Questions tagged [function-fields]

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On the complexity of global fields isomorphism

Let $L$ and $K$ be isomorphic global fields (i.e. either function fields of curves over a finite field, or number fields). What is the complexity of finding an isomorphism between them? Is there a ...
Reyx_0's user avatar
  • 139
2 votes
0 answers
51 views

Degeneracy maps of Drinfeld modular curves

Over number fields, we have two natural degeneracy maps $$X_0(N)\leftarrow X_0(pN) \rightarrow X_0(N)$$ between the (compactified) moduli space of elliptic curves with level $pN$ and $N$ respectively (...
curious math guy's user avatar
1 vote
0 answers
83 views

Finiteness of elliptic curves with trivial conductor over function fields

Let $K=\mathbb{F}_q(C)$ be the function field of a smooth projective curve $C$ over a finite field $\mathbb{F}_q$ with $\text{cha}(K)>3$ and let $E$ be an elliptic curve over $E$. To $E$ we may ...
MightyGuy's user avatar
  • 121
2 votes
1 answer
155 views

Selberg sieve for counting monic irreducible $P \in \mathbb{F}_q[t]$ such that $P + K$ is also irreducible

In a 1983 paper by William Webb (link below), the author gives a version of the Selberg sieve for function fields and uses it to prove that for a fixed $K \in \mathbb{F}_q[t]$, the number $\mathcal{N}(...
Owen Sharpe's user avatar
5 votes
1 answer
203 views

Higher weight modular forms in function fields

There exists a very nice analogue of modular forms of weight two of some level $N$ in the function field setting, namely (cuspidal) harmonic cochains on the Bruhat-Tits tree which are invariant under ...
curious math guy's user avatar
2 votes
1 answer
149 views

Automorphism of positive characteristic field

Suppose $K$ is a field with $\text{char}(K) \geq 0$. Let $L$ be a cyclic extension of $K$ with degree $2n$. We consider the generator $\sigma$ of the Galois group $\text{Gal}(L/K)$. I am interested in ...
Sky's user avatar
  • 913
4 votes
2 answers
262 views

Biquadratic extension of global function fields with cyclic decomposition groups

Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$. Question. What would be an example of a ...
Mikhail Borovoi's user avatar
2 votes
1 answer
99 views

Equations for $H_1(M)$ and $T$-Tate module of Anderson t-motive $M$ are equivalent: a reference?

What is a reference for the following construction? Let $M$ be an Anderson t-motive of rank $r$ dimension $n$, i.e. a module over the Anderson ring $\mathbb{C}_\infty[T]\{\tau\}$ satisfying some ...
Dmitry Logachev's user avatar
2 votes
0 answers
101 views

Action of Galois group on the lattice of a Drinfeld module - a reference?

What is a reference for the following construction? Let $K$ be a finite extension of $\Bbb F_q(\theta)$ and $M$ a Drinfeld module of rank $r$ defined over $K$. Its lattice $L(M)$ is a free $\Bbb F_q[\...
Dmitry Logachev's user avatar
5 votes
0 answers
183 views

Finite coefficients Langlands for function fields

Do we have a finite coefficients Langlands correspondence for function fields? By which I mean a bijection between Galois representations $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
curious math guy's user avatar
4 votes
0 answers
265 views

modularity of elliptic curves over function fields in positive characteristic

Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
Anwesh Ray's user avatar
6 votes
1 answer
460 views

Mordell conjecture over function fields

So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...
curious math guy's user avatar
1 vote
1 answer
95 views

Extension of morphisms in function fields

Let $k=\mathbb F_q\left(\left(\frac1T\right)\right)$, $\overline k$ be an algebraic closure of $k$ and $K$ be the completion of $\overline k$ for the $\frac1T$ valuation. Consider the morphism $\sigma:...
joaopa's user avatar
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1 vote
0 answers
150 views

$\mathbb{Z}_\ell$-extensions of global function fields

Given a number field $F$ and a prime $p$, it is natural to study Iwasawa theory over the cyclotomic $\mathbb{Z}_p$-extension of $F$, i.e., the unique $\mathbb{Z}_p$-extension of $F$ which is contained ...
Anwesh Ray's user avatar
2 votes
1 answer
200 views

Ramification and singular points

Let $ X $ and $ Y $ be integral curves over some perfect field $ k $ and suppose that $ X $ is smooth. Moreover, let $ \pi_1 : Y \to X $ and $ \pi_2 : Y \to X $ be finite morphisms such that the ...
diddy's user avatar
  • 327
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0 answers
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Monogenic function fields

Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
Andry's user avatar
  • 93
1 vote
1 answer
125 views

References on function fields over imperfect fields in positive characteristic

There are many references (good books, papers, ...) available that treat global function fields (over finite fields) of one independent variable. To name a few, Stichtenoth's book "Algebraic ...
Andry's user avatar
  • 93
6 votes
0 answers
421 views

Étale cohomology of the field with one element

In the function field - number field analogy, some expect progress on RH to come from reproducing various aspects of the Grothendieck program in a way where $\mathbb{Z}$ could be treated as a function ...
user avatar
14 votes
2 answers
516 views

Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?

Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
Jonathan Love's user avatar
3 votes
0 answers
165 views

Reference Request: CM Motives over Function Fields

Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface: $$ \mathcal{E} : y^2 = x^3 - 27ux - 54v \...
Angus McAndrew's user avatar
2 votes
0 answers
99 views

Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$

I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
LoneStar's user avatar
  • 153
4 votes
1 answer
482 views

Galois cohomology of separable closure

Let $K$ be a local field, $K^{sep}$ its separable closure, $G = Gal(K^{sep}/ K)$ the Galois group and $C := \overline{K^{sep}}$ the completion with respect to the induced valuation. In his paper on $p$...
Piotr Pstrągowski's user avatar
2 votes
1 answer
184 views

Wild ramification in composite fields

Let $ F $, $ F_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F_i \subseteq E $ and $ E = F_1 \cdot F_2 $ is the composite ...
diddy's user avatar
  • 327
1 vote
1 answer
331 views

An automorphism of a function field

I am looking for an explicit example of a function field other than rational with an automorphism which fixes places of different degrees. Also, is there a counterexample, namely a function field with ...
Engin Şenel's user avatar
4 votes
0 answers
122 views

Honda-Tate theorem and prescribing roots of $L$-functions

I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \...
A. Bailleul's user avatar
  • 1,174
1 vote
1 answer
111 views

Under what conditions is the compositum of two rational cubic Kummer extensions a rational function field?

Let $k$ be an algebraically closed field of characteristic $p > 3$ and $F = k(x)$ be the rational function field in the variable $x$. Consider two Kummer extensions $F_1 = F(\sqrt[3]{g_1})$, $F_2 = ...
Dimitri Koshelev's user avatar
3 votes
1 answer
533 views

Understanding moduli of shtukas of non-minuscule cocharacter

I have kind of a soft question. I've studied the basics of L. Lafforgue's proof of function field Langlands for GLn, and its use of the moduli of shtukas with two legs, and the cocharacters $[1,0,\...
xir's user avatar
  • 1,964
2 votes
2 answers
365 views

Computing the class group of a quadratic function field

I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
Lior Bary-Soroker's user avatar
3 votes
1 answer
222 views

Bounds on Artin conductors over function fields

Let $L/K$ be a geometric Galois extension of function fields over $\mathbb F_q$. Let $\chi$ be a non-trivial irreducible character of $\text{Gal}(L/K)$. According to Michael Rosen's Number Theory in ...
A. Bailleul's user avatar
  • 1,174
35 votes
2 answers
3k 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 ...
Terry Tao's user avatar
  • 109k
2 votes
0 answers
162 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 $v\in S$, ...
Will Dukeminier's user avatar
7 votes
0 answers
415 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 ...
Trihan Fabien's user avatar
3 votes
0 answers
80 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)\...
paul's user avatar
  • 365
4 votes
0 answers
448 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 ...
wkf's user avatar
  • 637
1 vote
1 answer
134 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 ...
Andry's user avatar
  • 93
0 votes
1 answer
143 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 ...
user221330's user avatar
6 votes
0 answers
154 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 ...
rpc's user avatar
  • 81
2 votes
1 answer
265 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 ...
darko's user avatar
  • 165
2 votes
1 answer
174 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 ...
Bib-lost's user avatar
  • 267
5 votes
2 answers
429 views

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

I would be thankful if someone had references to provide...
Dr. Pi's user avatar
  • 2,949
0 votes
1 answer
203 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 ...
352506's user avatar
  • 1,001
2 votes
0 answers
304 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 ...
Max CYLin's user avatar
  • 151
2 votes
1 answer
317 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 ...
352506's user avatar
  • 1,001
2 votes
1 answer
192 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/...
Angus Chung's user avatar
2 votes
1 answer
440 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 (...
wkf's user avatar
  • 637
1 vote
0 answers
163 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-...
Boris Bukh's user avatar
  • 7,746
4 votes
1 answer
224 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^\...
giladude's user avatar
  • 155
5 votes
1 answer
286 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 ...
David E Speyer's user avatar
3 votes
1 answer
322 views

Function field of a Drinfeld module and product formula

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 $\...
Hugo's user avatar
  • 181
8 votes
1 answer
326 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 ...
Marty's user avatar
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