An elliptic surface is a surface that admits an elliptic fibration. Its generic fiber is an elliptic curve over a function field but special fibers may be singular. After giving the Kodaira-Neron classification of possible singular fibers and related geometric invariants, we build the neat connection between the geometry of elliptic surfaces and arithmetic of elliptic curves. This allows us to reprove the finite generation of the Mordell-Weil group of an elliptic curve and to further classify its possible rank and torsion using techniques from intersection theory and lattice theory. The theory of Mordell-Weil lattices plays an important role in finding elliptic curves over of high rank via specialization on elliptic surfaces. Our main sources are  and . See also ,  and . This is a note prepared for the Baby Algebraic Geometry Seminar at Harvard.
By definition, the generic fiber of an elliptic surface is a smooth curve of genus 1 over the function field . Let be a section of . Then gives a rational point . Conversely, let . Let be the closure of in . We obtain a surjective birational morphism , which is an isomorphism since is smooth. In this way we have exhibited a bijection
Given an elliptic curve over , there are different ways to extend to an elliptic surface over giving rise to the generic fiber. However, all these models are birational, so if we require that is relatively minimal, i.e., the fibers do not contain -curves, then is unique up to isomorphism (the uniqueness will follow from the classification of singular fibers, see [4, II.1.2]). We obtain the following correspondence: The explicit description of the relatively minimal model is given by Kodaira in characteristic 0 and by Neron in general. The elliptic surface thus associated to is sometimes called the Kodaira-Neron model of .
As already alluded, the theme of this talk is to relate the geometry of the elliptic surface and the arithmetic of the elliptic curve .
How do the singular fibers of an elliptic surface look like? There are many ways to classify the possible singular fibers. Here we use the explicit equations.
It may not be a good idea if I keep on blowing-up for 3 hours. Let me tell you the result instead. In fact, one can determine the singular type from the equation using the so-called Tate's algorithm over any perfect field, as demonstrated above.
Write for short. A point determines a section of . Denote the image curve by . A point also gives us a divisor , the fiber above . Every divisor on the elliptic surfaces can be written as the sum of such horizontal and vertical divisors. Let be an divisor, then we can decompose . The horizontal part intersects at a divisor on , which gives a point using the group law. So we have a map The kernel of is where is the group of divisors algebraically equivalent to 0. To see that , we use the fact ([4, VII.1.1]) that is an isomorphism to conclude that any divisor in is linearly equivalent to a vertical divisor in . So we obtained that
This theorem relates the arithmetic of and the geometry of . It is well-known that the Neron-Severi group is finitely generated for any smooth projective variety (the theorem of the base). Consequently, we have reproved the Mordell-Weil theorem using an argument of geometric nature.
Our next goal is to further study the structure of (e.g., its rank and torsion) using the geometry of .
The number is called the Picard number of . By the Hodge index theorem, the lattice equipped with the intersection pairing has signature . We immediately find the following bound on the rank of :
As discussed above, is bounded above by , the second Betti number (it is even bounded by if ). So we need more knowledge about the trivial lattice .
Because all fibers are algebraically equivalent, we know that Note that the intersection matrix of is which is non-degenerate with determinant and signature . Also, is a root lattice of type , hence the intersection matrix, denoted by , is negative definite. Therefore, the above decomposition of is actually a direct sum. It follows that
Therefore we can really compute the rank of as long as we know all the singular fibers. Conversely, knowing the possible rank of will help us to classify configurations of singular fibers of elliptic surfaces.
How about the torsion? The crucial idea is to endow a height pairing. We already know that and possesses an intersection pairing. So it is natural to construct a splitting of this isomorphism so that we can embed into .
We need the following theorem due to Kodaira.
The following is not quite a ``splitting'', nonetheless is good enough for our purpose.
One can check that is also a group homomorphism. So we can define a pairing on using the pairing on .
The following is easily deduced.
Now from the explicit formula for , we can also write down the height pairing explicitly, where is a positive number only depending on the the fiber components of meeting and .
We can apply the height pairing to deduce some information about torsion groups. Recall that is the component group of the singular fiber . The map sending a section to the simple fiber components it meets is a group homomorphism.
In this way, the singular types of the elliptic surface impose very strong constraints on the torsion group of , and vice versa.
How can we identify with a sublattice of ? A bit of lattice-theoretic and intersection-theoretic computation gives the precise answer as long as is unimodular.
We now step toward the case study of rational elliptic surfaces, where the lattice-theoretic method has achieved huge success in classifying all possible structures of .
Here comes a clever way to classify relying on the fact there is only one unimodular even positive definite lattice of rank 8, namely the root lattice . Consider the complementary lattice of the rank 2 sublattice . Then is unimodular, even and positive definite, so it must be ! Since and each is a root lattice. We only need to find all possible embeddings of a root lattice into . If you know , it is just so simple — there are only 74 cases. All the possible shapes of are beautifully classified by Oguiso and Shioda .
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