Return to Question

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$? (EDIT: There's some ambiguity here: I mean $X$ and $Y$ to be the unique smooth curves (i.e. $\mathbf{P}^1$) with function fields $K(X)$ and $K(Y)$).

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: https://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$? (EDIT: There's some ambiguity here: I mean $X$ and $Y$ to be the unique smooth curves (i.e. $\mathbf{P}^1$) with function fields $K(X)$ and $K(Y)$).

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: https://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$? (EDIT: There's some ambiguity here: I mean $X$ and $Y$ to be the unique smooth curves (i.e. $\mathbf{P}^1$) with function fields $K(X)$ and $K(Y)$).

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: http://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$? (EDIT: There's some ambiguity here: I mean $X$ and $Y$ to be the unique smooth curves (i.e. $\mathbf{P}^1$) with function fields $K(X)$ and $K(Y)$).

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: https://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$?

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: http://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)

Suppose I chose two rational functions, say,

$$u = \frac{t(4+t)^5}{(1+4t)^5}, \qquad v = \frac{t^5(4+t)}{(1+4t)}.$$

Then I know that $K(X) = \mathbf{C}(u,v)$ is the function field of the projective line (Proof: If $K(Y) = \mathbf{C}(t)$, then there is an inclusion $K(X) \subseteq K(Y)$ and hence a surjection $\mathbf{P}^1 \simeq Y \rightarrow X$, and so $X$ must have genus $0$). From this it follows that $K(X) \simeq \mathbf{C}(s)$ for some $s \in \mathbf{C}(t)$.

Is there a practical easy algorithm to explicity construct such an $s$ and, moreover, write $s$ as a rational function of $u$ and $v$? Is there an easy way at least to determine the degree of the map $Y \rightarrow X$? (EDIT: There's some ambiguity here: I mean $X$ and $Y$ to be the unique smooth curves (i.e. $\mathbf{P}^1$) with function fields $K(X)$ and $K(Y)$).

In case you were wondering, the specific choice of $u$ and $v$ where motivated by the question:

Deciding whether a given power series is modular or not

(This question was posted on math.stackexchange a week ago; I am cross posting it here because it did not receive any replies: http://math.stackexchange.com/questions/17960/function-field-of-the-projective-line)