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Chris H
Chris H
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Let $f:X\rightarrow Y$ be an abelian galois coveringmap (not necessarily unramified) of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the characteristic of $k$. We can view the function field $K(X)$ as a $K(Y)$ vector space, and by galois theory we know its structure as a $k[A]$ module is given by $K(Y)[A]$.

So given our assumptions, we see that $K(X)$ has a basis of eigenvectors for $A$, corresponding the the distinct one dimensional representations of $A$ over $k$. Explicitly, it has a basis of $f_\lambda\in K(X)$ such that $g.f_\lambda=\lambda(g).f$ for $\lambda$ a character$\lambda(g)$ an element of $A$ with value in $k$$K(Y)$. So the divisors associated to these $f_\lambda$ are canonical up to principal divisors from $Y$, and we have one for each character of $A$.

My question is, can you describe these divisors more geometrically?

I am vaguely aware that all coverings of this form should come from isogenies of the jacobian, but my knowledge of abelian covers of curves is not great, so apologies if this question is something really simple.

Let $f:X\rightarrow Y$ be an abelian galois covering of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the characteristic of $k$. We can view the function field $K(X)$ as a $K(Y)$ vector space, and by galois theory we know its structure as a $k[A]$ module is given by $K(Y)[A]$.

So given our assumptions, we see that $K(X)$ has a basis of eigenvectors for $A$, corresponding the the distinct one dimensional representations of $A$ over $k$. Explicitly, it has a basis of $f_\lambda\in K(X)$ such that $g.f_\lambda=\lambda(g).f$ for $\lambda$ a character of $A$ with value in $k$. So the divisors associated these $f_\lambda$ are canonical, and we have one for each character of $A$.

My question is, can you describe these divisors more geometrically?

I am vaguely aware that all coverings of this form should come from isogenies of the jacobian, but my knowledge of abelian covers of curves is not great, so apologies if this question is something really simple.

Let $f:X\rightarrow Y$ be an abelian galois map (not necessarily unramified) of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the characteristic of $k$. We can view the function field $K(X)$ as a $K(Y)$ vector space, and by galois theory we know its structure as a $k[A]$ module is given by $K(Y)[A]$.

So given our assumptions, we see that $K(X)$ has a basis of eigenvectors for $A$, corresponding the the distinct one dimensional representations of $A$ over $k$. Explicitly, it has a basis of $f_\lambda\in K(X)$ such that $g.f_\lambda=\lambda(g).f$ for $\lambda(g)$ an element of $K(Y)$. So the divisors associated to these $f_\lambda$ are canonical up to principal divisors from $Y$, and we have one for each character of $A$.

My question is, can you describe these divisors more geometrically?

I am vaguely aware that all coverings of this form should come from isogenies of the jacobian, but my knowledge of abelian covers of curves is not great, so apologies if this question is something really simple.

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Chris H
Chris H
  • 1.9k
  • 9
  • 19

Given an abelian galois map of curves, what are the principal divisors on the source fixed by the galois group?

Let $f:X\rightarrow Y$ be an abelian galois covering of nonsingular complete curves over algebraically closed $k$, where the order of the galois group $A$ is coprime to the characteristic of $k$. We can view the function field $K(X)$ as a $K(Y)$ vector space, and by galois theory we know its structure as a $k[A]$ module is given by $K(Y)[A]$.

So given our assumptions, we see that $K(X)$ has a basis of eigenvectors for $A$, corresponding the the distinct one dimensional representations of $A$ over $k$. Explicitly, it has a basis of $f_\lambda\in K(X)$ such that $g.f_\lambda=\lambda(g).f$ for $\lambda$ a character of $A$ with value in $k$. So the divisors associated these $f_\lambda$ are canonical, and we have one for each character of $A$.

My question is, can you describe these divisors more geometrically?

I am vaguely aware that all coverings of this form should come from isogenies of the jacobian, but my knowledge of abelian covers of curves is not great, so apologies if this question is something really simple.