All Questions
Tagged with function-fields nt.number-theory
37 questions
0
votes
1
answer
115
views
Cancellation in correlations of the Möbius function over function fields
Let $p$ be an odd prime and $q$ a power of $p$. For a polynomial $f \in \mathbb{F}_q[T]$, let $\mu(f)$ be the Möbius function of $f$. For a positive integer $d$, let $M_d$ be the set of monic ...
- 1
2
votes
0
answers
108
views
Arithmetic interest of the Goss zeta function
I'm someone with more of a number fields background who recently started working on a project more in the function fields setting. I was reading Goss's book (Basic structures of function field ...
- 2,044
1
vote
0
answers
89
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 ...
- 121
2
votes
1
answer
169
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}(...
- 23
5
votes
1
answer
219
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 ...
- 4,398
4
votes
2
answers
296
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 ...
- 14.1k
2
votes
1
answer
110
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 ...
2
votes
0
answers
109
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[\...
5
votes
0
answers
185
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 $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
- 4,398
4
votes
0
answers
284
views
modularity of elliptic curves over function fields in positive characteristic
Let $F$ be a global function field over a finite field, and $E$ be an elliptic curve over $F$. Much like the number field case, it is natural to study the Galois representation on the Tate module of $...
- 685
6
votes
1
answer
512
views
Mordell conjecture over function fields
So I've read (for instance in the introduction to R.S de Jong's thesis ) that the naive adaptation of the proof of the Mordell conjecture over function fields fails, even using Arakelov intersection ...
- 4,398
1
vote
0
answers
158
views
$\mathbb{Z}_\ell$-extensions of global function fields
Given a number field $F$ and a prime $p$, it is natural to study Iwasawa theory over the cyclotomic $\mathbb{Z}_p$-extension of $F$, i.e., the unique $\mathbb{Z}_p$-extension of $F$ which is contained ...
- 685
14
votes
2
answers
566
views
Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?
Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
- 1,343
3
votes
0
answers
174
views
Reference Request: CM Motives over Function Fields
Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface:
$$
\mathcal{E} : y^2 = x^3 - 27ux - 54v \...
3
votes
1
answer
608
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,\...
- 2,044
3
votes
1
answer
246
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 ...
- 1,322
36
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 ...
- 114k
2
votes
0
answers
169
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 $v\in S$, ...
- 323
3
votes
0
answers
83
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)\...
- 375
0
votes
1
answer
145
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 ...
- 487
2
votes
1
answer
181
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 ...
- 277
5
votes
2
answers
436
views
Is there any work on the Gauss circle problem over function fields? [closed]
I would be thankful if someone had references to provide...
- 3,062
2
votes
1
answer
203
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/...
- 21
2
votes
1
answer
463
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 (...
- 647
5
votes
1
answer
296
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 ...
- 156k
2
votes
1
answer
160
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 $S = \{\infty\} \subset X$ ...
user81900
4
votes
0
answers
457
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
2
votes
0
answers
95
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]...
- 305
5
votes
1
answer
442
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
12
votes
2
answers
901
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?
- 11.3k
2
votes
0
answers
230
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 ...
- 29
4
votes
1
answer
590
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
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
0
votes
1
answer
410
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 $Z < G < Aut(F/K)$...
- 601
11
votes
2
answers
891
views
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 :
...
- 1,320
3
votes
1
answer
395
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
30
votes
5
answers
6k
views
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