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On a theorem by Girondo and Gonzalez-Diez
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In their textbook, Girondo and Gonzalez-Diez show the following theorem.
Let $S$ be a compact Riemann surface admitting an automorphism $tau$ of prime order such that $S/langle taurangle$ is isomorphic to the projective line. Assume that $tau$ fixes $r+1$ points $P_1,dots,P_r+1$ with rotation numbers $d_1,dots,d_r+1$. Then $S$ is isomorphic to the Riemann surface of an algebraic curve of the form $y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$, where $1 le m_i <p$ and $sumlimits_i=1^r m_i$ is prime to $p$. Moreover, there is an isomorphism $Phi: S to y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$ under which $tau in Aut(S)$ corresponds to $(x,y) to (x,xi_p y)$, the points $P_1,dots,P_r+1$ to $hat P_1=(a_1,0),dots,hat P_r=(a_r,0)$ and $hat P_r+1=infty$ and the integers $m_1,dots,m_r,m_r+1:=-sumlimits_i=1^r m_i$ are the inverses of $d_1,dots,d_r+1$ modulo $p$.
My question is the following: they are assuming that $tau$ must be of prime order. What happens when its order is not a prime number?
algebraic-geometry
asked Jul 18 at 17:20
TheWanderer
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In their textbook, Girondo and Gonzalez-Diez show the following theorem.
Let $S$ be a compact Riemann surface admitting an automorphism $tau$ of prime order such that $S/langle taurangle$ is isomorphic to the projective line. Assume that $tau$ fixes $r+1$ points $P_1,dots,P_r+1$ with rotation numbers $d_1,dots,d_r+1$. Then $S$ is isomorphic to the Riemann surface of an algebraic curve of the form $y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$, where $1 le m_i <p$ and $sumlimits_i=1^r m_i$ is prime to $p$. Moreover, there is an isomorphism $Phi: S to y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$ under which $tau in Aut(S)$ corresponds to $(x,y) to (x,xi_p y)$, the points $P_1,dots,P_r+1$ to $hat P_1=(a_1,0),dots,hat P_r=(a_r,0)$ and $hat P_r+1=infty$ and the integers $m_1,dots,m_r,m_r+1:=-sumlimits_i=1^r m_i$ are the inverses of $d_1,dots,d_r+1$ modulo $p$.
My question is the following: they are assuming that $tau$ must be of prime order. What happens when its order is not a prime number?
algebraic-geometry
asked Jul 18 at 17:20
TheWanderer
1,75511029
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
In their textbook, Girondo and Gonzalez-Diez show the following theorem.
Let $S$ be a compact Riemann surface admitting an automorphism $tau$ of prime order such that $S/langle taurangle$ is isomorphic to the projective line. Assume that $tau$ fixes $r+1$ points $P_1,dots,P_r+1$ with rotation numbers $d_1,dots,d_r+1$. Then $S$ is isomorphic to the Riemann surface of an algebraic curve of the form $y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$, where $1 le m_i <p$ and $sumlimits_i=1^r m_i$ is prime to $p$. Moreover, there is an isomorphism $Phi: S to y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$ under which $tau in Aut(S)$ corresponds to $(x,y) to (x,xi_p y)$, the points $P_1,dots,P_r+1$ to $hat P_1=(a_1,0),dots,hat P_r=(a_r,0)$ and $hat P_r+1=infty$ and the integers $m_1,dots,m_r,m_r+1:=-sumlimits_i=1^r m_i$ are the inverses of $d_1,dots,d_r+1$ modulo $p$.
My question is the following: they are assuming that $tau$ must be of prime order. What happens when its order is not a prime number?
algebraic-geometry
asked Jul 18 at 17:20
TheWanderer
1,75511029
In their textbook, Girondo and Gonzalez-Diez show the following theorem.
Let $S$ be a compact Riemann surface admitting an automorphism $tau$ of prime order such that $S/langle taurangle$ is isomorphic to the projective line. Assume that $tau$ fixes $r+1$ points $P_1,dots,P_r+1$ with rotation numbers $d_1,dots,d_r+1$. Then $S$ is isomorphic to the Riemann surface of an algebraic curve of the form $y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$, where $1 le m_i <p$ and $sumlimits_i=1^r m_i$ is prime to $p$. Moreover, there is an isomorphism $Phi: S to y^p=(x-a_1)^m_1 dots (x-a_r)^m_r$ under which $tau in Aut(S)$ corresponds to $(x,y) to (x,xi_p y)$, the points $P_1,dots,P_r+1$ to $hat P_1=(a_1,0),dots,hat P_r=(a_r,0)$ and $hat P_r+1=infty$ and the integers $m_1,dots,m_r,m_r+1:=-sumlimits_i=1^r m_i$ are the inverses of $d_1,dots,d_r+1$ modulo $p$.
My question is the following: they are assuming that $tau$ must be of prime order. What happens when its order is not a prime number?
algebraic-geometry
asked Jul 18 at 17:20
TheWanderer
1,75511029
asked Jul 18 at 17:20
TheWanderer
1,75511029
asked Jul 18 at 17:20
TheWanderer
1,75511029
asked Jul 18 at 17:20
TheWanderer
1,75511029
1,75511029
add a comment |Â
add a comment |Â
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