# 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

$$
\mathop{\rm NS}\nolimits({\cal E}_{\bar k})\otimes{\mathbb{Q}}_\ell\hookrightarrow H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)(1).
$$

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

$$
\#{\cal E}({\mathbb{F}}_q)=1+q^2-(1+q)\mathop{\rm Trace}\nolimits(F_q|H^1(C_{\bar k},{\mathbb{Q}}_\ell))+
\mathop{\rm Trace}\nolimits(F_q|H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)).
$$

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

$$
\det(1-T\cdot F_q|H^2({\cal E}_{\bar k},{\mathbb{Q}}_\ell)/\mathop{\rm im}\nolimits(N\otimes{\mathbb{Q}}_\ell)),
$$

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.