# Questions tagged [function-fields]

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**27**

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849 views

### What is the most direct proof of the Riemann hypothesis for Dirichlet L-functions over function fields?

Let $\mathbb{F}$ be a finite field of order $q$, let $m$ be an irreducible polynomial in the ring $\mathbb{F}[T]$, and let $\chi$ be a Dirichlet character modulo $m$. Define the function field ...

**2**

votes

**0**answers

123 views

### A version of weak approximation with S-integers

Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let
$$K_S = \prod_{v\in S} K_v$$
where $K_v$ is the completion of $K$ at $v$. For $v\in S$, ...

**7**

votes

**0**answers

214 views

### A mysterious number related to Hasse-Weil L-function of elliptic curve

Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial ...

**3**

votes

**0**answers

54 views

### The analogue of difference operator in Drinfeld module theory

The theory of Drinfeld modules is, in part, inspired by the analogy between Frobenius and the operator of differentiation. My question is: what should be the analogue of a difference operator $f(x)\...

**4**

votes

**0**answers

243 views

### Reference request: ramified and local geometric class field theory

There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...

**1**

vote

**1**answer

63 views

### On the maximal value of the valuation at infinity of elements in the ring of integers of a global function field

Let $F$ be a global function field with full constant field $\mathbb{F}_q$. We fix a place $\infty$ and let $A$ be the ring of elements of $F$ regular away from $\infty$. We denote by $v_\infty$ the ...

**0**

votes

**1**answer

121 views

### Understanding a valuation property of function fields

I came across this point in a paper recently and I'm having difficulty seeing why it's true. Any explanations or hints would be appreciated.
For any prime $\mathfrak{p}$ of $\mathbb{F}_q [t]$ such ...

**5**

votes

**0**answers

85 views

### Quadratic function fields that are norm-euclidean or PIDs

It is known that there are finitely many square-free integers $d$ for which the ring of integers of $\mathbf{Q}(\sqrt{d})$ is Euclidean under the norm function.
Is there an analogous result for ...

**2**

votes

**1**answer

115 views

### maximal pro-l-quotients of absolute Galois groups

Let $K$ be a field, preferably a function field of a variety $X$ over $\overline{\mathbb{F}}_p$. I am looking for an answer or existing literature on the following question:
What is known about the ...

**2**

votes

**1**answer

126 views

### $[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2?

Let $E/F$ be a quadratic extension of global fields. Denote by $J_F$ the idèle group of $F$, by $P_F$ the subgroup of principal idèles and by $N_{E/F} : J_E \to J_F$ the norm map between idèles. In ...

**5**

votes

**2**answers

335 views

### Is there any work on the Gauss circle problem over function fields? [closed]

I would be thankful if someone had references to provide...

**0**

votes

**1**answer

78 views

### Field of constants of a Galois extension of function fields

Let $F$ be a finite Galois extension of the rational function field $\mathbb Q(x)$. Let $k$ be the field of constants of $F$, i.e., the algebraic closure of $\mathbb Q$ in $F$. Is $k$ necessarily a ...

**2**

votes

**0**answers

253 views

### Rational point on variety over function field

This is one of the theorems in Field Arithmetic written by M. Fried and M. Jarden as following which is Proposition 13.4.6 in that monograph:
Every field K has a regular extension F which is PAC ...

**2**

votes

**1**answer

115 views

### Genus of a function field after an extension

Let $t$ and $x$ be indeterminates, and let $P\in\mathbb Q(t)[x]$ be irreducible. Adjoining a root of $P$ to $\mathbb Q(t)$ we obtain a function field $F/\mathbb Q$. Now suppose that $K$ is an ...

**2**

votes

**1**answer

131 views

### An estimate to show that Goss L functions are entire

In Section 8 of Goss' Basic Structure of Function Field Arithmetic, Goss computed an estimate in section 8.8 in order to show that his L-function is entire on $S_\infty$. The main tool is binomial/...

**2**

votes

**1**answer

264 views

### Modular parametrization in terms of the moduli of shtukas

The modular parametrization of an elliptic curve over $\mathbb{Q}$ (and maybe over a number field in general?) is well-known; also for an elliptic curve over global function field with some condition (...

**0**

votes

**0**answers

136 views

### Algorithm to test if a variety is rational

Is there an algorithm to test if a variety is rational? Equivalently, is there an algorithm to test if a field extension is purely transcendental?
I could not find anything in the book by Cox-Little-...

**3**

votes

**1**answer

153 views

### Puiseux decomposition over a field with positive characteristic

Let $K$ be an algebraically closed field with characteristic $p>0$, and let $f(t,x)\in K[t,x]$ be a polynomial separable in $x$. Denote:
\begin{equation} \Lambda = \bigcup_{i\in \mathbb{N}} K((t^\...

