In mathematics, there is the notion of fields. In brief, this is a set of "numbers" where you can do four usual operations; addition, subtraction, multiplication and division. For instance, the sets of rational numbers \(\mathbb{Q}\) and real numbers \(\mathbb{R}\) are fields, while the one of integers \(\mathbb{Z}\) is not a field, because one cannot divide an integer by another (the result is no longer an integer).
Among various kinds of fields, there is a weired kind of fields called ones of positive characteristic. In such a field, summing up several 1's gives zero. \[1+1+\cdots+1=0\] The simplest example is the field with \(p\) elements for each prime number \(p\), which is often written as \(\mathbb{F}_p\). (It is also called a Galois field and written as \(GF(p)\).) This correponds to considering congruence modulo \(p\). Namely, the remainder of a division by \(p\) is one of \(0,1,\dots,p-1\). When two natural numbers have the same remainder, one identifies them and gets \(p\) classes of numbers. Moreover, for each prime number \(p\) and positive integer \(e\), there is exactly one field having \(p^e\) elements, \(\mathbb{F}_{p^e}\).
Raising elements of \(\mathbb{F}_{p^e}\) to there \(p\)-th powers gives a self-map of \(\mathbb{F}_{p^e}\); it is called the Frobenius map and written as \(F\). \[ F:\mathbb{F}_{p^e} \to \mathbb{F}_{p^e},\, x\mapsto x^p \] This is an isomorphism of field. Namely, it is a bijection (one-to-one correspondence), and raising two element to \(p\)-th powers and then summing them up, and summing the given elements and then raising the sum to the \(p\)-th power give the same result, and similarly for the multiplication. Another important feature of the Frobenius map is; if \(f\) is a divisor of \(e\), then \(\mathbb{F}_{p^f}\) is a subfield of \(\mathbb{F}_{p^e}\) and the \(f\)-times iteration of the Frobenius map, \(F^f = F\circ \cdots \circ F\), fix the elments of \(\mathbb{F}_{p^f}\) and these are the only fixed elements. In other words, \(\mathbb{F}_{p^f}\) is the fixed point set of \(F^f\).
Now let us consider the equation \[ y^2 + x^2(x+1) = 0. \] It is a problem of the number theory to ask how many pairs \(x,y\in \mathbb{F}_{p^e}\) satisfy the equation.．Varying the equation, \(p\) and \(e\), one can consider any number of problems of the same kind. It is the famous "Weil Conjectures" which tries to answer this sort of problem in terms of cohomology groups, a notion from geometry. After Deligne solved the last part of the conjectures using the etale cohomology developped by Grothendieck, they are now theorems. In the theorems, the Frobenius map plays a crucial role. Thanks to the fact that \( F:\mathbb{F}_{p^e} \to \mathbb{F}_{p^e} \) is an isomorphism of fields, the Frobenius map gives a map from the figure \(X\) defined by an equation to itself, \(F:X\to X\); that is, if \((x,y)\in \mathbb{F}_{p^e} \times \mathbb{F}_{p^e} \) is a solution of the equation, then so is \((x^p,y^p)\). Regarding solutions in \(\mathbb{F}_{p^f}\) as fixed points of \(F^f:X\to X\) and using Lefschetz's fixed point theorem from geometry, one can express the number of fixed points in terms of the action of Frobenius maps on cohomology groups.
In the picture above, we consider the field with \(64=2^6\) elements,\(\mathbb{F}_{64}\). Points represent elements of \( \mathbb{F}_{64}\times \mathbb{F}_{64} = \{(x,y)\mid x,y\in \mathbb{F}_{64}\}. \) Between the fields \(\mathbb{F}_{64}\) and\(\mathbb{F}_2\), there lie two intermediate fields \(\mathbb{F}_8\) and \(\mathbb{F}_4\). There is no inclusion relation between the two and their intersection is \(\mathbb{F}_2\). The size of a point is the biggest when its \(x,y\) coordinates are both in \(\mathbb{F}_2\), the second biggest when the coordinates are in \(\mathbb{F}_4\), and so on. The bright points are the solutions of the above equation.
Please notice the following points:
© 2016 Takehiko Yasuda