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Fourier expansion at inequivalent cusps
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Let $Gammasubset SL(2,mathbbR)$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $Gamma.$
Consider $fin M_k(Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c_1$ is known.
Given the above expansion, is there an algorithm to compute (even numerically) the Fourier expansion of $f$ at the cusp $c_2$?
Thanks!
modular-forms automorphic-forms
asked 4 hours ago
Gabriele
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Let $Gammasubset SL(2,mathbbR)$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $Gamma.$
Consider $fin M_k(Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c_1$ is known.
Given the above expansion, is there an algorithm to compute (even numerically) the Fourier expansion of $f$ at the cusp $c_2$?
Thanks!
modular-forms automorphic-forms
asked 4 hours ago
Gabriele
212
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Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago
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up vote
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Let $Gammasubset SL(2,mathbbR)$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $Gamma.$
Consider $fin M_k(Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c_1$ is known.
Given the above expansion, is there an algorithm to compute (even numerically) the Fourier expansion of $f$ at the cusp $c_2$?
Thanks!
modular-forms automorphic-forms
asked 4 hours ago
Gabriele
212
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Let $Gammasubset SL(2,mathbbR)$ be a Fuchsian group of the first kind. Let $c_1, c_2$ be inequivalent cusps of $Gamma.$
Consider $fin M_k(Gamma)$ a weight $k$ holomorphic automorphic form, and suppose the Fourier expansion of $f$ at the cusp $c_1$ is known.
Given the above expansion, is there an algorithm to compute (even numerically) the Fourier expansion of $f$ at the cusp $c_2$?
Thanks!
modular-forms automorphic-forms
modular-forms automorphic-forms
asked 4 hours ago
Gabriele
212
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Gabriele
212
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Gabriele
212
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 4 hours ago
Gabriele
212
asked 4 hours ago
Gabriele
212
212
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Gabriele is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago
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If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago
If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago
If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago
add a comment |Â
2 Answers
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Pari/GP 2.11 does this for $Gamma_1(N)$ (more precisely for the spaces $M_k(Gamma_0(N),chi)$). The algorithm is based on a variant of a theorem of Borisov--Gunnells on the generation by products of two Eisenstein series.
answered 3 hours ago
Henri Cohen
2,834421
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This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$int_0^1 f|_sigma_2 (z) e(-nz) dx,$$ where $sigma_2$ is a scaling matrix for the cusp $c_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|_sigma_2$ is periodic with period $1$.
In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|_sigma_2$ at any $z in mathbbH$, so the above integral can be numerically calculated to any degree of precision.
answered 3 hours ago
Matt Young
3,40011525
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
Pari/GP 2.11 does this for $Gamma_1(N)$ (more precisely for the spaces $M_k(Gamma_0(N),chi)$). The algorithm is based on a variant of a theorem of Borisov--Gunnells on the generation by products of two Eisenstein series.
answered 3 hours ago
Henri Cohen
2,834421
add a comment |Â
up vote
2
down vote
Pari/GP 2.11 does this for $Gamma_1(N)$ (more precisely for the spaces $M_k(Gamma_0(N),chi)$). The algorithm is based on a variant of a theorem of Borisov--Gunnells on the generation by products of two Eisenstein series.
answered 3 hours ago
Henri Cohen
2,834421
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Pari/GP 2.11 does this for $Gamma_1(N)$ (more precisely for the spaces $M_k(Gamma_0(N),chi)$). The algorithm is based on a variant of a theorem of Borisov--Gunnells on the generation by products of two Eisenstein series.
answered 3 hours ago
Henri Cohen
2,834421
Pari/GP 2.11 does this for $Gamma_1(N)$ (more precisely for the spaces $M_k(Gamma_0(N),chi)$). The algorithm is based on a variant of a theorem of Borisov--Gunnells on the generation by products of two Eisenstein series.
answered 3 hours ago
Henri Cohen
2,834421
answered 3 hours ago
Henri Cohen
2,834421
answered 3 hours ago
Henri Cohen
2,834421
answered 3 hours ago
Henri Cohen
2,834421
2,834421
add a comment |Â
add a comment |Â
up vote
1
down vote
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$int_0^1 f|_sigma_2 (z) e(-nz) dx,$$ where $sigma_2$ is a scaling matrix for the cusp $c_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|_sigma_2$ is periodic with period $1$.
In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|_sigma_2$ at any $z in mathbbH$, so the above integral can be numerically calculated to any degree of precision.
answered 3 hours ago
Matt Young
3,40011525
add a comment |Â
up vote
1
down vote
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$int_0^1 f|_sigma_2 (z) e(-nz) dx,$$ where $sigma_2$ is a scaling matrix for the cusp $c_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|_sigma_2$ is periodic with period $1$.
In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|_sigma_2$ at any $z in mathbbH$, so the above integral can be numerically calculated to any degree of precision.
answered 3 hours ago
Matt Young
3,40011525
add a comment |Â
up vote
1
down vote
up vote
1
down vote
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$int_0^1 f|_sigma_2 (z) e(-nz) dx,$$ where $sigma_2$ is a scaling matrix for the cusp $c_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|_sigma_2$ is periodic with period $1$.
In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|_sigma_2$ at any $z in mathbbH$, so the above integral can be numerically calculated to any degree of precision.
answered 3 hours ago
Matt Young
3,40011525
This can be done numerically. The $n$-th Fourier coefficient around the cusp is given by an integral of the form $$int_0^1 f|_sigma_2 (z) e(-nz) dx,$$ where $sigma_2$ is a scaling matrix for the cusp $c_2$. See p.43 of Iwaniec's Topics in Classical Automorphic Forms for definitions. In the above, $z=x+iy$ and the integral is independent of $y>0$ which generally one picks to make the calculation as efficient as possible. Also, I am assuming that the multiplier system is such that $f|_sigma_2$ is periodic with period $1$.
In principle, the Fourier expansion of $f$ allows one to numerically approximate $f|_sigma_2$ at any $z in mathbbH$, so the above integral can be numerically calculated to any degree of precision.
answered 3 hours ago
Matt Young
3,40011525
answered 3 hours ago
Matt Young
3,40011525
answered 3 hours ago
Matt Young
3,40011525
answered 3 hours ago
Matt Young
3,40011525
3,40011525
add a comment |Â
add a comment |Â
Gabriele is a new contributor. Be nice, and check out our Code of Conduct.
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If I remember correctly, the PhD thesis of Christophe Delaunay talks about this. See III.2 of delaunay.perso.math.cnrs.fr/these.pdf
– Siksek
3 hours ago