Questions tagged [function-fields]
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83
questions
35
votes
2
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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 ...
- 109k
30
votes
5
answers
6k
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Global fields: What exactly is the analogy between number fields and function fields?
Note: This comes up as a byproduct of Qiaochu's question "What are examples of good toy models in mathematics?"
There seems to be a general philosophy that problems over function fields are easier to ...
user709
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(...
- 1,158
12
votes
2
answers
888
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?
- 11.2k
11
votes
2
answers
862
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Why the roots of unity are the analogs of constants ?
Hello,
Joel Dogde, in a comment on his question "Roots of unity in different completions of a number field", says the following, about the analogy between number fields and function fields :
...
- 1,310
9
votes
2
answers
812
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 (...
- 1,873
9
votes
1
answer
790
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 ...
- 1,537
8
votes
1
answer
727
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 ...
- 83
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 ...
- 13.1k
7
votes
0
answers
416
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 ...
6
votes
2
answers
2k
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(...
- 601
6
votes
1
answer
486
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 ...
- 4,454
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 ...
- 4,246
6
votes
0
answers
422
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 ...
user30211
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 ...
- 81
6
votes
0
answers
466
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 ...
- 61
5
votes
2
answers
1k
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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 $\...
- 2,199
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 ...
- 4,246
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...
- 2,949
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 ...
- 151k
5
votes
1
answer
427
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,\...
- 93
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)...
- 4,246
5
votes
0
answers
300
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)$?...
- 585
5
votes
0
answers
369
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 ...
- 468
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 ...
- 13.6k
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^\...
- 155
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$...
- 2,157
4
votes
1
answer
572
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 ...
- 43
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 $...
- 543
4
votes
0
answers
122
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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 \...
- 1,174
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 ...
- 637
4
votes
0
answers
435
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 ...
user76513
4
votes
0
answers
175
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
...
- 17.3k
3
votes
3
answers
780
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))$?
- 31
3
votes
1
answer
534
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,\...
- 1,964
3
votes
1
answer
287
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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 ...
- 11.2k
3
votes
1
answer
391
views
ray class field of rational function field
Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module....
user19475
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 ...
- 1,174
3
votes
1
answer
322
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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 $\...
- 181
3
votes
0
answers
50
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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 ...
- 139
3
votes
0
answers
165
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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 \...
3
votes
0
answers
80
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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)\...
- 365
3
votes
0
answers
269
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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 ...
- 13.7k
3
votes
0
answers
198
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 ...
- 13.7k
2
votes
2
answers
501
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 ...
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 ...
- 267
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 ...
- 913
2
votes
1
answer
201
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 ...
- 327
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 ...
- 327
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 (...
- 637