Why is $12$ the smallest weight for which a cusp form exists?
On wikipedia (here) I have read the following:
Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?
number-theory modular-forms
edited Nov 28 at 15:01
Micah
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
add a comment |
On wikipedia (here) I have read the following:
Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?
number-theory modular-forms
edited Nov 28 at 15:01
Micah
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
1
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40
add a comment |
On wikipedia (here) I have read the following:
Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?
number-theory modular-forms
edited Nov 28 at 15:01
Micah
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
On wikipedia (here) I have read the following:
Twelve is the smallest weight for which a cusp form exists. [...] This fact is related to a constellation of interesting appearances of the number twelve in mathematics ranging from the value of the Riemann zeta function at $−1$ i.e. $zeta(-1) = -1/12$, the fact that the abelianization of $SL(2,mathbb{Z})$ has twelve elements, and even the properties of lattice polygons.
I know the dimension formula and the weight formula for modular forms as well as the proof of the weight formula via the residue theorem, but I do not understand how this relates to any of the three concepts mentioned in the quote above. Can anybody tell me more?
number-theory modular-forms
number-theory modular-forms
edited Nov 28 at 15:01
Micah
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
edited Nov 28 at 15:01
Micah
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
edited Nov 28 at 15:01
Micah
29.6k1363105
edited Nov 28 at 15:01
Micah
29.6k1363105
edited Nov 28 at 15:01
Micah
29.6k1363105
29.6k1363105
asked Jun 3 '16 at 12:43
Martin
268111
asked Jun 3 '16 at 12:43
Martin
268111
asked Jun 3 '16 at 12:43
Martin
268111
268111
Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
1
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40
add a comment |
Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
1
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40
Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
1
1
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40
add a comment |
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Do you know the valence formula? (I think you might be calling it the weight formula, but I want to be sure)
– mdave16
Mar 16 '17 at 1:50
1
You need to be careful with such numerology. Some people define weight to be twice the usual value to avoid half integer weights. The values of the zeta function are, in general, comples numbers. Why pick the value at $-1$? I doubt lattice polygons have anything to do with modular forms. I think the answer to your question is that there is no connection.
– Somos
May 9 '17 at 18:40