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edited Dec 21, 2009 at 0:10

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I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well controlled, so that you can instead focus on arithmetic issues. I might describe the philosophy this way, even though it is not my area and not my philosophy: If number theory is like playing music, and algebraic geometry is like juggling, then general arithmetic geometry is like playing music while juggling the instruments. I.e., it's a great thing to do, but it combines two problems that already hard enough separately.

To the extent that you adopt this emphasis, it makes sense that you would learn more from complex curves than from higher-dimensional complex geometry. But without the one-dimensional emphasis, it is not true. For instance, the Weil conjectures are a deep result in higher-dimensional number theory, if you can call finite fields number theory, and they are both motivated by and informed by higher-dimensional complex geometry. (If number theory is music, finite fields could be music mainly in one note. If I had to juggle instruments, I might rather juggle tambourines or triangles.)

I should add that restricting to one dimension isn't a completely consistent vision of geometry. To give two examples, if $C$ is a complex curve of genus $g > 1$, it comes from a higher-dimensional moduli space, and it has a higher-dimensional Jacobian variety. These issues also arise in positive characteristic. The philosophy that algebraic number theory is one-dimensional is there, but it is not a completely serious philosophy, and not just because higher-dimensional generalizations exist. The Langlands program eventually leads you to the opposite philosophy.

Finally there are also connections between number theory and manifolds with three real dimensions, because of the fact that the boundary of hyperbolic space $\mathbb{H}^3$ is a Riemann sphere and $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.

I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well controlled, so that you can instead focus on arithmetic issues. I might describe the philosophy this way, even though it is not my area and not my philosophy: If number theory is like playing music, and algebraic geometry is like juggling, then general arithmetic geometry is like playing music while juggling the instruments. I.e., it's a great thing to do, but it combines two problems that already hard enough separately.

To the extent that you adopt this emphasis, it makes sense that you would learn more from complex curves than from higher-dimensional complex geometry. But without the one-dimensional emphasis, it is not true. For instance, the Weil conjectures are a deep result in higher-dimensional number theory, if you can call finite fields number theory, and they are both motivated by and informed by higher-dimensional complex geometry. (If number theory is music, finite fields could be music mainly in one note. If I had to juggle instruments, I might rather juggle tambourines or triangles.)

Finally there are also connections between number theory and manifolds with three real dimensions, because of the fact that the boundary of hyperbolic space $\mathbb{H}^3$ is a Riemann sphere and $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.

I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well controlled, so that you can instead focus on arithmetic issues. I might describe the philosophy this way, even though it is not my area and not my philosophy: If number theory is like playing music, and algebraic geometry is like juggling, then general arithmetic geometry is like playing music while juggling the instruments. I.e., it's a great thing to do, but it combines two problems that already hard enough separately.

To the extent that you adopt this emphasis, it makes sense that you would learn more from complex curves than from higher-dimensional complex geometry. But without the one-dimensional emphasis, it is not true. For instance, the Weil conjectures are a deep result in higher-dimensional number theory, if you can call finite fields number theory, and they are both motivated by and informed by higher-dimensional complex geometry. (If number theory is music, finite fields could be music mainly in one note. If I had to juggle instruments, I might rather juggle tambourines or triangles.)

I should add that restricting to one dimension isn't a completely consistent vision of geometry. To give two examples, if $C$ is a complex curve of genus $g > 1$, it comes from a higher-dimensional moduli space, and it has a higher-dimensional Jacobian variety. These issues also arise in positive characteristic. The philosophy that algebraic number theory is one-dimensional is there, but it is not a completely serious philosophy, and not just because higher-dimensional generalizations exist. The Langlands program eventually leads you to the opposite philosophy.

Finally there are also connections between number theory and manifolds with three real dimensions, because of the fact that the boundary of hyperbolic space $\mathbb{H}^3$ is a Riemann sphere and $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.

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created Dec 20, 2009 at 5:44

56.6k
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203
282

I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number theory on elliptic curves, function fields, etc. The reason is that algebraic geometry in one dimension is relatively well controlled, so that you can instead focus on arithmetic issues. I might describe the philosophy this way, even though it is not my area and not my philosophy: If number theory is like playing music, and algebraic geometry is like juggling, then general arithmetic geometry is like playing music while juggling the instruments. I.e., it's a great thing to do, but it combines two problems that already hard enough separately.

To the extent that you adopt this emphasis, it makes sense that you would learn more from complex curves than from higher-dimensional complex geometry. But without the one-dimensional emphasis, it is not true. For instance, the Weil conjectures are a deep result in higher-dimensional number theory, if you can call finite fields number theory, and they are both motivated by and informed by higher-dimensional complex geometry. (If number theory is music, finite fields could be music mainly in one note. If I had to juggle instruments, I might rather juggle tambourines or triangles.)

Finally there are also connections between number theory and manifolds with three real dimensions, because of the fact that the boundary of hyperbolic space $\mathbb{H}^3$ is a Riemann sphere and $\text{Isom}(\mathbb{H}^3) = \text{PSL}(2,\mathbb{C})$.