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edited Mar 19 2011 at 12:49 |
The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.
Neukirch says that Langlands' Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?
Addendum (28/02/2010) Berger's Berger's recent Bourbaki exposé 1017 (arXiv:1002.4111) arXiv:1002.4111 says that cases of the Fontaine-Mazur conjecture have been proved by Matthew Emerton as well. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...
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edited Jan 3 2011 at 10:27 |
The question is about characterising the sets $S(K)$ of primes which split completely in a given galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?
Addendum (28/02/2010) Berger's recent Bourbaki exposé 1017 (arXiv:1002.4111) says that cases of the Fontaine-Mazur conjecture have been proved by Matthew Emerton as well. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...
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edited Jul 24 2010 at 12:53 |
The question is about characterising the sets $S(K)$ of primes which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?
Addendum (28/02/2010) Berger's recent Bourbaki exposé 1017 (arXiv:1002.4111) says that cases of the Fontaine-Mazur conjecture have been proved by Mathew Matthew Emerton as well. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...
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edited Jul 24 2010 at 3:10 |
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edited Feb 28 2010 at 13:43 |
The question is about characterising the sets $S(K)$ of primes which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?
Addendum (28/02/2010) Berger's recent Bourbaki exposé 1017 (arXiv:1002.4111) says that cases of the Fontaine-Mazur conjecture have been proved by Mathew Emerton as well. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...
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edited Feb 28 2010 at 13:29 |
The question is about characterising the sets $S(K)$ of primes which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $\mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $\mathbb{P}$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $\mathbb{P}$. Can this remark now be illustrated by some striking example ?
Addendum (28/02/2010) Berger's recent Bourbaki exposé 1017 (arXiv:1002.4111) says that cases of the Fontaine-Mazur conjecture have been proved by Mathew Emerton. I didn't know this at the time of asking the question, and the unique answerer did not let on that he'd had something to do with Fontaine-Mazur...
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edited Jan 15 2010 at 9:02 |
The question is about characterising the sets of primes $S(K)$ of primes which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $P$ \mathbb{P}$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset P$ S(K)\subset\mathbb{P}$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset P$ T\subset\mathbb{P}$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recently, a subset $T\subset P$ T\subset\mathbb{P}$ is abelian if and only if it is defined by congruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $P$.\mathbb{P}$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $P$. \mathbb{P}$. Can this remark now be illustrated by some striking example ?
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edited Jan 15 2010 at 4:12 |
The question is about characterising sets of primes $S(K)$ which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $P$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset P$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset P$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed here recentlyin Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?, a subset $T\subset P$ is abelian if and only if it is defined by congruence conditionscongruences. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$. "Being defined by congruences" can be made precise, and counts as a characterisation of abelian subsets of $P$.
Neukirch says that Langlands Philosophy provides a characterization characterisation of all galoisian subsets of $P$. Can this remark now be illustrated by some striking example ?
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edited Jan 14 2010 at 14:14 |
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asked Jan 14 2010 at 11:28 |
Galoisian sets of prime numbers
The question is about characterising sets of primes $S(K)$ which split completely in a galoisian extension $K|\mathbb{Q}$. Do recent results such as Serre's modularity conjecture (as proved by Khare-Wintenberger), or certain cases of the Fontaine-Mazur conjecture (as proved by Kisin), have anything to say about such subsets, beyond what Class Field Theory has to say ?
I'll now introduce some terminology and recall some background.
Let $P$ be the set of prime numbers. For every galoisian extension $K|\mathbb{Q}$, we have the subset $S(K)\subset P$ consisting of those primes which split (completely) in $K$. The question is about characterising such subsets; we call them galoisian subsets.
If $T\subset P$ is galoisian, there is a unique galoisian extension $K|\mathbb{Q}$ such that $T=S(K)$, cf. Neukirch (13.10). We say that $T$ is abelian if $K|\mathbb{Q}$ is abelian.
As discussed recently in MO11688, a subset $T\subset P$ is abelian if and only if it is defined by congruence conditions. For example, the set of primes $\equiv1\pmod{l}$ is the same as $S(\mathbb{Q}(\zeta_l))$.
Neukirch says that Langlands Philosophy provides a characterisation of all galoisian subsets of $P$. Can this remark now be illustrated by some striking example ?
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