What is Arithmetic Geometry?


Arithmetic geometry focuses on the study of various height functions. The most common types of height functions that are being studied include:

Honestly, I am still (and perhaps always will be) in the process of learning these notions. However, they all developed naturally and consecutively from the simplest form of height: canonical heights, where the very reason of arithmetic geometry lies.

Canonical height is motivated by the idea of evaluating the arithmetic (or equivalently, algebraic) complexity of a rational point. For any \(\alpha\in\mathbb{Q}\), we want to define a so-called "size" of it, denoted by \(\text{size}(\alpha)\), so that the "size" can reflect the arithmetic complexity of \(\alpha\). Canonical height is a generalization of this "size".

Importantly, we want that there are only finitely many number of rational points of bounded "size", i.e. we want the set \(\{\alpha\in\mathbb{Q}:\text{size}(\alpha)\leq B\}\) to be finite for any constant constant \(B\). We require this finiteness due to technical and yet natural reasons. For example, a classical problem is to show that some specific diophantine equation has finitely many solutions. However, one could show that the height of the solutions is bounded, and since there are only finitely many points of bounded height, the set of solutions is finite. Similarly, this finiteness condition gives height function much more power to be utilized in many problems in both classical and modern algebraic geometry.

From Integers to Rationals

Let us firstly consider an integer \(a\in\mathbb{Z}\). An usual way to determine the arithmetic complexity of an integer is the length of this binary expansion. For example, consider \(10\in\mathbb{Z}\), since \(10=2+8=0\cdot 2^{0}+1\cdot 2^{1}+0\cdot 2^{2}+1\cdot 2^{3},\) the binary expansion of it is \(1010\). Since the binary length of a positive integer \(a\) equals to \(\lfloor \log_{2}(a)\rfloor+1\), a natural choice of \(\text{size}(a)\) is then to define $$\text{size}(a):=\log|a|\approx \text{binary length}(a)-1.$$ Moreover, as there are only finitely many of integers of bounded absolute values, the set \(\{a\in\mathbb{Z}:\text{size}(a)\leq B\}\) is finite for any \(B\).

When \(\alpha\in\mathbb{Q}\) is a rational number, one may consider to define \(\text{size}(\alpha):= \log|\alpha|\). However, this is a bad choice. To begin with, there are infinitely many rational numbers of bounded absolute values, and thus this size does not satisfy the finiteness condition. Furthermore, it does not reveal the arithmetic complexity of \(\alpha\). For example, \(\frac{10000001}{10000000}\) and \(1\) have really close absolute values, but the former is much more arithmetically complex than the later. Write \(\alpha=\frac{a}{b}\) with \(\gcd(a,b)=1\). One way to modify the definition of this size is to determine the arithmetic complexity of \(\alpha\) by the arithmetic complexity of its numerator \(a\) and denominator \(b\) as integers. Hence, one way to do this is to define $$\text{size}(\alpha):=\max\{\text{size}(a),\text{size}(b)\}=\log\max\{|a|,|b|\}.$$ In other words, between the numerator and the denominator, we choose the one that is more arithmetically complex to represent the arithmetic complexity of \(\alpha\). Moreover, it satisfies the finiteness condition, since there are only finitely many integers of bounded absolute value. Note that this is not the only feasible construction, you can also define the size to be \(\log|a|+\log|b|\), which defines the complexity of \(\alpha\) to be the total complexity of \(a\) and \(b\), and it just turned out the above definition has been working better in the field of arithmetic geometry.

From Rationals to Rational Projective Points

We now generalize this idea to homogenous vectors \(P\in\mathbf{P}^{n}(\mathbb{Q})\) with rational coordinates. Write \(P=[\alpha_{0}:\dots:\alpha_{n}]=[\frac{a_{0}}{b_{0}}:\dots:\frac{a_{n}}{b_{n}}].\) We can multiply the common denominator, which is \(m:=\text{lcm}(b_{0},\dots, b_{n})\), then $P$ can be rewritten as $$P=[ma_{0}:\dots:ma_{n}].$$ Now, we can factor out all the common divisors among \(ma_{0},\dots, ma_{n}\), which is divided by \(g:=\gcd(ma_{0},\dots, ma_{n})\), so that $$P=[c_{0}:\dots:c_{n}],$$ where \(c_{i}=ma_{i}/g\in\mathbb{Z}\) and \(c_{i},c_{j}\) are pairwise coprime. Then, we can define $$\text{size}(P)=\text{size}([\alpha_{0}:\dots:\alpha_{n}])=\text{size}([c_{0}:\dots:c_{n}]):=\log\max\{|c_{0}|,\dots,|c_{n}|\}.$$

For \(P\) a rational point in the projective space and for \(\text{size}(P)\) expressed defined above, for any constant \(B\), the set \(\{P\in\mathbf{P}^{n}(\mathbb{Q}):\text{size}(P)\leq B\}\) has at most \((2 \lceil e^{B}\rceil +1)^{n+1}\) elements. Hence, the size defined above satisfies the finiteness condition.

From Rational Projective Points to Canonical Heights

Canonical Height is a generalization of the \(\text{size}(P)\) when \(P\) is a homogenous vector with rational coordinates. From rational, we want to generalize the above notion of "size" to homogenous vector with algebraic coordinates.

However, an algebraic number lies in number field, and it invovles the theory of absolute values to extend the absolute value \(|\ \ |\) to some absolute values to the number fields. There is not an easy way to make it clear in this small page. But one could immeidately see the similarity after I define the canonical height.

Let \(\overline{\mathbb{Q}}\) be a choice of an algebraic closure of \(\mathbb{Q}\). We consider the projective space \(\mathbf{P}^{n}(\overline{\mathbb{Q}})\). Let \(P\) be a point of \(\mathbf{P}^{n}(\overline{\mathbb{Q}})\) be represented by a homogenous non-zero vector \(\mathbf{x}:=[x_{0}:\dots:x_{n}]\) with coordinates in a number field \(K\). Then, we set $$h(\mathbf{x}):=\sum_{v\in M_{K}}\log\max_{j}|x_{j}|_{v},$$ where \(M_{K}\) is the set of all non-trivial inequivalent absolute values on the number field \(K\), i.e. the set of places in \(K\).

We will talk about this it much more details in need a link to canonical height page. For now, let us see if the above definition makes sense in a particular number field we just saw: the field of rational number \(\mathbb{Q}\). Do not forget that \(\mathbb{Q}\) not only as the usual absolute value \(|\ \ |\), but also has the \(p\)-adic absolute value \(|\ \ |_{p}\) for each prime \(p\).

For \(P\in\mathbf{P}^{n}(\mathbb{Q})\), this definition of size of \(P\) coincides with \(h(P)\).

One more side note. As you can see above, the reason why all the \(p\)-adic absolute values contribute trivially is that using the property of homogenous coordinate, we can cancel the \(gcd\) and \(lcm\) of the rational coordinates so that they are relatively prime. In another view, we prime factorize the denominators and numerators of each coordinate, and then cancel all the common primes and their common powers. This notion cannot be fully generalized to a number field, unless the number field is a unique factorization domain. In this case, every algebraic number has a unique way to be factorized as a unit multiplying prime elements in the number field. Then, in a similar fasion, we can cancel all the common prime elements and their powers, so that the homogenous coordinates are relatively prime. Then, all the non-archimedean absolute values contribute trivially. This is because every non-archimedean absolute value in a number field uniquely correponds to a prime element in the ring of integer of the number field. Since now coordinates are relatively prime, a similar argument shows that the height function is only a summation over all the archimedean absolute values on the number field.