How to interpret action of $SL_2(mathcal{O}_d)$
$begingroup$
Given a lattice $wedge = {omega_1, omega_2 }$ in $mathbb{C}$, $omega_1 / omega_2 notin mathbb{R}$, we know that $wedge' = {omega_1', omega_2' }$ defines the same lattice precisely when
$$begin{pmatrix} omega_1' \ omega_2' end{pmatrix} = begin{pmatrix} a & b \ c & dend{pmatrix} begin{pmatrix} omega_1 \ omega_2 end{pmatrix}$$
for the acting matrix an element in $SL_2(mathbb{Z})$.
I am currently reading about extensions of $SL_2(mathbb{Z})$ to larger groups, specifically $SL_2(mathcal{O}_d)$, $mathcal{O}_d$ the ring of quadratic integers in a quadratic field $K$, for some $d < 0$ (namely Bianchi groups).
In most of the literature I have read, I have not found an intuitive reason for why we might want to study such an extension. We care about $SL_2(mathbb{Z})$ because lattices in $mathbb{C}$ are invariant under an action by a fractional linear transformation described by elements of $SL_2(mathbb{Z})$, and this in turn relates to studying elliptic curves and modular forms and so on.
In a nutshell, here are some of my questions:
- Is the space that we are acting on with $SL_2(mathcal{O}_d)$ still the upper half plane, or do we extend that as well?
- Is there such an analogue reason for why we might want to consider the action of $SL_2(mathcal{O}_d)$? For example, are lattices, or specific sets of lattices, now invariant under this transformation? If so, what kinds of consequences result from that?
- How does the class number of the field $K$ that $mathcal{O}_d$ is derived from affect our interpretation here? I ask because not all rings for arbitrary $d$ have characteristic 1, and this should affect whether or not the ring is a P.I.D., and then a unique factorization domain. In turn, I believe that this would affect the resulting structure of whatever analogue of a Hecke Algebra (if we can form one) we could form.
- Is there a general way to understand the action of $SL_2(mathcal{O}_d)$ for arbitrary $d$? Currently I mostly care about $d < 0$ (generating an imaginary quadratic field), but if there is a general framework / intuition that motivates all $d$, I would be interested to hear it. I am aware that for $d > 0$, we generate a real quadratic field, and such actions described lead to the study of Hilbert Modular forms, but I am unsure if this is too far removed or too large of topic to tackle all at once.
- If you have any good sources to read about this from, I would appreciate references! Currently I am reading through John Cremona's thesis, Modular Symbols, although I find that he has omitted many details, and that is making it hard for me to work through his text without the proper motivation or background.
I apologize for the long winded questions, but I would really appreciate answers to them!
abstract-algebra modular-forms algebraic-numbers quadratic-integer-rings
asked Dec 25 '18 at 6:23
SethSeth
1478
$endgroup$
|
show 3 more comments
$begingroup$
Given a lattice $wedge = {omega_1, omega_2 }$ in $mathbb{C}$, $omega_1 / omega_2 notin mathbb{R}$, we know that $wedge' = {omega_1', omega_2' }$ defines the same lattice precisely when
$$begin{pmatrix} omega_1' \ omega_2' end{pmatrix} = begin{pmatrix} a & b \ c & dend{pmatrix} begin{pmatrix} omega_1 \ omega_2 end{pmatrix}$$
for the acting matrix an element in $SL_2(mathbb{Z})$.
I am currently reading about extensions of $SL_2(mathbb{Z})$ to larger groups, specifically $SL_2(mathcal{O}_d)$, $mathcal{O}_d$ the ring of quadratic integers in a quadratic field $K$, for some $d < 0$ (namely Bianchi groups).
In most of the literature I have read, I have not found an intuitive reason for why we might want to study such an extension. We care about $SL_2(mathbb{Z})$ because lattices in $mathbb{C}$ are invariant under an action by a fractional linear transformation described by elements of $SL_2(mathbb{Z})$, and this in turn relates to studying elliptic curves and modular forms and so on.
In a nutshell, here are some of my questions:
- Is the space that we are acting on with $SL_2(mathcal{O}_d)$ still the upper half plane, or do we extend that as well?
- Is there such an analogue reason for why we might want to consider the action of $SL_2(mathcal{O}_d)$? For example, are lattices, or specific sets of lattices, now invariant under this transformation? If so, what kinds of consequences result from that?
