An Overview of Relevant Theory#
An elliptic curve over \(K = k(C)\) may be regarded as a surface \({\cal E}\) over \(k\) with a map \(\pi:{\cal E}\rightarrow C\) (in other words, an elliptic surface); the generic fibre of \({\cal E}\) is \(E\). Under this interpretation, elements of the Mordell–Weil group \(E(K)\) are in one-to-one correspondence with sections of \(\pi\). (A section is a morphism \(s:C\rightarrow{\cal E}\) such that \(\pi\circ s = \mathop{\rm Id}\nolimits_C\).) This means that one may study the Mordell–Weil group by studying the geometry of the surface.
Given \(E\), there is a unique \({\cal E}\) 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 \({\cal E}_{\bar k}\) the elliptic surface considered over \(\bar k\). The Néron–Severi group \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\) of \({\cal E}_{\bar k}\) is the group of divisors of \({\cal E}_{\bar k}\) 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 \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\) that is generated by all components of all the fibres of \(\pi\) together with the section corresponding to the zero point of \(E(\bar k(C))\); this is known as the trivial lattice of \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\). 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 \(N\)
- (i)
the image of the section corresponding to the zero point;
- (ii)
one complete fibre; and
- (iii)
the components of all the reducible fibres, with one component from each fibre omitted.
It is known that the quotient \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})/N\) is generated by images of sections of \(\pi\), and that \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})/N\cong E(\bar k(C))\) (via the identification of sections with points). In particular, this implies the Shioda–Tate formula \(\mathop{\rm rank}\nolimits(E(\bar k(C))) + 2 + \sum_{v\in C(\bar k)} (m_v-1) = \mathop{\rm rank}\nolimits(\mathop{\rm NS}\nolimits({\cal E}_{\bar k}))\) where \(m_v\) denotes the number of components of the fibre \(\pi^{-1}(v)\).
The Galois group \(G=G_{\bar k/k}\) acts on \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\), and it maps \(N\) to itself. Moreover, after extending scalars to \({\mathbb{Q}}\) one can split the Galois representation. That is, there exists \(M\subset\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\otimes{\mathbb{Q}}\) such that \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\otimes{\mathbb{Q}}\cong M\oplus (N\otimes{\mathbb{Q}})\) and \(M\cong E(\bar k(C))\otimes{\mathbb{Q}}\) as \(G\)-modules. In particular, \(M^G\cong E(K)\otimes{\mathbb{Q}}\). 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 \(\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\) as a \(G\)-module, one can embed it in the \(\ell\)-adic cohomology group \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)\). To get a \(G\)-equivariant map one must slightly change the \(G\)-action on \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)\). Let \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)(1)\) denote the \((1)\)-Tate twist of \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)\). The main property that we need to know about this twist is that it transforms the \(q\)-eigensubspace in \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)\) of some \(q\)-Frobenius element to the \(1\)-eigenspace of this Frobenius in \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)(1)\). The cycle class map then yields a \(G\)-equivariant embedding
It is conjectured by Tate that the image of \((\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\otimes{\mathbb{Q}}_\ell)^G\) under this map exactly equals \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)(1)^G\).
In the case that \(k\) is a finite field one can in principal determine \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)(1)^G\) via the Lefschetz trace formula. Suppose that \(k\subset{\mathbb{F}}_q\) and let \(F_q\) denote the \(q\)-th power Frobenius map. Then
The trace on \(H^1(C_{\bar k},{\mathbb{Q}}_\ell)\) is zero if \(C\) is a rational curve, and it can be determined by counting \({\mathbb{F}}_q\)-rational points on \(C\) in the general case. Hence one can determine \(\mathop{\rm Trace}\nolimits(F_q|H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell))\) by counting \({\mathbb{F}}_q\)-rational points on \({\cal E}\). By doing this for various powers of \(q\) one can determine the characteristic polynomial of \(F_q\) acting on \(H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)\), 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
where \(\mathop{\rm im}\nolimits(N\otimes{\mathbb{Q}}_\ell)\) stands for the image of \(N\otimes{\mathbb{Q}}_\ell\) 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 \(a_i\) are polynomials of degree at most \(i\), then \({\cal E}_{\bar k}\) is a rational surface and \(\mathop{\rm rank}\nolimits(\mathop{\rm NS}\nolimits({\cal E}_{\bar k}))=10\). In this case \(\mathop{\rm rank}\nolimits(E(\bar k(C)))=10-\mathop{\rm rank}\nolimits(N)\) 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.