Miscellaneous preliminaries on arithmetic geometry

One definition of a hyperelliptic curve is a curve C over an algebraically closed field k whose function field K is a degree 2 extension of a purely transcendental extension of k. 1. (a) Show that every hyperelliptic curve is birational to a curve of the form y 2 = f (x) where f ∈ k[x] is a monic squarefree polynomial. (b) Conversely, show that every squarefree f ∈ k[x] gives rise to a hyperelliptic curve in this way. (c) Give an example to show that two distinct monic squarefree f ∈ k[x] can lead to isomorphic curves. 2. (a) Given a hyperelliptic curve C : y 2 = f (x) as above, let D be the divisor arising from the function x on C. Show that the degree of D is 2 and that dim H 0 (C, D) = 2 if g > 0. (b) Show that a curve C that has a degree 2 divisor D with dim H 0 (C, D) = 2 is hyperelliptic. 3. Let K be the function field of a curve C over F q. Show that the degree 0 part of the class group of K is finite. The following is summarized from [Poonen, §2.4]. Recall that a closed point of a scheme X is a point x ∈ X such that {x} is Zariski closed in X. For example, over an algebraically closed field k, there is a bijection between X(k) and the closed points of X. 4. Let X be a variety over a field k and let x ∈ X. Prove that x is a closed point if and only if the residue field κ(x) is a finite extension of k. Let X be a variety over the field k. The degree of a closed point x on X is [κ(x) : k]. 5. (a) Let X be the plane conic over Q cut out by f (x, y, z) = 3x 2 + 4y 2 + 5z 2. What is the minimal degree of a closed point on X? (b) Let Y be the plane cubic over Q cut out by g(x, y, z) = x 3 + y 3 + z 3. What is the minimal degree of a closed point on X? 6. Let k = F q and let X = Spec F q n over k. (a) What is #X (as a set)? …