About Sturm's bound
$begingroup$
The next theorem is known as Sturm's bound.
Theorem:Let $mathfrak{m}$ be a prime ideal in the ring of integers $mathcal{O}$ of a number field $K$, and let $Gamma$ be a congruence subgroup of of index $m$ and level $N$. Suppose $fin M_k(Gamma,mathcal{O})$ is a modular form and
begin{align}
mathrm{ord}_{mathfrak{m}}(f)>dfrac{km}{2}.
end{align}
Then $fequiv 0;(mathrm{mod},mathfrak{m})$.
I want to know if all the sophisticated techniques that Sturm uses in his proof are necessary? In other words, is there a elementary proof? I found, in the case of $mathrm{SL}_2(mathbb{Z})$, a sketch of a proof in a book (Problems In The Theory of modular forms by M. Ram Murty ) but i can't realize a complete proof.
Especifically, I'm struggling with the width of cusps, why the existence of a represent of a coset associated to a cusp with $h$ as the value of the width implies the existence of other $h-1$ representatives with the same value of width.
algebraic-number-theory modular-forms elliptic-functions
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
$endgroup$
add a comment |
$begingroup$
The next theorem is known as Sturm's bound.
Theorem:Let $mathfrak{m}$ be a prime ideal in the ring of integers $mathcal{O}$ of a number field $K$, and let $Gamma$ be a congruence subgroup of of index $m$ and level $N$. Suppose $fin M_k(Gamma,mathcal{O})$ is a modular form and
begin{align}
mathrm{ord}_{mathfrak{m}}(f)>dfrac{km}{2}.
end{align}
Then $fequiv 0;(mathrm{mod},mathfrak{m})$.
I want to know if all the sophisticated techniques that Sturm uses in his proof are necessary? In other words, is there a elementary proof? I found, in the case of $mathrm{SL}_2(mathbb{Z})$, a sketch of a proof in a book (Problems In The Theory of modular forms by M. Ram Murty ) but i can't realize a complete proof.
Especifically, I'm struggling with the width of cusps, why the existence of a represent of a coset associated to a cusp with $h$ as the value of the width implies the existence of other $h-1$ representatives with the same value of width.
algebraic-number-theory modular-forms elliptic-functions
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
$endgroup$
add a comment |
$begingroup$
The next theorem is known as Sturm's bound.
Theorem:Let $mathfrak{m}$ be a prime ideal in the ring of integers $mathcal{O}$ of a number field $K$, and let $Gamma$ be a congruence subgroup of of index $m$ and level $N$. Suppose $fin M_k(Gamma,mathcal{O})$ is a modular form and
begin{align}
mathrm{ord}_{mathfrak{m}}(f)>dfrac{km}{2}.
end{align}
Then $fequiv 0;(mathrm{mod},mathfrak{m})$.
I want to know if all the sophisticated techniques that Sturm uses in his proof are necessary? In other words, is there a elementary proof? I found, in the case of $mathrm{SL}_2(mathbb{Z})$, a sketch of a proof in a book (Problems In The Theory of modular forms by M. Ram Murty ) but i can't realize a complete proof.
Especifically, I'm struggling with the width of cusps, why the existence of a represent of a coset associated to a cusp with $h$ as the value of the width implies the existence of other $h-1$ representatives with the same value of width.
algebraic-number-theory modular-forms elliptic-functions
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
$endgroup$
The next theorem is known as Sturm's bound.
Theorem:Let $mathfrak{m}$ be a prime ideal in the ring of integers $mathcal{O}$ of a number field $K$, and let $Gamma$ be a congruence subgroup of of index $m$ and level $N$. Suppose $fin M_k(Gamma,mathcal{O})$ is a modular form and
begin{align}
mathrm{ord}_{mathfrak{m}}(f)>dfrac{km}{2}.
end{align}
Then $fequiv 0;(mathrm{mod},mathfrak{m})$.
I want to know if all the sophisticated techniques that Sturm uses in his proof are necessary? In other words, is there a elementary proof? I found, in the case of $mathrm{SL}_2(mathbb{Z})$, a sketch of a proof in a book (Problems In The Theory of modular forms by M. Ram Murty ) but i can't realize a complete proof.
Especifically, I'm struggling with the width of cusps, why the existence of a represent of a coset associated to a cusp with $h$ as the value of the width implies the existence of other $h-1$ representatives with the same value of width.
algebraic-number-theory modular-forms elliptic-functions
algebraic-number-theory modular-forms elliptic-functions
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
edited Dec 18 '18 at 13:23
A. Gomez
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
asked Dec 18 '18 at 6:50
A. GomezA. Gomez
256
256
add a comment |
add a comment |
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