**5**

votes

**1**answer

234 views

### Elementary symmetric functions of reciprocals of monic polynomials in function fields

Let $q$ be a prime power and $\mathbb{F}_q$ the field of cardinality $q$. Let $A = \mathbb{F}_q[T]$ and let $A_+ \subset A$ be the monic polynomials. Choose any ordering $<$ of $A_+$ and let $k$ be ...

**3**

votes

**1**answer

206 views

### Function field of a Drinfeld module, product formula, or Chow group

I am learning about Drinfeld modules, and I have a few questions. There is an analogue that Drinfeld modules are like elliptic curves, which are projective, or are compact Riemann surfaces over $\...

**7**

votes

**1**answer

252 views

### Lexicographic distribution of irreducible polynomials

Let $A = {\mathbb F}_2[X]$, though the following can be adapted to $p \neq 2$ too. Order the elements of $A$ lexicographically. Equivalently, take a polynomial such as $P = X^4 + X + 1$, write its ...

**2**

votes

**1**answer

151 views

### What is the cokernel of $O_S \to F_\infty/O_\infty$?

Let $k$ be a field of characteristic $\neq 2$ and consider the quadratic extension $F$ of $k(T)$ generated by $\sqrt{T^3 + 1}$. Let $X$ be the set of all places of $F$. Let $S = \{\infty\} \subset X$ ...

**4**

votes

**0**answers

288 views

### Fermat's Little Theorem in function fields

There is a well-known generalisation of Fermat's Little Theorem as for any $a\in\mathbb{Z}$ and $n\in\mathbb{N}$ we have $$\sum_{d\mid n} \mu(n/d) a^d\equiv 0\pmod n.$$ where $\mu$ is the Möbius ...

**0**

votes

**0**answers

210 views

### Extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is
$v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...

**2**

votes

**0**answers

76 views

### Pairing for non-uniformizable Anderson T-motives

Let $M$ be an Anderson T-motive (the simplest case, i.e. abelian in the meaning of [G] Goss, Basic structures of function field arithmetic, Def. 5.4.12, over $A^1$, having $N=0$), and let $H_1(E)$ ([G]...

**0**

votes

**0**answers

57 views

### Generators of fixed function fields under involutions

I am trying to prove something for function fields in two generators and only one involution but I don't know if it is true, if true, I would like to generalize, here it is.
Let $K=k(\eta_1,\eta_2)$ ...

**4**

votes

**0**answers

205 views

### Ranks of elliptic curves over Q(t)

I have an elliptic curve $E/\mathbb{Q}(t)$, and I want to compute its rank. Does knowing the rank over $\mathbb{F}_p(t)$ for some prime of good reduction give a bound on the rank over $\mathbb{Q}(t)$?...

**2**

votes

**0**answers

62 views

### Action of a Drinfeld modular group on a Drinfeld symmetric space

Let $\Bbb C_\infty$ be functional field case analog of $\Bbb C$, i.e. $\Bbb C_\infty$ is the completion of the algebraic closure of the field of Laurent series $\Bbb F_q((\theta^{-1}))$, where $q$ is ...

**5**

votes

**1**answer

387 views

### Relation between ramification locus of a tower and of its constant field extension

I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth.
In that remark he considers a tower $\mathcal{F} = (F_0,F_1,F_2,\...

**12**

votes

**2**answers

722 views

### Infinitely many irreducible polynomials of the form f(X^2) + X mod 3?

Are there infinitely many polynomials $f \in \mathbb{F}_3[X]$ for which $f(X^2) + X$ is irreducible?

**2**

votes

**0**answers

221 views

### Genus of $k(T)$ is $0$ without using Riemann-Roch

Let $F$ be a function field with one variable with total constant field $k$, and let $X$ be the set of all places of $F$. How do I show that the genus of $k(T)$ is $0$ without using Riemann-Roch? Is ...

**8**

votes

**1**answer

447 views

### Abelian varieties with good reduction everywhere over function fields

There is a famous theorem due to J.-M. Fontaine,
Il n'y a pas de variété abélienne sur Z
(and independently by V.A. Abrashkin) that there are no abelian varieties over Z. I was wondering whether ...

**3**

votes

**1**answer

258 views

### Special linear groups over function fields

Let $p$ be a prime number, and let $q$ be a finite power of $p$. Denote by $F_q$ the unique field with $q$ elements.
What is known about the structure and properties of $\mathrm{SL}_2(F_q[t])$ as ...

**2**

votes

**0**answers

231 views

### The uniform boundedness of rational torsion for traceless abelian surfaces over a function field

The function field analog of the theorem of Mazur-Kamienny-Merel (giving a universal bound in $[K:\mathbb{Q}]$ for the size of the $K$-rational torsion of an elliptic curve) is immediate just from the ...

