Mixing
Algebraic dependence of any 3 integral weighted modular forms
Let $M_k(Gamma)$ be the integral weight $k$ modular forms defined over discrete subgroup $Gammaleq SL_2(R)$. One is given $dim_C M_k(Gamma)leq frac{kV}{4pi}+1$ with $V$ some constant number.
Book says the following. "If $f,g,h$ were algebraically independent modular forms, then for large $k$, dimension $M_k(Gamma)$ would be at least the number of monomials in $f,g,h$ of total weight $k$, which is bigger than some positive multiple of $k^2$. This contradicts dimension of $M_k(Gamma)$ growth in $k$ by above."
$textbf{Q:}$ Suppose $f,g,h$ are weighted $l_1,l_2,l_3$ modular forms. I need to look for number of solutions asymptotic solution $(a_1,a_2,a_3)in Z_{geq 0}^3$ $a_1l_1+a_2l_2+a_3l_3=k$. It is clear that in general the defining equation cuts off a hyperplane in $R^3$ with a surface area asymptotic $k^2$. If I further assume the points are evenly distributed(very wrong hypothesis), then indeed I expect number of points lying on that hypersurface should be asymptotic to $k^2$ for large $k$. How do I argue that statement rigorously?
Ref. Zagier 1-2-3 Modular Forms pg 12. Proposition 3
abstract-algebra complex-analysis number-theory
asked Nov 21 '18 at 2:14
user45765
2,5952722
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Let $M_k(Gamma)$ be the integral weight $k$ modular forms defined over discrete subgroup $Gammaleq SL_2(R)$. One is given $dim_C M_k(Gamma)leq frac{kV}{4pi}+1$ with $V$ some constant number.
Book says the following. "If $f,g,h$ were algebraically independent modular forms, then for large $k$, dimension $M_k(Gamma)$ would be at least the number of monomials in $f,g,h$ of total weight $k$, which is bigger than some positive multiple of $k^2$. This contradicts dimension of $M_k(Gamma)$ growth in $k$ by above."
$textbf{Q:}$ Suppose $f,g,h$ are weighted $l_1,l_2,l_3$ modular forms. I need to look for number of solutions asymptotic solution $(a_1,a_2,a_3)in Z_{geq 0}^3$ $a_1l_1+a_2l_2+a_3l_3=k$. It is clear that in general the defining equation cuts off a hyperplane in $R^3$ with a surface area asymptotic $k^2$. If I further assume the points are evenly distributed(very wrong hypothesis), then indeed I expect number of points lying on that hypersurface should be asymptotic to $k^2$ for large $k$. How do I argue that statement rigorously?
Ref. Zagier 1-2-3 Modular Forms pg 12. Proposition 3
abstract-algebra complex-analysis number-theory
asked Nov 21 '18 at 2:14
user45765
2,5952722
add a comment |
Let $M_k(Gamma)$ be the integral weight $k$ modular forms defined over discrete subgroup $Gammaleq SL_2(R)$. One is given $dim_C M_k(Gamma)leq frac{kV}{4pi}+1$ with $V$ some constant number.
Book says the following. "If $f,g,h$ were algebraically independent modular forms, then for large $k$, dimension $M_k(Gamma)$ would be at least the number of monomials in $f,g,h$ of total weight $k$, which is bigger than some positive multiple of $k^2$. This contradicts dimension of $M_k(Gamma)$ growth in $k$ by above."
$textbf{Q:}$ Suppose $f,g,h$ are weighted $l_1,l_2,l_3$ modular forms. I need to look for number of solutions asymptotic solution $(a_1,a_2,a_3)in Z_{geq 0}^3$ $a_1l_1+a_2l_2+a_3l_3=k$. It is clear that in general the defining equation cuts off a hyperplane in $R^3$ with a surface area asymptotic $k^2$. If I further assume the points are evenly distributed(very wrong hypothesis), then indeed I expect number of points lying on that hypersurface should be asymptotic to $k^2$ for large $k$. How do I argue that statement rigorously?
Ref. Zagier 1-2-3 Modular Forms pg 12. Proposition 3
abstract-algebra complex-analysis number-theory
asked Nov 21 '18 at 2:14
user45765
2,5952722
Let $M_k(Gamma)$ be the integral weight $k$ modular forms defined over discrete subgroup $Gammaleq SL_2(R)$. One is given $dim_C M_k(Gamma)leq frac{kV}{4pi}+1$ with $V$ some constant number.
Book says the following. "If $f,g,h$ were algebraically independent modular forms, then for large $k$, dimension $M_k(Gamma)$ would be at least the number of monomials in $f,g,h$ of total weight $k$, which is bigger than some positive multiple of $k^2$. This contradicts dimension of $M_k(Gamma)$ growth in $k$ by above."
$textbf{Q:}$ Suppose $f,g,h$ are weighted $l_1,l_2,l_3$ modular forms. I need to look for number of solutions asymptotic solution $(a_1,a_2,a_3)in Z_{geq 0}^3$ $a_1l_1+a_2l_2+a_3l_3=k$. It is clear that in general the defining equation cuts off a hyperplane in $R^3$ with a surface area asymptotic $k^2$. If I further assume the points are evenly distributed(very wrong hypothesis), then indeed I expect number of points lying on that hypersurface should be asymptotic to $k^2$ for large $k$. How do I argue that statement rigorously?
Ref. Zagier 1-2-3 Modular Forms pg 12. Proposition 3
abstract-algebra complex-analysis number-theory
abstract-algebra complex-analysis number-theory
asked Nov 21 '18 at 2:14
user45765
2,5952722
asked Nov 21 '18 at 2:14
user45765
2,5952722
asked Nov 21 '18 at 2:14
user45765
2,5952722
asked Nov 21 '18 at 2:14
user45765
2,5952722
asked Nov 21 '18 at 2:14
user45765
2,5952722
2,5952722
add a comment |
add a comment |
1 Answer
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In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(Gamma)$ contains all the homogeneous polynomials of degree $k$ of $mathbb{C}[f,g,h]$ which is a space of dimension $frac{(k+2)(k+1)}{2}$.
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
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1 Answer
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1 Answer
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In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(Gamma)$ contains all the homogeneous polynomials of degree $k$ of $mathbb{C}[f,g,h]$ which is a space of dimension $frac{(k+2)(k+1)}{2}$.
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
add a comment |
In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(Gamma)$ contains all the homogeneous polynomials of degree $k$ of $mathbb{C}[f,g,h]$ which is a space of dimension $frac{(k+2)(k+1)}{2}$.
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
add a comment |
In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(Gamma)$ contains all the homogeneous polynomials of degree $k$ of $mathbb{C}[f,g,h]$ which is a space of dimension $frac{(k+2)(k+1)}{2}$.
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
In your example, you can assume that $l_1 = l_2 = l_3$ by replacing $f,g$ and $h$ by $f^a, g^b$ and $h^c$ for positive integers $a,b,c$ such that $l = al_1 = bl_2 = cl_3$.
Then $M_{lk}(Gamma)$ contains all the homogeneous polynomials of degree $k$ of $mathbb{C}[f,g,h]$ which is a space of dimension $frac{(k+2)(k+1)}{2}$.
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
answered Nov 21 '18 at 21:44
Arnaud ETEVE
1
1
add a comment |
add a comment |
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