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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
- (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 NS(Ekbar)/N is generated by images
of sections of π, and that NS(Ekbar)/N isomorphic to E(bar k(C))
(via the identification of sections with points). In
particular, this implies the Shioda--Tate formula
rank(E(bar k(C))) + 2 + ∑v∈C(bar k) (mv - 1) =
rank(NS(Ekbar)) where mv denotes the number of components of the
fibre π - 1(v).
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.
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