Questions tagged [function-fields]

Question 1

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 ...

Question 2

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 $\...

Question 3

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 ...

Question 4

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 $\...

Question 5

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[...

Question 6

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 ...

Question 7

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 $Aut(E/K)$...

Question 8

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 ...

Question 9

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 ...

Question 10

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 ...

Question 11

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 ...

Question 12

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 $\...

Question 13

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, ...

Question 14

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, ...

Question 15

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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Questions tagged [function-fields]

Do places ever become inert in the Hermitian function field over $\mathbb{F}_4$ ($q = 2$)?

Analogue for Kronecker-Weber for Drinfeld Modules

Finding the degree of a morphism

Compute an integral closure

reference for a fact about height change under a morphism over a global function field

About the height of a constant polynomial in some finite extension

Is the automorphism group of algebraic function field (of a curve) isomorphism to the automorphism group of the curve?

What's the geometric interpretation of separable extension for algebraic function field?

How to compute function field (for a curve family)?

Is the fixed field of automorphism group (of an algebraic function field) still a function field?

Galois group and subfield of a polynomial with coefficients in a bivariate function field

Computing the inverses of two linear maps

Evaluation of Riemann Roch basis in Magma

Function field and rational maps to the projective line correspondence

The galois group of $p^{k}$-torsions of an elliptic curve defined over a global function field of characteristic $p \ge 5$

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