Divisors and the Picard Group

In this post, once again focusing on the subject of algebraic geometry, we will consider a “curve”, which, confusingly, refers what we usually think of as a surface. The reason for this is that if we are considering complex numbers $x$ and $y$, an equation such as $y^{2}=x^{3}-x$, which we would normally think of as a “curve” if $x$ and $y$ were real numbers, actually refers to something that looks like a surface, in the same way the real numbers form a line and complex numbers form a plane. We will rely on this intuition and leave the more formal definitions of curves, surfaces, and dimension to the references for now.

A divisor is a finite “linear combination” of points on the curve, with integer coefficients. For example, if we have points $P_{1}$ and $P_{2}$ on the curve, we can have something like

$\displaystyle 5P_{1}-3P_{2}$.

The degree of a divisor is the sum of its coefficients. For the example above, the degree is equal to $2$.

A special kind of divisor called a principal divisor comes from  so-called “rational functions” (which, despite the name, may not really be “functions” in the set-theoretic sense but merely expressions involving a “fraction” whose numerator and denominator are both polynomials) on the curve. We let the coefficients of each point denote the “order of vanishing” of the function. For instance, the function

$\displaystyle \frac{x(x-1)^{2}}{(x-3)^5}$

gives rise to the principal divisor

$(f)=P_{1}+2P_{2}-5P_{3}$

where $P_{1}$ is the point $x=0$$P_{2}$ is the point $x=1$, and $P_{3}$ is the point $x=3$.

The Picard group of a curve is a group (whose law of composition is given by addition – see also Groups) obtained from the divisors by considering two divisors $D$ and $D'$ equivalent (see Modular Arithmetic and Quotient Sets) if their difference $D-D'$ is a  principal divisor. An element of the Picard group is also called a divisor class.

The Picard group of a curve can say a lot of things about the curve. For instance, it can be used to prove that on the curve $y^{2}=x^{3}-x$, which is an example of what is called an elliptic curve, the points form a group. The group structure on the elliptic curve, along with other properties such as its being a Riemann surface (a surface which “locally” looks like the complex plane), makes it one of the most interesting objects in mathematics.

The Picard group is also important because its elements, the divisor classes on the curve, correspond to line bundles (vector bundles of dimension $1$ – see Vector Fields, Vector Bundles, and Fiber Bundles – but do keep in mind our discussion earlier regarding complex numbers and how this changes our conventions regarding dimension, as in the case of the line and the plane, and curves and surfaces) on the curve. Line bundles are also related to sheaves, in particular those called “locally free sheaves of rank $1$” (more general vector bundles correspond to locally free sheaves of finite rank). There is, therefore, a relation between the concept of divisors, the concept of vector bundles, and the concept of sheaves.

We now relate the theory of divisors and the Picard group to number theory. We have mentioned in Localization that we can obtain a scheme out of the integers $\mathbb{Z}$; the points of this scheme are the prime ideals of $\mathbb{Z}$, and the set of all these points (prime ideals) we call $\text{Spec }\mathbb{Z}$. As we can make a scheme out of a more general ring, we can therefore make a scheme out of the ring of integers $\mathcal{O}_{K}$ of an algebraic number field $K$ (see Algebraic Numbers); its points will be the prime ideals of $\mathcal{O}_{K}$, and the “rational functions” on this scheme will be the elements of $K$.

In this case, the divisors are made up of “linear combinations” of prime ideals. The principal divisors, which come from rational functions, then correspond, accordingly, to principal fractional ideals, ideals which are generated by a single element of $K$, which as we have mentioned above correspond to the rational functions. Finally, the Picard group is none other than the ideal class group, which “measures” the failure of unique factorization in an algebraic number field!

More explicitly, an example of a divisor may be written in this way:

$\displaystyle 5\mathfrak{p}_{1}-3\mathfrak{p}_{2}$

for prime ideals $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$, which as we have said correspond to points. For a principal divisor, we may have, for example, the following element of the rational numbers $\mathbb{Q}$

$\displaystyle \frac{63}{64}$

which generates the principal fractional ideal

$\displaystyle (\frac{63}{64})=\{...,-\frac{189}{64},-\frac{126}{64},-\frac{63}{64},0,\frac{63}{64},\frac{126}{64},\frac{189}{64},...\}$

which in turn gives us the principal divisor

$\displaystyle 2\mathfrak{p}_{1}+\mathfrak{p}_{2}-6\mathfrak{p}_{3}$

where $\mathfrak{p}_{1}=(3)$$\mathfrak{p}_{2}=(7)$, and $\mathfrak{p}_{3}=(2)$, the principal ideals generated by $3$, $7$, and $2$ respectively. Note that if we “factorize” the numerator and denominator of $\frac{63}{64}$, we obtain

$\displaystyle \frac{63}{64}=\frac{3^{2}\cdot 7}{2^{6}}$.

More generally, we should “factorize” in terms of ideals, in case we don’t have unique factorization:

$\displaystyle (\frac{63}{64})=\frac{(3)^{2}(7)}{(2)^{6}}$.

The coefficients of a principal divisor, measuring “how much” of a certain prime is in the factorization of the principal fractional ideal it corresponds to, are called valuations. The theory of valuations offers us another way to develop the entire field of algebraic number theory under a new perspective.

References:

Divisor on Wikipedia

Picard Group on Wikipedia

Algebraic Geometry by Robin Hartshorne

Algebraic Number Theory by Jurgen Neukirch