An elliptic curve over K = k(C) may be regarded as a surface E over k with a map π:E -> C (in other words, an elliptic surface); the generic fibre of E is E. Under this interpretation, elements of the Mordell--Weil group E(K) are in one-to-one correspondence with sections of π. (A section is a morphism s:C -> E such that π s = IdC.) This means that one may study the Mordell--Weil group by studying the geometry of the surface.
Given E, there is a unique E up to isomorphism that is projective, regular, and relatively minimal. This is called the Kodaira--Néron model, and we will always assume that we are working with this model of the surface.
Let bar k denote the separable closure of k, and Ebar k the elliptic surface considered over bar k. The Néron--Severi group NS(Ekbar) of Ekbar is the group of divisors of Ekbar modulo algebraic equivalence. It is a finitely generated group and is closely connected with the Mordell--Weil group E(K).
Let N be the subgroup of NS(Ekbar) that is generated by all components of all the fibres of π together with the section corresponding to the zero point of E(bar k(C)); this is known as the trivial lattice of NS(Ekbar). It can easily be determined since the number of components in reducible fibres can be computed by Tate's algorithm. The following divisor classes together form a basis of N
The Galois group G=Gbar k/k acts on NS(Ekbar), and it maps N to itself. Moreover, after extending scalars to Q one can split the Galois representation. That is, there exists M⊂NS(Ekbar) tensor Q such that NS(Ekbar) tensor Q isomorphic to M direct-sum (N tensor Q) and M isomorphic to E(bar k(C)) tensor Q as G-modules. In particular, MG isomorphic to E(K) tensor Q. In the case that k is a finite field, the Frobenius action on N can be determined with the functions FrobeniusActionOnReducibleFiber and FrobeniusActionOnTrivialLattice.
In order to study NS(Ekbar) as a G-module, one can embed it in the ell-adic cohomology group H2(Ekbar, Qell). To get a G-equivariant map one must slightly change the G-action on H2(Ekbar, Qell). Let H2(Ekbar, Qell)(1) denote the (1)-Tate twist of H2(Ekbar, Qell). The main property that we need to know about this twist is that it transforms the q-eigensubspace in H2(Ekbar, Qell) of some q-Frobenius element to the 1-eigenspace of this Frobenius in H2(Ekbar, Qell)(1). The cycle class map then yields a G-equivariant embedding NS(Ekbar) tensor Qell -> H2(Ekbar, Qell)(1). It is conjectured by Tate that the image of (NS(Ekbar) tensor Qell)G under this map exactly equals H2(Ekbar, Qell)(1)G.
In the case that k is a finite field one can in principal determine H2(Ekbar, Qell)(1)G via the Lefschetz trace formula. Suppose that k⊂Fq and let Fq denote the q-th power Frobenius map. Then #E(Fq)=1 + q2 - (1 + q)Trace(Fq|H1(Cbar k, Qell)) + Trace(Fq|H2(Ekbar, Qell)).
The trace on H1(Cbar k, Qell) is zero if C is a rational curve, and it can be determined by counting Fq-rational points on C in the general case. Hence one can determine Trace(Fq|H2(Ekbar, Qell)) by counting Fq-rational points on E. By doing this for various powers of q one can determine the characteristic polynomial of Fq acting on H2(Ekbar, Qell), and hence the conjectural ranks of E(bar k(C)) and E(K) by using Tate's conjectures. The conjectural ranks obtained in this way give unconditional upper bounds on the true ranks.
As the Galois action on N can be determined, the difficult part is to compute det(1 - T.Fq|H2(Ekbar, Qell)/im(N tensor Qell)), where im(N tensor Qell) stands for the image of N tensor Qell under the cycle class map. It can be shown that this polynomial is equal to the L-function of E over K; it follows that this L-function is a polynomial. This shows that Tate's conjectures are linked to a geometric version of the Birch and Swinnerton-Dyer conjecture. Just as in the number field case, this conjecture expresses the rank and the product of the order of the Tate--Shafarevich group and the regulator of E in terms of its L-function. The L-function can be computed with the function LFunction and the conjectural information on the rank, Tate--Shafarevich group, and regulator can be obtained with the function AnalyticInformation.
If E can be defined by a Weierstrass equation in which the coefficients ai are polynomials of degree at most i, then Ekbar is a rational surface and rank(NS(Ekbar))=10. In this case rank(E(bar k(C)))=10 - rank(N) can be easily determined. In the case that k is a finite field then E(bar k(C)) and E(K) can be computed using functions in this section.