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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 ...
curious math guy's user avatar
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 ...
Mikhail Borovoi's user avatar
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)...
curious math guy's user avatar
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 $v\in S$, ...
Will Dukeminier's user avatar
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 ...
Bib-lost's user avatar
  • 277
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,\...
Bharath's user avatar
  • 93
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 ...
user75810's user avatar