Math 254B. Homework #1

Due Wednesday 3 February, online

1(nc). For α ∈ \overline{ℚ}, show that h_{ℚ}(α) = 0 if and only if α = 0 or α is a root of unity.

2. Show that the following versions of Roth’s theorem are all equivalent (in the sense that proving any one of them suffices to prove them all).

Theorem. Let α ∈ \overline{ℚ} \ ℚ, ϵ > 0 , and C ∈ ℝ be given. Then there are only finitely many ordered pairs (p, q) , with p, q ∈ ℤ and q ≠ 0 , such that

|\frac{p}{q} − α| ≤ \frac{C}{|q|^{2+ϵ}} .

Theorem. Let α ∈ \overline{ℚ} \ ℚ, ϵ > 0 , and C ∈ ℝ be given. Then there are only finitely many x ∈ ℚ such that

|x − α| ≤ \frac{C}{H_{ℚ}(x)^{2+ϵ}} .

Theorem. Let α ∈ \overline{ℚ}\ℚ and ϵ > 0 be given. Then there are only finitely many x ∈ ℚ

such that

|x − α| ≤ \frac{1}{H_{ℚ}(x)^{2+ϵ}} .

3. Let X be a variety over a field k, let U = Spec A be a nonempty open affine subset of X, and let P ∈ X be a point not in U . Then there is a function f ∈ A that does not extend to a regular function at P (i.e., f ∉ 𝒪_{X,P} ). Moreover, f can be chosen from any given generating set for A over k.

[Hint: Hartshorne II Ex. 4.2 or Vakil 10.2.A may be useful. But if you use one of them, say which you are using (if at all) and how you are using it.]