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Results tagged with algebraic-number-theory
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user 11765
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
1
vote
1
answer
636
views
About points on affine curves defined over finite fields
So let $\mathbf{F}_q$ be a finite field with $q$ elements where $q=p^m$,
$p$ a prime number and $m\in\mathbf{Z}_{\geq 1}$. Let $f(x,y)\in \mathbf{F}_q[x,y]$
be a smooth non-constant polynomial and …
- 7,609
7
votes
2
answers
344
views
explicit uniformizer for the false Tate extension
Let $p$ be an odd prime and let $n\geq 1$. Set $K=\mathbb{Q}_p(\zeta_{p^n})$,
$L=\mathbb{Q}_p(\sqrt[p^n]{p})$, and $M=KL$. I claim that $M$ is totally ramified of degree $\phi(p^n)p^n$ (the proof simu …
- 7,609
12
votes
1
answer
774
views
Intersection of a ring class field of a quadratic field K with the cyclotomic extension of K
Let $K$ be a quadratic field. Let $f\in\mathbb{Z}_{\geq 1}$. Let $\mathcal{O}_f=\mathbf{Z}+f\mathcal{O}_K$ be the unique order of $K$ of index $f$ in $\mathcal{O}_K$. Let $H_f^{ring}$ denote the ring …
- 7,609
3
votes
2
answers
531
views
What is the exact meaning of the real period in the $p$-adic formulation of BSD?
Let $E$ be an elliptic curve over $\mathbf{Q}$ which has split multiplicative reduction at $p$ (a prime). If one chooses a global Neron model of $E$ over $\mathbf{Z}$ (unique up to unique isomorphism …
- 7,609
9
votes
2
answers
3k
views
Is there a Riemann-Roch for smooth projective curves over an arbitrary field?
Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical
line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$.
(1) When $k$ is alg …
- 7,609
5
votes
1
answer
1k
views
Effective Chebotarev density results for arbitrary number fields
So let $f(x)\in\mathbf{Z}[x]$ be a monic polynomial of degree $d$ and let $K$ be the splitting field of $f$. Let us define
the "heigt of $f$" $:=||f||$ to be the maximum of the abolute values of
the …
5
votes
0
answers
392
views
What is the shape of the zeta function of a singular hypersurface?
So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$.
Assume that
(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,
(b) and that $X(\mathbb{C …
- 7,609
6
votes
4
answers
2k
views
Criteria for topologically finitely generated profinite groups
Q1: Do we have a criterion which allows us to say when is a profinite group $G$ topologically finitely generated?
For example, if $G$ is topologically finitely generated then, for a fixed integer $N$ …
- 7,609
2
votes
1
answer
214
views
Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme
Let $R$ be a commutative ring and let $I\subseteq R$ be an ideal. We have a natural projection map
$$
\pi:SL_n(R)\rightarrow SL_n(R/I)
$$
(In the original question I had put $GL_n$ instead of $SL_n$ w …
- 7,609
1
vote
Criteria for the surjectivity of the reduction map of the $SL_n$-group scheme
The answers to Q2 and Q3 are positive. See Luc Guyot comment in the following MO Question:
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
In general, if $R$ is a noetherian one dimensional dom …
- 7,609
5
votes
2
answers
806
views
On Weil's characters of type (A)
In Weil's paper
"On a certain type of characters of the idele-class group of an algebraic number field",
Weil introduces a class of characters on the Idele class group (of not necessarily finite o …
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3
votes
0
answers
142
views
$\mathbb{Z}$-linear independence of arguments of units in non-CM number fields
Playing a little bit with Groessencharacters a stumbled on the following question:
Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$ and a …
- 7,609
2
votes
1
answer
461
views
On the conductor of the Groessencharacter of a CM elliptic curve
Let $K$ be a quadratic imaginary field. Let $L$ be a number field which contains $K$ and let $E/L$ be an elliptic curve defined over $L$ with complex multiplication by $K$, i.e. such that $End_{\overl …
- 7,609
3
votes
Accepted
Fields generated by torsion points of CM elliptic curves
The proof of Corollary 1.7 is fine. I had misunderstood his proof. His proof uses in a crucial way his assumption (ii) which appears on the top of p. 41. As is explained on p. 41, this assumption impl …
- 7,609
5
votes
4
answers
682
views
Examples of discrete subgroups of $PSL_2(\mathbf{R})$ with finite covolume and which are not...
Is there a natural example of a discrete subgroup $\Gamma\leq PSL_2(\mathbf{R})$ such that
(1) $\Gamma$ has finite covolume
(2) $\mathfrak{h}/\Gamma$ is not compact ($\mathfrak{h}$ being the upper h …
- 7,609