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**5**

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204 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 ...

**4**

votes

**1**answer

169 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 ...

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**0**answers

88 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 ...

**3**

votes

**0**answers

69 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

49 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 ...

**3**

votes

**0**answers

99 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

281 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 ...

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**0**answers

293 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 ...

**7**

votes

**2**answers

434 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 ...

**4**

votes

**2**answers

903 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 ...

**0**

votes

**1**answer

201 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 < ...

**2**

votes

**2**answers

354 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 ...

**6**

votes

**1**answer

400 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 ...

**2**

votes

**0**answers

307 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 ...

**1**

vote

**1**answer

292 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

478 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'. ...

**2**

votes

**1**answer

190 views

### behavior of places of a function field under automorphism

if $P_{1}$ and $P_{2}$ are distinct places of equal degree of the function field F/K, and $\sigma$ is a K-field automorphism, such that $\sigma(P_{1})=P_{2}$. then, does $\deg (P_{1}\cap K(x))=\deg ...

**9**

votes

**2**answers

613 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 :
...

**3**

votes

**1**answer

335 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 ...

**22**

votes

**5**answers

2k 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 ...