- How does the class number of the field $K$ that $mathcal{O}_d$ is derived from affect our interpretation here? I ask because not all rings for arbitrary $d$ have characteristic 1, and this should affect whether or not the ring is a P.I.D., and then a unique factorization domain. In turn, I believe that this would affect the resulting structure of whatever analogue of a Hecke Algebra (if we can form one) we could form.
- Is there a general way to understand the action of $SL_2(mathcal{O}_d)$ for arbitrary $d$? Currently I mostly care about $d < 0$ (generating an imaginary quadratic field), but if there is a general framework / intuition that motivates all $d$, I would be interested to hear it. I am aware that for $d > 0$, we generate a real quadratic field, and such actions described lead to the study of Hilbert Modular forms, but I am unsure if this is too far removed or too large of topic to tackle all at once.
- If you have any good sources to read about this from, I would appreciate references! Currently I am reading through John Cremona's thesis, Modular Symbols, although I find that he has omitted many details, and that is making it hard for me to work through his text without the proper motivation or background.
I apologize for the long winded questions, but I would really appreciate answers to them!
abstract-algebra modular-forms algebraic-numbers quadratic-integer-rings
asked Dec 25 '18 at 6:23
SethSeth
1478
$endgroup$
6
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
1
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
3
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
2
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
1
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26
|
show 3 more comments
$begingroup$
Given a lattice $wedge = {omega_1, omega_2 }$ in $mathbb{C}$, $omega_1 / omega_2 notin mathbb{R}$, we know that $wedge' = {omega_1', omega_2' }$ defines the same lattice precisely when
$$begin{pmatrix} omega_1' \ omega_2' end{pmatrix} = begin{pmatrix} a & b \ c & dend{pmatrix} begin{pmatrix} omega_1 \ omega_2 end{pmatrix}$$
for the acting matrix an element in $SL_2(mathbb{Z})$.
I am currently reading about extensions of $SL_2(mathbb{Z})$ to larger groups, specifically $SL_2(mathcal{O}_d)$, $mathcal{O}_d$ the ring of quadratic integers in a quadratic field $K$, for some $d < 0$ (namely Bianchi groups).
In most of the literature I have read, I have not found an intuitive reason for why we might want to study such an extension. We care about $SL_2(mathbb{Z})$ because lattices in $mathbb{C}$ are invariant under an action by a fractional linear transformation described by elements of $SL_2(mathbb{Z})$, and this in turn relates to studying elliptic curves and modular forms and so on.
In a nutshell, here are some of my questions:
- Is the space that we are acting on with $SL_2(mathcal{O}_d)$ still the upper half plane, or do we extend that as well?
- Is there such an analogue reason for why we might want to consider the action of $SL_2(mathcal{O}_d)$? For example, are lattices, or specific sets of lattices, now invariant under this transformation? If so, what kinds of consequences result from that?
- How does the class number of the field $K$ that $mathcal{O}_d$ is derived from affect our interpretation here? I ask because not all rings for arbitrary $d$ have characteristic 1, and this should affect whether or not the ring is a P.I.D., and then a unique factorization domain. In turn, I believe that this would affect the resulting structure of whatever analogue of a Hecke Algebra (if we can form one) we could form.
- Is there a general way to understand the action of $SL_2(mathcal{O}_d)$ for arbitrary $d$? Currently I mostly care about $d < 0$ (generating an imaginary quadratic field), but if there is a general framework / intuition that motivates all $d$, I would be interested to hear it. I am aware that for $d > 0$, we generate a real quadratic field, and such actions described lead to the study of Hilbert Modular forms, but I am unsure if this is too far removed or too large of topic to tackle all at once.
- If you have any good sources to read about this from, I would appreciate references! Currently I am reading through John Cremona's thesis, Modular Symbols, although I find that he has omitted many details, and that is making it hard for me to work through his text without the proper motivation or background.
I apologize for the long winded questions, but I would really appreciate answers to them!
abstract-algebra modular-forms algebraic-numbers quadratic-integer-rings
asked Dec 25 '18 at 6:23
SethSeth
1478
$endgroup$
Given a lattice $wedge = {omega_1, omega_2 }$ in $mathbb{C}$, $omega_1 / omega_2 notin mathbb{R}$, we know that $wedge' = {omega_1', omega_2' }$ defines the same lattice precisely when
$$begin{pmatrix} omega_1' \ omega_2' end{pmatrix} = begin{pmatrix} a & b \ c & dend{pmatrix} begin{pmatrix} omega_1 \ omega_2 end{pmatrix}$$
for the acting matrix an element in $SL_2(mathbb{Z})$.
