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Question 1

1vote

1answer

46views

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 ...

Engin Şenel

11

asked15 hours ago

Question 2

3votes

1answer

175views

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 $\...

Vincent Tran

349

askedSep 17 at 3:35

Question 3

2votes

1answer

196views

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 ...

mathfan24

822

askedJul 3 at 10:14

Question 4

1vote

1answer

134views

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 $\...

ANT

57

askedApr 23 at 5:24

Question 5

0votes

0answers

43views

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[...

jawlang

715

askedMar 29 at 3:37

Question 6

1vote

0answers

33views

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 ...

oleout

1,242

askedMar 3 at 15:28

Question 7

0votes

1answer

37views

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 $Aut(E/K)$...

ZWJ

591

askedFeb 16 at 7:56

Question 8

1vote

0answers

110views

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 ...

ZWJ

591

askedFeb 15 at 11:11

Question 9

0votes

1answer

65views

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 ...

ZWJ

591

askedFeb 14 at 13:04

Question 10

0votes

1answer

83views

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 ...

ZWJ

591

askedFeb 8 at 16:47

Question 11

0votes

0answers

43views

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 isp[i][j][k], using ...

Parcly Taxel

106k

askedJan 15 at 12:41

Question 12

1vote

0answers

99views

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 $\...

cnks

21

askedDec 25, 2024 at 2:21

Question 13

1vote

0answers

78views

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, ...

cnks

21

askedDec 20, 2024 at 0:49

Question 14

0votes

1answer

131views

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, ...

Lorenzo Andreaus

652

askedOct 25, 2024 at 10:03

Question 15

1vote

0answers

42views

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} -...

jawlang

715

askedJun 14, 2024 at 0:51

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Stack Exchange Network

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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