To develop a point already mentioned: To some extent algebraic geometry, including complex algebraic geometry, is a part of number theory. The reason for this is that any algebraic variety, say over $\mathbb{C}$, is defined by polynomial equations involving only finitely many coefficients, so the coefficients live in a ring which is a finitely generated $\mathbb{Z}$-algebra $A$, and $A$ belong to the domain of number theory. For example, if $m$ is a maximal ideal of $A$, then $A/m$ is a finite field, and the intersection of all maximal ideals of $A$ is $(0)$, so studying "varieties" over $A$ can be reduced in principle to studying varieties over finite fields.

Applications of this method to prove results in complex algebraic geometry by means of number-theoretic methods are many. For a few, relatively elementary but still striking, examples (including the theorem of Ax-Grothendieck and the existence of a fixed point for a $p$-group acting algebraically on $\mathbb{C}^n$), see e.g. this survey paper of Serre. For a more advanced example, consider the beautiful theorem of Batyrev, that birational Calabi-Yau $n$-folds have equal Betti numbers, which is proved by reducing the question to a question over finite field solved using Deligne's proof of the (last) Weil's conjecture, one of the jewel of modern algebraic number theory.

To develop a point already mentioned: To some extent algebraic geometry, including complex algebraic geometry, is a part of number theory. The reason for this is that any algebraic variety, say over $\mathbb{C}$, is defined by polynomial equations involving only finitely many coefficients, so the coefficients live in a ring which is a finitely generated $\mathbb{Z}$-algebra $A$, and $A$ belongs to the domain of number theory. For example, if $m$ is a maximal ideal of $A$, then $A/m$ is a finite field, and the intersection of all maximal ideals of $A$ is $(0)$, so studying "varieties" over $A$ can be reduced in principle to studying varieties over finite fields.

Applications of this method to prove results in complex algebraic geometry by means of number-theoretic methods are many. For a few, relatively elementary but still striking, examples (including the theorem of Ax-Grothendieck and the existence of a fixed point for a $p$-group acting algebraically on $\mathbb{C}^n$), see e.g. this survey paper of Serre. For a more advanced example, consider the beautiful theorem of Batyrev, that birational Calabi-Yau $n$-folds have equal Betti numbers, which is proved by reducing the question to a question over finite field solved using Deligne's proof of the (last) Weil's conjecture, one of the jewel of modern algebraic number theory.