Construction of Bases for Modular Functions for
$\Gamma_0(121)$ related to $p(11n+6)$
Supplement material to
the
article
by
Ralf Hemmecke,
Peter Paule, and
Silviu Radu.
This article is also available as a
RISC report.
$t
= \frac{\eta(\tau)}{\eta(121\tau)}
= q^{-5}-{q^{-4}}-{q^{-3}}+O(q^{-2})$
$u
= \frac{\eta(11\tau)^{12}}{\eta(\tau)\eta(121\tau)^{11}}
= q^{-50}+q^{-49}+2\,q^{-48}+O(q^{-47})$
$z = \frac{1}{11}(u - t^{10})
= q^{-49}-3\,q^{-48}+3\,q^{-47}+O(q^{-46})$
$f
= \frac{\eta(\tau)^{11}\eta(11\tau)}{\eta(121\tau)^{11}}
q^{\frac{13}{24}} \sum_{n=1}^\infty p(11n+6)q^n
= 11\,q^{-54}+176\,q^{-53}+935\,q^{-52}+O(q^{-51})$g
- dc.input
-
The polynomials $d_1, \ldots, d_5$ with
$d=d_1\,d_2^2\,d_3^2\,d_4\,d_5$ and the polynomials
$c_0,\ldots,c_4$ such that
$d(t)f = \sum_{k=0}^4 c_k(t) z^k$.
- bf.input
-
The order-complete basis $B^{(f)} =
\{1,b^{(f)}_1,\ldots,b^{(f)}_4\}$ such that
$\mathbb{Q}[t, u, f]=\langle B^{(f)} \rangle_{\mathbb{Q}[t]}$.
- f2.input
-
Relation showing that $f_2 \in \mathbb{Q}[t, u, f]$.
- bj.input
-
The integral basis $B^{(j)} =
\{1,b^{(j)}_1,\ldots,b^{(j)}_4\}$ such that
$\mathbb{Q}[t,u,j^\infty_0,j^\infty_2]=\langle B^{(j)}
\rangle_{\mathbb{Q}[t]}=M^\infty(121)$ together with a
representations of the basis elements in terms of $t$, $u$,
$j(\tau)$ (variable $J0$) and $j_2(\tau)=j(11^2\tau)$
(variable $J2$).
Note that $j^\infty_0 := t^{25}\,u\,j$ and
$j^\infty_2 := t\, u\,j_2$.
- fj.input
-
Relation showing that $f$ can be expressed as a polynomial in
terms of $t$, $u$, $j(\tau)$ (variable $J0$) and
$j_2(\tau)=j(11^2\tau)$ (variable $J2$).
Note that $j^\infty_0 := t^{25}\,u\,j$ and
$j^\infty_2 := t\, u\,j_2$.
- bg.input
-
The order-complete basis $B^{(g)} =
\{1,b^{(g)}_1,\ldots,b^{(g)}_4\}$ such that
$\mathbb{Q}[t, u, g]=\langle B^{(g)} \rangle_{\mathbb{Q}[t]}$
together with a representations of the basis elements in
terms of $t$, $u$, and $g$.
- bfg.input
-
The integral basis $B^{(fg)} =
\{1,b^{(fg)}_1,\ldots,b^{(fg)}_4\}$ such that
$\mathbb{Q}[t,u,f,g]=\langle B^{(fg)}
\rangle_{\mathbb{Q}[t]}=M^\infty(121)$ together with a
representations of the basis elements in terms of $t$, $u$,
$f$, and $g$.
- bgh.input
-
The integral basis $B^{(gh)} =
\{1,b^{(gh)}_1,\ldots,b^{(gh)}_4\}$ such that $\mathbb{Q}[t, u,
g, h]=\langle B^{(gh)} \rangle_{\mathbb{Q}[t]=M^\infty(121)}$
together with a representations of the basis elements in terms
of $t$, $u$, $g$, and $h$.
- bh.input
-
The integral basis $B^{(h)} =
\{1,b^{(h)}_1,\ldots,b^{(h)}_4\}$ such that $\mathbb{Q}[t, u,
h]=\langle B^{(h)} \rangle_{\mathbb{Q}[t]=M^\infty(121)}$
together with a representations of the basis elements in terms
of $t$, $u$, and $h$.
- ds.input
-
The polynomials $d_1, \ldots, d_6$ with
$d^*=d_1\,{d_2}^2\,d_3^2\,d_4\,d_5\,d_6$
such that
$d^*(t)h \in \mathbb{Q}[t,u]$.
- dz.input
-
The factors of the discriminant $D(T)$ (with respect to $Z$)
of the polynomial $p(T,Z)$ such that $p(t,z)=0$ together with
their exponents.
$D(T) =
dz_1^4\,dz_2^2\,dz_3^4\,dz_4^4\,dz_5\,dz_6^2\,dz_7^2\,dz_8^2
=
T^4\,dz_5 \, d^*$.
- p.maple
- Minimal polynomial p(T, U) and Maple commands to compute an
integral basis for $M^\infty(121)$.
- v.input
-
The polynomial $v(T,U)$ from the basis
$\{1,U,U^2,U^3,v(T,U)\}$ returned by Maple.
- bv.input
-
The integral basis $B^{(v)} =
\{1,b^{(v)}_1,\ldots,b^{(v)}_4\}$ such that $\mathbb{Q}[t, u,
v(t,u)]=\langle B^{(v)} \rangle_{\mathbb{Q}[t]}=M^\infty(121)$
together with a representations of the basis elements in terms
of $t$, $u$, and $v(t,u)$ given by the polynomials
$p_{i,j}(T)$, $i=1,\ldots,4$, $j=0,\ldots,4$.
The Computation has been mostly done by means of the
QEta package.
The scripts to compute the above data are available upon request.
Ralf Hemmecke
Last modified: Thu Sep 10 14:25:00 CEST 2019