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How to compute explicitly the covering map in the modularity theorem?
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The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $pi:X_0(N) to E$ for every $E$, an elliptic curve defined over $mathbb{Q}$.
If $y^2 = x^3 + a x + b$ is the Weierstraß equation of $E$, this amounts to finding power-series $x(q)$, $y(q)$ in $q = exp(2 pi i z)$ where $z$ is the standard coordinate of $mathbb{C}$. The $x(q), y(q)$ are from the function field of $X_0(N)$, so they are modular functions with respect to $Gamma_0(N)$ (is this true?).
In the system Pari/GP there is the function elltaniyama to compute such $x(q)$, $y(q)$, and in Sage there is the function modular_parametrization. But all searching with Google did not find me a paper, where the algorithm involved is described. I found some articles describing how to compute the modularity conductor $N$, but not, how $x(q)$ and $y(q)$ can be calculated.
Could someone direct me to a paper (or book) where to find an explanation of the algorithm used?
algebraic-geometry elliptic-curves arithmetic-geometry
edited Jan 21 at 19:55
André 3000
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
$endgroup$
add a comment |
$begingroup$
The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $pi:X_0(N) to E$ for every $E$, an elliptic curve defined over $mathbb{Q}$.
If $y^2 = x^3 + a x + b$ is the Weierstraß equation of $E$, this amounts to finding power-series $x(q)$, $y(q)$ in $q = exp(2 pi i z)$ where $z$ is the standard coordinate of $mathbb{C}$. The $x(q), y(q)$ are from the function field of $X_0(N)$, so they are modular functions with respect to $Gamma_0(N)$ (is this true?).
In the system Pari/GP there is the function elltaniyama to compute such $x(q)$, $y(q)$, and in Sage there is the function modular_parametrization. But all searching with Google did not find me a paper, where the algorithm involved is described. I found some articles describing how to compute the modularity conductor $N$, but not, how $x(q)$ and $y(q)$ can be calculated.
Could someone direct me to a paper (or book) where to find an explanation of the algorithm used?
algebraic-geometry elliptic-curves arithmetic-geometry
edited Jan 21 at 19:55
André 3000
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
$endgroup$
1
$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
4
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42
add a comment |
$begingroup$
The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $pi:X_0(N) to E$ for every $E$, an elliptic curve defined over $mathbb{Q}$.
If $y^2 = x^3 + a x + b$ is the Weierstraß equation of $E$, this amounts to finding power-series $x(q)$, $y(q)$ in $q = exp(2 pi i z)$ where $z$ is the standard coordinate of $mathbb{C}$. The $x(q), y(q)$ are from the function field of $X_0(N)$, so they are modular functions with respect to $Gamma_0(N)$ (is this true?).
In the system Pari/GP there is the function elltaniyama to compute such $x(q)$, $y(q)$, and in Sage there is the function modular_parametrization. But all searching with Google did not find me a paper, where the algorithm involved is described. I found some articles describing how to compute the modularity conductor $N$, but not, how $x(q)$ and $y(q)$ can be calculated.
Could someone direct me to a paper (or book) where to find an explanation of the algorithm used?
algebraic-geometry elliptic-curves arithmetic-geometry
edited Jan 21 at 19:55
André 3000
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
$endgroup$
The modularity theorem (original Shimura-Taniyama-Weil conjecture) asserts the existence of a covering (uniformization) map $pi:X_0(N) to E$ for every $E$, an elliptic curve defined over $mathbb{Q}$.
If $y^2 = x^3 + a x + b$ is the Weierstraß equation of $E$, this amounts to finding power-series $x(q)$, $y(q)$ in $q = exp(2 pi i z)$ where $z$ is the standard coordinate of $mathbb{C}$. The $x(q), y(q)$ are from the function field of $X_0(N)$, so they are modular functions with respect to $Gamma_0(N)$ (is this true?).
In the system Pari/GP there is the function elltaniyama to compute such $x(q)$, $y(q)$, and in Sage there is the function modular_parametrization. But all searching with Google did not find me a paper, where the algorithm involved is described. I found some articles describing how to compute the modularity conductor $N$, but not, how $x(q)$ and $y(q)$ can be calculated.
Could someone direct me to a paper (or book) where to find an explanation of the algorithm used?
algebraic-geometry elliptic-curves arithmetic-geometry
algebraic-geometry elliptic-curves arithmetic-geometry
edited Jan 21 at 19:55
André 3000
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
edited Jan 21 at 19:55
André 3000
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
edited Jan 21 at 19:55
André 3000
12.7k22243
edited Jan 21 at 19:55
André 3000
12.7k22243
edited Jan 21 at 19:55
André 3000
12.7k22243
12.7k22243
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
asked Apr 8 '16 at 22:47
Jürgen BöhmJürgen Böhm
2,288512
2,288512
1
$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
4
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42
add a comment |
1
$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
4
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42
1
1
$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
4
4
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42
add a comment |
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$begingroup$
I don't think there exists a general algorithm to do this. In fact the proof of the fact that such a map exists is deeply non-effective, as it just shows that the $p$-adic Galois representation of $E$ is isomorphic to the $p$-adic Galois representation attached to a newform of level= the conductor of $E$. Probably for low genuses of $X_0(N)$ there are some special tricks.
$endgroup$
– Ferra
Apr 9 '16 at 9:45
4
$begingroup$
have you read Cremona's book homepages.warwick.ac.uk/~masgaj/book/amec.html
$endgroup$
– mercio
Apr 9 '16 at 22:42