On to higher genus curves

Let us summarize what we have discussed about Elliptic curves. First of all, we can parameterize Elliptic curves by either studying the moduli of lattices in the complex plane, or by looking at the Hodge decomposition of their first cohomology group, H^1(E, \mathbb{C}).

We also discussed how maps between elliptic curves are in fact determined by maps on the underlying lattices, in effect reducing the problem of studying maps to one of linear algebra. Let us discuss this somewhat further.

The first thing to note is that any map between two elliptic curves is unramified. That is, a map E_1 \to E_2 is in fact a covering map. This can be seen quickly with the Riemann-Hurwitz formula: given a map f : C \to C' of curves, we have the relationship

\chi(C) = d \cdot \chi(C') - \deg R

where d is the degree of the map, and R is the ramification divisor. Since \chi(E) = 0, it follows that \deg R = 0, and so the map is unramified.

Consequently, through the theory of covering spaces, there is a Galois correspondence between covers of an Elliptic curve E and subgroups of the fundamental group \pi_1(E) \cong \mathbb{Z} \oplus \mathbb{Z}. It then follows that degree d covers of an elliptic curve are in bijective correspondence with sublattices of \mathbb{Z} \oplus \mathbb{Z} of index d. As an aside, it is easy enough to show that there are exactly

\displaystyle \sigma_1(d) = \sum_{k \mid d} k

such covers.

So how can we carry this on to higher genus curves? Since maps between higher genus curves may be ramified (for example, any map of a higher genus curve to an elliptic curve necessarily has ramification), we can no longer look directly to the theory of covering spaces and the resulting analysis of the fundamental group. Instead what we will do, in some sense, is to linearize our maps of curves. To do so, we will introduce the Jacobian of a curve. It will naturally follow that Elliptic curves are those curves which are isomorphic to their Jacobians, and so all of the above discussion can be seen in some sense as a toy model of what is to follow.

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About charlesflorian

Mathematician. Climber. Living in Copenhagen.
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