Show that $textSL_2(mathbbZ) backslash mathbbH$ is an algebraic variety
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I'd like to show that $textSL_2(mathbbZ) backslash mathbbH$ is a variety. Is it even true that $mathbbH = x + iy : y > 0 $ is an algebraic variety? There's no metric, so it's just the half of the affine plane $mathbbA^2$.
Do we have that $ q in mathbbQ : q > 0$ is a variety? We have that $ xy = 1 subset mathbbA^2$ is a variety, and there's a group action such as $(x,y) mapsto (-x,-y)$ preserving that curve.
Then we have the group action of $textSL_2(mathbbZ)$ (which are integer-valued matrices) - is the quotient space of an algebraic variety another one?
I'd like to be able to say that $big[textSL_2(mathbbZ) backslash mathbbHbig](mathbbQ)$ is a variety over $K = mathbbQ$ and discuss the rational points.
Depending on which theorems we admit the problem is straightfoward. It's well-known that a modular curve is an algebraic curve.
We know this is an instance of a Shimura Variety. This is like assuming the result we want to prove. See also: Hilbert modular surfaces,
algebraic-geometry sheaf-theory group-actions
asked Aug 27 at 16:48
cactus314
15.1k41862
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I'd like to show that $textSL_2(mathbbZ) backslash mathbbH$ is a variety. Is it even true that $mathbbH = x + iy : y > 0 $ is an algebraic variety? There's no metric, so it's just the half of the affine plane $mathbbA^2$.
Do we have that $ q in mathbbQ : q > 0$ is a variety? We have that $ xy = 1 subset mathbbA^2$ is a variety, and there's a group action such as $(x,y) mapsto (-x,-y)$ preserving that curve.
Then we have the group action of $textSL_2(mathbbZ)$ (which are integer-valued matrices) - is the quotient space of an algebraic variety another one?
I'd like to be able to say that $big[textSL_2(mathbbZ) backslash mathbbHbig](mathbbQ)$ is a variety over $K = mathbbQ$ and discuss the rational points.
Depending on which theorems we admit the problem is straightfoward. It's well-known that a modular curve is an algebraic curve.
We know this is an instance of a Shimura Variety. This is like assuming the result we want to prove. See also: Hilbert modular surfaces,
algebraic-geometry sheaf-theory group-actions
asked Aug 27 at 16:48
cactus314
15.1k41862
$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55
add a comment |Â
up vote
1
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up vote
1
down vote
favorite
I'd like to show that $textSL_2(mathbbZ) backslash mathbbH$ is a variety. Is it even true that $mathbbH = x + iy : y > 0 $ is an algebraic variety? There's no metric, so it's just the half of the affine plane $mathbbA^2$.
Do we have that $ q in mathbbQ : q > 0$ is a variety? We have that $ xy = 1 subset mathbbA^2$ is a variety, and there's a group action such as $(x,y) mapsto (-x,-y)$ preserving that curve.
Then we have the group action of $textSL_2(mathbbZ)$ (which are integer-valued matrices) - is the quotient space of an algebraic variety another one?
I'd like to be able to say that $big[textSL_2(mathbbZ) backslash mathbbHbig](mathbbQ)$ is a variety over $K = mathbbQ$ and discuss the rational points.
Depending on which theorems we admit the problem is straightfoward. It's well-known that a modular curve is an algebraic curve.
We know this is an instance of a Shimura Variety. This is like assuming the result we want to prove. See also: Hilbert modular surfaces,
algebraic-geometry sheaf-theory group-actions
asked Aug 27 at 16:48
cactus314
15.1k41862
I'd like to show that $textSL_2(mathbbZ) backslash mathbbH$ is a variety. Is it even true that $mathbbH = x + iy : y > 0 $ is an algebraic variety? There's no metric, so it's just the half of the affine plane $mathbbA^2$.
Do we have that $ q in mathbbQ : q > 0$ is a variety? We have that $ xy = 1 subset mathbbA^2$ is a variety, and there's a group action such as $(x,y) mapsto (-x,-y)$ preserving that curve.
Then we have the group action of $textSL_2(mathbbZ)$ (which are integer-valued matrices) - is the quotient space of an algebraic variety another one?
I'd like to be able to say that $big[textSL_2(mathbbZ) backslash mathbbHbig](mathbbQ)$ is a variety over $K = mathbbQ$ and discuss the rational points.
Depending on which theorems we admit the problem is straightfoward. It's well-known that a modular curve is an algebraic curve.
We know this is an instance of a Shimura Variety. This is like assuming the result we want to prove. See also: Hilbert modular surfaces,
algebraic-geometry sheaf-theory group-actions
asked Aug 27 at 16:48
cactus314
15.1k41862
asked Aug 27 at 16:48
cactus314
15.1k41862
asked Aug 27 at 16:48
cactus314
15.1k41862
asked Aug 27 at 16:48
cactus314
15.1k41862
15.1k41862
$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55
add a comment |Â
$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55
$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55
$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55
add a comment |Â
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$BbbH$ is not an affine algebraic variety, see here. Pete Clark says more in his answer to this MO-question.
– Dietrich Burde
Aug 27 at 16:55