Questions tagged [function-fields]
This tag is for questions related to function field, a finitely generated field extension of transcendence degree $n>0$ of a field of constants $k$.
240 questions
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Do places ever become inert in the Hermitian function field over $\mathbb{F}_4$ ($q = 2$)?
I consider the Hermitian function field
$H = \mathbb{F}_4(x,y)$, given by $y^2 + y = x^3$,
which is a quadratic extension of $F = \mathbb{F}_4(x)$.
Let $P$ be a place of $F$. If $v_P(x^3) \ge 0$, then ...
- 11
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Analogue for Kronecker-Weber for Drinfeld Modules
I was reading some notes on Drinfeld modules, and it states that the analogue of the Kronecker-Weber theorem is (for a Drinfeld module $\phi$)
Every abelian extension of $K$ in which the place $\...
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Finding the degree of a morphism
Let $X$ and $Y$ be varieties of the same dimension. Define the degree of a dominant morphism $f:X \to Y$ as the degree of the function field extension $|K(X):K(Y)|$.
I am finding it hard to apply this ...
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Compute an integral closure
This question is from the 15th Yau Contest.
Let $k$ be a field of characteristic $p>0$ and consider $k(t) \subset k((t))$. Let $\alpha \in k[[t]]$ which is transcendental over $k(t)$, and write $\...
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reference for a fact about height change under a morphism over a global function field
I'm looking for a refence containing a global function field analogue of the following theorem.
For a morphism $F: \mathbb{P}^{n} \to \mathbb{P}^{m}$ of degree $d$ (i.e., each $F_{i}$ of $F = \left[...
- 715
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About the height of a constant polynomial in some finite extension
Let $k$ be an algebraically closed field of characteristic zero and $K$ be an algebraic function field with field of constants $k$, i.e., $K$ is a finite extension of $k(t)$. Let $f(X) = \sum_{i=0}^n ...
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Is the automorphism group of algebraic function field (of a curve) isomorphism to the automorphism group of the curve?
Seems to be an easy question (arising from a paper I'm reading)...
For elliptic curve $C$ defined over $K=F_q$, let $Aut(C/K)$ be its automorphism group (automorphisms defined over $K$).
And $Aut(E/K)$...
- 591
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What's the geometric interpretation of separable extension for algebraic function field?
It's known that there is a correspondence between algebraic curves and algebraic function fields. Many concepts can be transferred from one to the other one.
Is there any interpretation (in the ...
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How to compute function field (for a curve family)?
This might be a stupid question... I‘m not good at Algebraic Geometry...
I'm reading a paper on Algebraic geometry code. There is a family of curve named as 'Garcia-Stichtenoth Curves'.
The function ...
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Is the fixed field of automorphism group (of an algebraic function field) still a function field?
Say, $C$ is elliptic curve defined over a finite field $F_q$. It's function field is $E$.
The automorphism group $Aut(E/F_q)$ is the group consisting of field automorphisms of $E$
while fixing ...
- 591
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Galois group and subfield of a polynomial with coefficients in a bivariate function field
Consider the following polynomial $p$ in three variables, a main one called $x$ and two secondary ones $b$ and $c$. The $x^ib^jc^k$ coefficient is p[i][j][k], using ...
- 106k
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Computing the inverses of two linear maps
I have two linear maps $X: \mathcal{L}(G) \rightarrow \mathbb{F}_q^k$ and $Y: \mathcal{L}(2G) \rightarrow \mathbb{F}_{q^n}$, where $G$ is a divisor of the rational function field $F_q(x)$ over $\...
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Evaluation of Riemann Roch basis in Magma
I'm trying to evaluate the basis of a Riemann-Roch space of the divisor $G$ of the rational function field $F$ over $GF(2^{16})$ at 8 places of $F$ using Magma. When building the sequence of outputs, ...
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Function field and rational maps to the projective line correspondence
Let $V/k$ be a variety. There seems to be a well-known correspondence between non constant elements of the function field $K(V)$ and dominant rational maps $V\dashrightarrow\mathbb{P}^1$. Furthermore, ...
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The galois group of $p^{k}$-torsions of an elliptic curve defined over a global function field of characteristic $p \ge 5$
I'm looking for an analogy of Corollary 7.5.3 in 『A first course in modular forms』(Diamond and Shurman) over a global function field. The colloary follows:
For the elliptic curve $$
E: y^{2} = x^{3} -...
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