Connection between definitions in function fields and on curves
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(Sorry for the weird title, I really don't know how to describe this question in a line)
I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all the subject, and I was trying to get some sketchy introduction to the geometrical perspective. I got the book "Codes and Curves" by Judy Walker and I am trying to make the translation between definitions and objects in each book.
Rosen define divisors on a function field $F$ over a base field $k=mathbb{F}_q$ to be the free abelian group over the set of all primes, and Walker's definition for a divisor on a projective, non-singular algebraic curve $f(x,y)=0$ is similar but not exactly the same, she defines it as the free abelian group over the set of "points with multiplicity"- if we have a point $P$ on the curve we're looking at its orbit under the froubenius map $xmapsto x^q$, and this set is the "point with multiplicity".
Also, the principal divisor of an element $ain F$ is defined (in Rosen's book) by $sum_P v_P (x)P$ where $P$ are the primes of the function field, and $v_P$ are the valuations of these primes. Walker's definition is of divisors of rational functions $h/g$ of the curve, and it is done by taking the intersection divisors of $C_f cap C_g, C_fcap C_h$ and substracting them (where $C_f,C_g,C_h$ are the curves obtained by the zero locus of the equations of $f,g,h$). Finally, the definitions for the degree of a prime / point are different - the degree of $P$ is $[O_P/PO_P:k(x)]$ (where $O_P$ is the place of $P$) and the degree of the point $P$ on the curve is the size of its orbit under the Frobenius map $xmapsto x^q$.
How each pair of definitions is connected to each other? (Thanks for taking the time to reading this!)
algebraic-number-theory algebraic-curves function-fields
asked Nov 27 at 21:07
Madarb
187111
add a comment |
up vote
2
down vote
favorite
(Sorry for the weird title, I really don't know how to describe this question in a line)
I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all the subject, and I was trying to get some sketchy introduction to the geometrical perspective. I got the book "Codes and Curves" by Judy Walker and I am trying to make the translation between definitions and objects in each book.
Rosen define divisors on a function field $F$ over a base field $k=mathbb{F}_q$ to be the free abelian group over the set of all primes, and Walker's definition for a divisor on a projective, non-singular algebraic curve $f(x,y)=0$ is similar but not exactly the same, she defines it as the free abelian group over the set of "points with multiplicity"- if we have a point $P$ on the curve we're looking at its orbit under the froubenius map $xmapsto x^q$, and this set is the "point with multiplicity".
Also, the principal divisor of an element $ain F$ is defined (in Rosen's book) by $sum_P v_P (x)P$ where $P$ are the primes of the function field, and $v_P$ are the valuations of these primes. Walker's definition is of divisors of rational functions $h/g$ of the curve, and it is done by taking the intersection divisors of $C_f cap C_g, C_fcap C_h$ and substracting them (where $C_f,C_g,C_h$ are the curves obtained by the zero locus of the equations of $f,g,h$). Finally, the definitions for the degree of a prime / point are different - the degree of $P$ is $[O_P/PO_P:k(x)]$ (where $O_P$ is the place of $P$) and the degree of the point $P$ on the curve is the size of its orbit under the Frobenius map $xmapsto x^q$.
How each pair of definitions is connected to each other? (Thanks for taking the time to reading this!)
algebraic-number-theory algebraic-curves function-fields
asked Nov 27 at 21:07
Madarb
187111
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
(Sorry for the weird title, I really don't know how to describe this question in a line)
I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all the subject, and I was trying to get some sketchy introduction to the geometrical perspective. I got the book "Codes and Curves" by Judy Walker and I am trying to make the translation between definitions and objects in each book.
Rosen define divisors on a function field $F$ over a base field $k=mathbb{F}_q$ to be the free abelian group over the set of all primes, and Walker's definition for a divisor on a projective, non-singular algebraic curve $f(x,y)=0$ is similar but not exactly the same, she defines it as the free abelian group over the set of "points with multiplicity"- if we have a point $P$ on the curve we're looking at its orbit under the froubenius map $xmapsto x^q$, and this set is the "point with multiplicity".
Also, the principal divisor of an element $ain F$ is defined (in Rosen's book) by $sum_P v_P (x)P$ where $P$ are the primes of the function field, and $v_P$ are the valuations of these primes. Walker's definition is of divisors of rational functions $h/g$ of the curve, and it is done by taking the intersection divisors of $C_f cap C_g, C_fcap C_h$ and substracting them (where $C_f,C_g,C_h$ are the curves obtained by the zero locus of the equations of $f,g,h$). Finally, the definitions for the degree of a prime / point are different - the degree of $P$ is $[O_P/PO_P:k(x)]$ (where $O_P$ is the place of $P$) and the degree of the point $P$ on the curve is the size of its orbit under the Frobenius map $xmapsto x^q$.
How each pair of definitions is connected to each other? (Thanks for taking the time to reading this!)
algebraic-number-theory algebraic-curves function-fields
asked Nov 27 at 21:07
Madarb
187111
(Sorry for the weird title, I really don't know how to describe this question in a line)
I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all the subject, and I was trying to get some sketchy introduction to the geometrical perspective. I got the book "Codes and Curves" by Judy Walker and I am trying to make the translation between definitions and objects in each book.
Rosen define divisors on a function field $F$ over a base field $k=mathbb{F}_q$ to be the free abelian group over the set of all primes, and Walker's definition for a divisor on a projective, non-singular algebraic curve $f(x,y)=0$ is similar but not exactly the same, she defines it as the free abelian group over the set of "points with multiplicity"- if we have a point $P$ on the curve we're looking at its orbit under the froubenius map $xmapsto x^q$, and this set is the "point with multiplicity".
Also, the principal divisor of an element $ain F$ is defined (in Rosen's book) by $sum_P v_P (x)P$ where $P$ are the primes of the function field, and $v_P$ are the valuations of these primes. Walker's definition is of divisors of rational functions $h/g$ of the curve, and it is done by taking the intersection divisors of $C_f cap C_g, C_fcap C_h$ and substracting them (where $C_f,C_g,C_h$ are the curves obtained by the zero locus of the equations of $f,g,h$). Finally, the definitions for the degree of a prime / point are different - the degree of $P$ is $[O_P/PO_P:k(x)]$ (where $O_P$ is the place of $P$) and the degree of the point $P$ on the curve is the size of its orbit under the Frobenius map $xmapsto x^q$.
How each pair of definitions is connected to each other? (Thanks for taking the time to reading this!)
algebraic-number-theory algebraic-curves function-fields
algebraic-number-theory algebraic-curves function-fields
asked Nov 27 at 21:07
Madarb
187111
asked Nov 27 at 21:07
Madarb
187111
edited Nov 27 at 21:13
asked Nov 27 at 21:07
Madarb
187111
asked Nov 27 at 21:07
Madarb
187111
asked Nov 27 at 21:07
Madarb
187111
187111
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