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edited Jul 25 2011 at 23:16 |
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Assume $j$ is algebraic, i.e., $X$ can be defined over a number field.
Question. Can the function $\log\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded (from above or below) in terms of the (height of the) $j$-invariant?
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
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edited Jul 25 2011 at 23:03 |
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Assume $j$ is algebraic, i.e., $X$ can be defined over a number field.
Question. Can the function $\log\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded in terms of the (height of the) $j$-invariant?
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
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edited Jul 25 2011 at 16:49 |
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Question. Can the function $\Vert \log\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded in terms of the (height of the) $j$-invariant?
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
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edited Jul 25 2011 at 15:36 |
Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Question. Can the function $\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded in terms of the (height of the) $j$-invariant.j$-invariant?
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
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edited Jul 25 2011 at 12:00 |
Bounding the modular discriminant of an elliptic curve in the j-invariant
Suppose that we are given Consider an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert $\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert$, 1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.
Question.
I would like to bound Can the function $\Vert\Delta\Vert(X)$ from below and above using constants depending \Vert \Delta \Vert(-)$ (on $d$ and the ramification moduli space of $\pi$. (Note that Hurwitz gives that elliptic curves over $2d = \deg R$ with \mathbf{C}$) be bounded in terms of the $R$ j$-invariant.
Firstly, one should be able to answer this question ineffectively. That is, to give a yes or no answer to the ramification divisor.question. An effective bound (if it exists) might be a bit harder to obtain.
I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.
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edited Dec 8 2010 at 9:42 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) = \vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, 1-q^k)^{24}\vert$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
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edited Dec 8 2010 at 8:16 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \vert \log \Vert \Delta \Vert(X) \vert \leq 1020323 \cdot d^{10} \cdot s_1\cdot \ldots \cdot s_n\cdot \vert S\vert^6. $$
Example. If $\pi$ is the Weierstrass function then it is etale above ${0,1,\infty,s}$ and of degree $2$. The modular discriminant is given by $\Vert\Delta\Vert(X) = s^2(s-1)^2$. This is precisely what I'm looking for.
EDIT: I think I just answered my own question! One simply performs some automorphisms of the projective line and uses the above formula. Whoooops.
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edited Dec 7 2010 at 11:25 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \vert \log \Vert \Delta \Vert(X) \vert \leq 1020323 \cdot d^{10} \cdot s_1\cdot \ldots \cdot s_n\cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
Example. If $\pi$ is the Weierstrass function then it is etale above ${0,1,\infty,s}$ and of degree $2$. The modular discriminant is given by $\Vert\Delta\Vert(X) = s^2(s-1)^2$. This is precisely what I'm looking for. Note that here we get an equality whereas in the general case
EDIT: I expect only an inequality involving $d$ and the branch points of $\pi$.
So just to be clear. think I want to bound $\Vert\Delta\Vert(X)$ from below and above in $d$ and constants depending on the branch locus $S=\{s_1,\ldots,s_n\}$.
Note: This question is analogous to the same just answered my own questionfor number fields, where one can use ! One simply performs some automorphisms of the ramification to bound projective line and uses the discriminantabove formula.
Note: I've edited the question a couple of times in the past hourWhoooops.
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edited Dec 7 2010 at 9:47 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ S=\{s_1,\ldots,s_n\}\subset\mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \vert \log \Vert \Delta \Vert(X) \vert \leq 1020323 \cdot d^{10} \cdot s_1\cdot \ldots \cdot s_n\cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
For example, if
Example. If $\pi$ is the Weierstrass function then it is etale above ${0,1,\infty,\lambda}$. {0,1,\infty,s}$ and of degree $2$. The modular discriminant is given by $\Delta(X) \Vert\Delta\Vert(X) = \lambda^2(\lambda-1)^2$. s^2(s-1)^2$. This is precisely what I'm looking for. (In this case Note that here we get an equality . But whereas in the general case I expect only an inequality of this form involving $d$ and the branch points of $\pi$. )
So just to be clear. I want to bound $\Vert\Delta\Vert(X)$ from below and above in $d$ and constants depending on the branch locus $S=\{s_1,\ldots,s_n\}$.
Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.
Note: I've edited the question a couple of times in the past hour.
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edited Dec 7 2010 at 9:36 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \vert \log \Vert \Delta \Vert(X) \vert \leq 1020323 \cdot d^{10} \cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
For example, if $\pi$ is the Weierstrass function then it is etale above ${0,1,\infty,\lambda}$ then it shouldn't be very hard using relations with {0,1,\infty,\lambda}$. The modular discriminant is given by $\Delta(X) = \lambda^2(\lambda-1)^2$. This is precisely what I'm looking for. (In this case we get an equality. But in the general case I expect only an inequality of this form involving $j$-invariant d$ and the Weierstrass equation. But again, how?branch points of $\pi$. )
Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.
Note: I've edited the question a couple of times in the past hour.
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edited Dec 7 2010 at 8:31 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ from below and above using constants depending on $d$ and the ramification of $\pi$. (Note that Hurwitz gives that $2d = \deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \vert \log \Vert \Delta \Vert(X) \vert \leq 1020323 \cdot d^{10} \cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
For example, if $\pi$ is etale above ${0,1,\infty,\lambda}$ then it shouldn't be very hard using relations with the $j$-invariant and the Weierstrass equation. But again, how?
Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.
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edited Dec 7 2010 at 8:25 |
Suppose that we are given an elliptic curve $E$ over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ using constants depending on $d$, $K$ d$ and the ramification of $\pi$ \pi$. (and Note that Hurwitz gives that $\pi \times_K 2d = \mathbf{C}$). deg R$ with $R$ the ramification divisor.)
For example, the bound could be something like
$$ \log \Vert \Delta \Vert(X) \leq [K:\mathbf{Q}]\cdot d^{10} \cdot \vert d_K\vert \cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
For example, if $\pi$ is etale above ${0,1,\infty,\lambda}$ then it shouldn't be very hard. But again, how?
Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.
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edited Dec 7 2010 at 8:12 |
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Bounding the modular discriminant of an elliptic curve
Suppose that we are given an elliptic curve over a number field $K$ and a finite morphism $\pi:E\longrightarrow \mathbf{P}^1_K$ of degree $d$. Assume $\pi$ is unramified outside a finite set $S\subset \mathbf{P}^1_K$ of points. Fix an embedding $K\longrightarrow \mathbf{C}$.
Consider $X=E\times_K \mathbf{C}$. We can write $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $\Vert \Delta\Vert(X) =\vert (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}$, where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant.
I would like to bound $\Vert\Delta\Vert(X)$ using constants depending on $d$, $K$ and the ramification of $\pi$ (and $\pi \times_K \mathbf{C}$).
For example, the bound could be something like
$$ \log \Vert \Delta \Vert(X) \leq [K:\mathbf{Q}]\cdot d^{10} \cdot \vert d_K\vert \cdot \vert S\vert^6. $$
This should be fairly standard. It comes down to choosing $\tau$ properly using $\pi$. How does one do this?
Note: This question is analogous to the same question for number fields, where one can use the ramification to bound the discriminant.
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