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This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. It is possible to answer this for any particular curve with some computational facility, but we don't have to! . We have Mazur's Theorem, which says that the answer is `no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.

2 added 9 characters in body

This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. Treating the curve in isolation, this would amount It is possible to solving answer this for any particular curve with some nasty equations. But, fortunatelycomputational facility, but we don't have to! We have Mazur's Theorem, which says that the answer is `no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.

1 [made Community Wiki]

This is not exactly what you wanted, but in algebraic geometry it is often easier to prove something for a particular object by considering the moduli space parametrizing such objects. The example I have in mind is the following: Suppose you picked some random elliptic curve over $\mathbb{Q}$ and were wondering if it has a rational point of order $11$. Treating the curve in isolation, this would amount to solving some nasty equations. But, fortunately, we have Mazur's Theorem, which says that the answer is `no, it doesn't.' Mazur does this essentially by showing that the corresponding moduli space of elliptic curves with a choice of $11$-torsion point (which is a nice modular curve) has no rational points: so you can never have an elliptic curve over $\mathbb{Q}$ with an $11$-torsion point.
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