I am currently reading about extensions of $SL_2(mathbb{Z})$ to larger groups, specifically $SL_2(mathcal{O}_d)$, $mathcal{O}_d$ the ring of quadratic integers in a quadratic field $K$, for some $d < 0$ (namely Bianchi groups).
In most of the literature I have read, I have not found an intuitive reason for why we might want to study such an extension. We care about $SL_2(mathbb{Z})$ because lattices in $mathbb{C}$ are invariant under an action by a fractional linear transformation described by elements of $SL_2(mathbb{Z})$, and this in turn relates to studying elliptic curves and modular forms and so on.
In a nutshell, here are some of my questions:
- Is the space that we are acting on with $SL_2(mathcal{O}_d)$ still the upper half plane, or do we extend that as well?
- Is there such an analogue reason for why we might want to consider the action of $SL_2(mathcal{O}_d)$? For example, are lattices, or specific sets of lattices, now invariant under this transformation? If so, what kinds of consequences result from that?
- How does the class number of the field $K$ that $mathcal{O}_d$ is derived from affect our interpretation here? I ask because not all rings for arbitrary $d$ have characteristic 1, and this should affect whether or not the ring is a P.I.D., and then a unique factorization domain. In turn, I believe that this would affect the resulting structure of whatever analogue of a Hecke Algebra (if we can form one) we could form.
- Is there a general way to understand the action of $SL_2(mathcal{O}_d)$ for arbitrary $d$? Currently I mostly care about $d < 0$ (generating an imaginary quadratic field), but if there is a general framework / intuition that motivates all $d$, I would be interested to hear it. I am aware that for $d > 0$, we generate a real quadratic field, and such actions described lead to the study of Hilbert Modular forms, but I am unsure if this is too far removed or too large of topic to tackle all at once.
- If you have any good sources to read about this from, I would appreciate references! Currently I am reading through John Cremona's thesis, Modular Symbols, although I find that he has omitted many details, and that is making it hard for me to work through his text without the proper motivation or background.
I apologize for the long winded questions, but I would really appreciate answers to them!
abstract-algebra modular-forms algebraic-numbers quadratic-integer-rings
abstract-algebra modular-forms algebraic-numbers quadratic-integer-rings
asked Dec 25 '18 at 6:23
SethSeth
1478
asked Dec 25 '18 at 6:23
SethSeth
1478
edited Dec 26 '18 at 1:56
Seth
asked Dec 25 '18 at 6:23
SethSeth
1478
asked Dec 25 '18 at 6:23
SethSeth
1478
asked Dec 25 '18 at 6:23
SethSeth
1478
1478
6
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
1
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
3
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
2
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
1
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26
|
show 3 more comments
6
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
1
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
3
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
2
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
1
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26
6
6
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
1
1
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
3
3
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
2
2
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
1
1
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26
|
show 3 more comments
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6
$begingroup$
The Bianchi groups are generally taken to act on the hyperbolic upper half-space. This can be taken as a subset of the quaternions, namely ${a+bi+cj:,a,b,cinBbb R,c>0}$ (that's right, there's no $j$). link.springer.com/book/10.1007/978-3-662-03626-6 is a fairly standard reference.
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:27
1
$begingroup$
Make that "no $k$" in the comment above!
$endgroup$
– Lord Shark the Unknown
Dec 25 '18 at 6:38
3
$begingroup$
Regarding motivation: the quotient space $mathrm{SL}_2(mathbb{Z}) backslash mathfrak{h}^2$ is modular curve whose points classify elliptic curves. (More precisely, it is a coarse moduli space.) I believe there is a similar moduli-theoretic picture for $mathrm{SL}_2(mathcal{O}_d) backslash mathfrak{h}^3$ and abelian varieties with multiplication by $mathcal{O}_d$. For more on the connection with quaternions, see Ch. 36 of the text found here.
$endgroup$
– André 3000
Dec 25 '18 at 7:56
2
$begingroup$
You might enjoy seeing what some Bianchi groups look like. Check out: Frohman, Charles; Fine, Benjamin Some amalgam structures for Bianchi groups. Proc. Amer. Math. Soc. 102 (1988), no. 2, 221–229.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:22
1
$begingroup$
Serre did some nice work on Bianchi Groups. He proved that the corresponding orbifold has one cusp for every element of the ideal class group of the corresponding ring of integers. His book trees explores the action on lattices you are talking about. Richard Swann found fundamental domains. I based my work on that. Rob Riley had early software for exploring Bianchi groups.
$endgroup$
– Charlie Frohman
Dec 26 '18 at 2:26