Questions tagged [function-fields]
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27
votes
2answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
votes
0answers
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
2answers
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
3answers
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
0answers
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'. (...
