A foundational result in Diophantine approximation is Roth's theorem, which asserts that for a given algebraic number $\alpha$ and $\varepsilon>0$, the inequality $|\alpha-p/q|<1/q^{2+\varepsilon}$ has only finitely many solutions in rational numbers $p/q\in \mathbb{Q}$. Ridout and Lang subsequently proved a general form of Roth's theorem allowing for arbitrary absolute values (including $p$-adic absolute values) and permitting arbitrary (fixed) number fields $k$ in place of the rational numbers. Wirsing proved a generalization of Roth's theorem, where the approximating elements are taken from varying number fields of degree $d$, and the quantity $2+\varepsilon$ in Roth's theorem is replaced by $2d+\varepsilon$. As a consequence of a deep inequality of Vojta, Wirsing's theorem, appropriately formulated, may be extended to a Diophantine approximation result for algebraic points of degree $d$ on a nonsingular projective curve.The main theorem of this thesis improves on this general form of Wirsing's theorem further, but only in the cases where $d=2$ and $d=3$. More specifically, if we let $C$ be a nonsingular projective curve over a number field $k$, $S$ a finite set of places of $k$ and $P_1, \ldots, P_n \in C(k)$ be distinct and define the divisor $D:=\sum_{i=1}^n P_i$, then letting $R\in C(k)$ and $\varepsilon > 0$, we get\[m_{D,S}(P)\leq (N_d(D)+\varepsilon) h_R(P) +O(1)\]for all $P\in C(\overline{k})$ with $[k(P):k]=d\in \{2,3\}$, where $m_{D,S}(P)$ is a sum of local heights associated to $D$ and $S$, $h_R$ is a global height associated to $R$, and \[N_d(D):=\max \left\{\left|\left(\sigma^{-1}(0)\cup \sigma^{-1}(\infty)\right)\cap \mathrm{Supp}(D)\right|\right\}\]taken over all finite $k$-morphisms $\sigma:C\rightarrow \mathbb{P}^1$ of degree $d$. As $N_d(D)\leq 2d$, the main result gives a refinement of Wirsing's theorem depending on the divisor $D$.In much the same way that Roth's theorem can be used to prove Siegel's Theorem about integral points on a genus-0 affine curve, the main theorem of this dissertation implies Levin's generalization of Siegel's theorem for algebraic points of degree $d$ in the cases $d=2,3$.After a review of some properties of global heights and local heights, we also show that the main theorem is sharp, in the sense that counterexamples exist when $N_d(D)+\varepsilon$ is replaced with $N_d(D)-\varepsilon$.To prove the main theorem, we first show that for curves of large enough genus, the number of morphisms $\sigma:C\rightarrow \mathbb{P}^1$ of degree $d=2,3$ defined over $k$ is finite. We then prove that curves for which the number of such morphisms is finite satisfy the main theorem. Finally, we prove the main theorem for the remaining small genus curves on a case by case basis.We transfer the Diophantine approximation problem for points of degree $d$ on $C$ to a Diophantine approximation problem for rational points on $\mathrm{Sym}^d(C )$ (the $d$th symmetric power of $C$). We exploit the map from $\mathrm{Sym}^d(C)$ to $\mathrm{Jac}(C)$, the Jacobian of $C$, and its associated geometry. We use Diophantine approximation results for abelian varieties (when we're on $\mathrm{Jac}(C)$), Diophantine approximation results for projective spaces (fibers of $\mathrm{Sym}^d(C)\mapsto \mathrm{Jac}(C)$), and Diophantine approximation results on $\mathrm{Sym}^d(C)$ directly, plus algebraic geometry to connect all of these. In the course of the proof we make use of Schmidt's subspace theorem and its generalizations due to Ru-Wong and Evertse-Ferretti as well as a version of Roth's theorem for nonreduced divisors and Faltings' approximation theorem for rational points on abelian varieties.
Read
