All Questions
Tagged with function-fields algebraic-number-theory
13 questions
5
votes
1
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219
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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
5
votes
1
answer
442
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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
185
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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 $$\pi_1(X)\to \mathrm{GL}_n\left(\overline{\mathbb{F}_p}\right)...
- 4,398
4
votes
2
answers
296
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Biquadratic extension of global function fields with cyclic decomposition groups
Let $F$ be a global function field, for example $F={\mathbb F}_q(t)$, the field of rational functions in one variable over a finite field ${\mathbb F}_q\,$.
Question. What would be an example of a ...
- 14.1k
4
votes
0
answers
464
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Reference request: ramified and local geometric class field theory
There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's ...
- 647
2
votes
1
answer
181
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$[J_F : P_FN_{E/F}J_E] = 2$ for quadratic extensions of global fields of characteristic 2?
Let $E/F$ be a quadratic extension of global fields. Denote by $J_F$ the idèle group of $F$, by $P_F$ the subgroup of principal idèles and by $N_{E/F} : J_E \to J_F$ the norm map between idèles. In ...
- 277
2
votes
2
answers
410
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Computing the class group of a quadratic function field
I am asking for a reference in which I can find tools to answer questions like the following: Let $K=\mathbb{F}_q(X)$ be a rational function field over the finite field with $q$ elements. Let $E/K$ be ...
- 3,362
2
votes
0
answers
56
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Degeneracy maps of Drinfeld modular curves
Over number fields, we have two natural degeneracy maps
$$X_0(N)\leftarrow X_0(pN) \rightarrow X_0(N)$$
between the (compactified) moduli space of elliptic curves with level $pN$ and $N$ respectively (...
- 4,398
2
votes
0
answers
169
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Monogenic function fields
Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
- 103
2
votes
0
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109
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Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$
I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
- 153
2
votes
0
answers
169
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A version of weak approximation with S-integers
Let $k$ be a finite field. Let $K$ a finite extension of $k(t)$. Let $S$ be a finite set of places of $K$. Let
$$K_S = \prod_{v\in S} K_v$$
where $K_v$ is the completion of $K$ at $v$. For $v\in S$, ...
- 323
2
votes
0
answers
230
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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
1
vote
1
answer
100
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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:...
- 3,996