q-expansion principle and the constant term of modular form
1
$begingroup$
If I have a modular form $f=sum_{n=0}^infty a_nq^n$
of weight k, level N and character $chi$. Assume all $a_n$ except $a_0$ generate a number field, must $a_0$ also lie in this number field?
Motivation：I want to understand why the constant terms of Eisenstein series i.e some special values of Dirichlet series are algebraic number in general.
number-theory modular-forms
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
$endgroup$
add a comment |
1
$begingroup$
If I have a modular form $f=sum_{n=0}^infty a_nq^n$
of weight k, level N and character $chi$. Assume all $a_n$ except $a_0$ generate a number field, must $a_0$ also lie in this number field?
Motivation：I want to understand why the constant terms of Eisenstein series i.e some special values of Dirichlet series are algebraic number in general.
number-theory modular-forms
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
$endgroup$
$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47
add a comment |
1
1
1
$begingroup$
If I have a modular form $f=sum_{n=0}^infty a_nq^n$
of weight k, level N and character $chi$. Assume all $a_n$ except $a_0$ generate a number field, must $a_0$ also lie in this number field?
Motivation：I want to understand why the constant terms of Eisenstein series i.e some special values of Dirichlet series are algebraic number in general.
number-theory modular-forms
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
$endgroup$
If I have a modular form $f=sum_{n=0}^infty a_nq^n$
of weight k, level N and character $chi$. Assume all $a_n$ except $a_0$ generate a number field, must $a_0$ also lie in this number field?
Motivation：I want to understand why the constant terms of Eisenstein series i.e some special values of Dirichlet series are algebraic number in general.
number-theory modular-forms
number-theory modular-forms
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
asked Dec 6 '18 at 18:27
zzyzzy
2,6331420
2,6331420
$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47
add a comment |
$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47
$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47
add a comment |
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$begingroup$
What about $h = (f-a_0) Delta in S_{k+12}(Gamma_0(N),chi)oplus S_{12}(Gamma_0(1))$ ? It has algebraic coefficients and its projection on the (normalized) eigenforms basis $(g_j)$ can be found from finitely many coefficients of $h$ and $g_j$, so $h = sum_j c_j g_j$ with $c_j$ algebraic
$endgroup$
– reuns
Dec 6 '18 at 20:48
$begingroup$
($g_1 = Delta, c_1 = -a_0$)
$endgroup$
– reuns
Dec 6 '18 at 20:56
$begingroup$
@reuns So you are considering the direct sum of spaces of cusp forms with different level, how to choose $g_j$ such that their coefficients are contained in the field generated by $a_i(i>0)$?
$endgroup$
– zzy
Dec 7 '18 at 3:36
$begingroup$
Try replacing $S_{k+12}(Gamma_0(N),chi)$ by $S_{k+12}(Gamma_1(N))$ ? In that space $g_i^sigma$ in one of the $g_j$, so you can find a basis for the $K_f$ vector space of the forms with coefficients in $K_f$
$endgroup$
– reuns
Dec 7 '18 at 12:32
$begingroup$
@reuns OK, thank you!
$endgroup$
– zzy
Dec 7 '18 at 14:47