**5**

votes

**2**answers

619 views

### Are Anderson $T$-motives motives for the function field analogy?

this question is related to this one Geometry for Anderson's motives?, though the previous one doesn't answer exactly my question.
Let $\mathbb{C}_{\infty}$ be the function field analog of $\...

**6**

votes

**1**answer

340 views

### Comparison of finite field extensions of $\mathbb{C}(t)$

Let K be a finite field extension of $\mathbb{C}(t)$. Then $K$ is isomorphic to the field of meromorphic functions on a compact Riemann surface $X$ with genius $g$. By an argument similar to the proof ...

**3**

votes

**0**answers

141 views

### Lang's height conjecture over $\mathbb{F}_q(T)$?

Is the canonical height of a non-torsion $\mathbb{F}_q(T)$-rational point of an elliptic curve over $\mathbb{F}_q(T)$ known or supposed to be bounded from below by an absolute positive number (or ...

**4**

votes

**0**answers

138 views

### Rational cohomology of $S$-arithmetic groups over function fields and Gauss-Bonnet

I have a question on the ranks of rational cohomology groups of
$S$-arithmetic groups over function fields. To fix the situation, $G$
is a simple Chevalley group of rank $r$, $k=\mathbb{F}_q$ a finite
...

**1**

vote

**1**answer

61 views

### Function field Towers of larger depth of recursion

A function field tower is a sequence of function fields
$$\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 \dots \subset \mathcal{F}_{n} \subset \mathcal{F}_{n+1} \subset \dots $$
over a base ...

**2**

votes

**0**answers

146 views

### Automorphisms of function fields under constant reduction

Let $K=\mathbb{Q}(x,y)$ be a function field of genus at least 2, with defining equation $f(x,y)=0$ (say, absolutely irreducible and with coefficients not divisible by $p$), and let $k$ be the mod-$p$ ...

**4**

votes

**1**answer

432 views

### To which automorphic forms/rep's over a function field can we associate a Galois representation?

As far as I understand it, by the work of Lafforgue (cf. Laumon, "Cohom. of Drinfeld ... II", Thm 12.4.1) there is a Galois representation associated to an irreducible cuspidal automorphic ...

**6**

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**0**answers

391 views

### elliptic curves over function fields

Let $K$ be a number field, $E$ an elliptic curve over $K$, and $p$ an odd prime. If $v$ is a place of $K$, we know by the Kummer injection that $E(K_v)/pE(K_v) \hookrightarrow H^1 (K_v, E[p])$ for ...

**8**

votes

**2**answers

653 views

### What is the advantage of the approach of valuations to the Riemann-Roch Theorem for curves (a la Chevalley)? AKA theory of algebraic functions in one variable

Hello,
Some books and courses take the approach to Riemann-Roch for curves considering only the algebraic viewpoint of algebraic extensions of a field k(t) without going into any algebraic geometry (...

**5**

votes

**2**answers

1k views

### Computing the fixed field of an automorphism of a function field

Let say we have a function field $k(x,y)$ defined by $f(x,y)$ over $k$, with $\sigma \in Aut(k(x,y)/k)$ and. Suppose, I'm not that out of luck, so that either of $\prod \sigma^i(x)$ or $\sum \sigma^i(...

**0**

votes

**1**answer

303 views

### The image of generator under an automorphism of a cyclic function field

I'm reading the proof of Lemma 4.1 [1] which says:
"Let $F = K(x,y), y^q = f(x)$, where $q$ is a prime different from characteristic of $K$.
Let $Z := Gal(F/K(x))$ and we have $Z < G < Aut(F/K)$...

**2**

votes

**2**answers

459 views

### About the ring used in the definition of drinfeld modules. Why is that ring a dedekind domain?

Let $K/F$ be a function field with exact field of constants $F$ ($F$ is a finite field of characteristic $p$ prime). A prime in $K$ is a discrete valuation in $K$ containing $F$. It has a unique ...

**8**

votes

**1**answer

519 views

### Splitting a polynomial with one root

Suppose we have an irreducible polynomial $f\in K[x]$. Is there some way to sometimes tell whether $f$ splits completely after adjoining just one root of $f$ to $K$?
I am mostly interested in the ...

**5**

votes

**0**answers

344 views

### Sheaf Cohomology on Zariski-Riemann Spaces

Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...

**3**

votes

**3**answers

541 views

### Implicit Function Theorem over arbitrary fields

Are there any restrictions on the ground field over which the implicit function theorem holds? In particular, does the theorem hold over function fields like $F_q((1/t))$?

**2**

votes

**0**answers

622 views

### Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. (...