since it is equal to the order of vanishing at $ s = 1$ of the $ L$ -function. Further, in the same paper Faltings proved that it suffices to show that for all but finitely many places $ v$ of $ K$ , the local factors agree.

Thus, the $ K$ -isogeny class of $ A/K$ can be seen to ‘know’ important arithmetic information about $ A$ , and in the case when $ A$ is the Jacobian of a curve $ C$ , also important information about $ C$ .

However, we know that there exist many examples (elliptic curves, say) of abelian varieties $ A_1, A_2$ which are isomorphic over $ \overline{\mathbb{Q}}$ (for example, quadratic twists of curves) but not isomorphic over their field of definition $ K$ , such that $ A_1(K), A_2(K)$ have different ranks. For example, it is known that for any elliptic curve $ E/\mathbb{Q}$ , $ E$ has many quadratic twists with rank at least two. Starting from a curve with rank 0 say, this produces many examples of elliptic curves that observe the aforementioned phenomenon. Therefore, the $ \overline{\mathbb{Q}}$ -isomorphism class of an abelian variety $ A$ need not see much of the arithmetic over $ K$ .

Nevertheless, if $ A_1, A_2$ are isomorphic over $ \overline{\mathbb{Q}}$ then they will become isomorphic over some finite extension $ M/K$ . Thus, the $ L$ -series of $ A_1, A_2$ over $ M$ ought to become identical, since they will then be isogenous over $ M$ .

It thus seems that the $ L$ -function of an abelian variety $ A/K$ depends subtly on the field $ K$ itself. The question is, for a given abelian variety $ A/K$ , how does the $ L$ -function $ L_M(A,s)$ , given by viewing $ A$ as an abelian variety over $ M/K$ for a finite extension of $ M$ , behave as a function of $ M$ ?