关于模曲线$Y(\Gamma)$的仿射性及其他志村簇相关问题的技术咨询

Hey, great question—let's unpack this together, since it's a common point of confusion when first diving into modular forms and Shimura varieties.

First, why are modular curves $Y(\Gamma)$ affine? Let's start with what $Y(\Gamma)$ actually is: it's the moduli space classifying elliptic curves equipped with a level $\Gamma$ structure (think of extra symmetries or marked points on the curve). Its compactification, $X(\Gamma)$, adds the so-called "cusps"—these are the degenerate limits of elliptic curves, like nodal cubics or pointed rational curves.

The key insight here is that $Y(\Gamma)$ is just the projective curve $X(\Gamma)$ with a finite set of points (the cusps) removed. For any smooth projective curve, removing finitely many points always leaves you with an affine variety. Let me make that tangible: the ring of regular functions on $Y(\Gamma)$ is finitely generated over your base field (usually $\mathbb{Q}$ or a number field), and since projective curves have no non-constant global regular functions, removing those cusps gives us enough non-constant regular functions to satisfy the definition of an affine variety. You can also think in terms of open covers: every point on $Y(\Gamma)$ has an affine open neighborhood, and these glue together perfectly to cover the whole space.

Now, moving on to your second question: do other Shimura varieties share this affine property? The short answer is no, not all of them. Shimura varieties come in a wide range of dimensions and types, and their compactifications are way more complex than just adding a handful of points.

Let's break down some examples:

• Shimura curves: These are 1-dimensional Shimura varieties associated to quaternion algebras, and they behave just like modular curves—their open version is affine, since their compactification only adds finitely many cusps (points). So they're the exception that follows the same rule as modular curves.
• Siegel modular varieties ($A_g(\Gamma)$): These parametrize $g$-dimensional principally polarized abelian varieties with level $\Gamma$ structure. When $g \geq 2$, the open Shimura variety $A_g(\Gamma)$ is not affine. Its compactifications (like the Satake or Baily-Borel compactifications) add entire divisors (codimension-1 subvarieties), not just points. In dimensions 2 or higher, removing a divisor from a projective variety gives you a quasi-projective space, but not an affine one—think of $\mathbb{P}^2$ minus a line: it's not affine, even though it's open in projective space.
• Orthogonal Shimura varieties: These classify quadratic forms or related geometric objects, and their open versions are also typically quasi-projective but not affine, since their compactifications add divisorial components rather than isolated points.

To wrap it up:

• Modular curves $Y(\Gamma)$ are affine because they're projective curves minus a finite set of points—a special case that only works in dimension 1.
• 1-dimensional Shimura varieties (like Shimura curves) follow the same rule and are affine, but higher-dimensional Shimura varieties are generally not affine. Their open forms are quasi-projective, but removing divisors (instead of points) from projective varieties in dimension $\geq 2$ doesn't result in an affine space.

备注：内容来源于stack exchange，提问作者user966218