replaced the link to the arXiv front end; see https://meta.mathoverflow.net/questions/5124/is-it-time-to-replace-links-to-the-ucdavis-arxiv-frontend

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edited Dec 25, 2021 at 7:50

Martin Sleziak

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I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper paper or Bloom's paper paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$$1 \to \langle i\rangle \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$$\langle j\rangle$, and $<k>$$\langle k\rangle$.

1 1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \langle i\rangle \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $\langle j\rangle$, and $\langle k\rangle$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

corrected link - see https://meta.mathoverflow.net/questions/3193/edits-with-links-to-material-under-restricted-access#comment14185_3193

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edit approved Jul 14, 2017 at 15:11

Martin Sleziak

4.7k
4
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40

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

added 441 characters in body; edited tags; edited title

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edited Apr 2, 2017 at 12:31

mdave16

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On the images Subgroup structure of modular and geometric three-dimensional Galois representationscertain maximal subgroups of PSL$(3,q)$

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

We have an exact sequence, $1 \to R \to P(G) \to U \to 1$, whereLet P(G) isbe an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

On the images of modular and geometric three-dimensional Galois representations

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

We have an exact sequence, $1 \to R \to P(G) \to U \to 1$, where P(G) is contained in a maximal subgroup of PSL$(3,l)$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.

Subgroup structure of certain maximal subgroups of PSL$(3,q)$

I have been reading the paper "On the images of modular and geometric three-dimensional Galois representations" (see also the earlier arXiv preprint). But there is a detail in section 6.2 case (c) and case (d) that I do not follow, and would be very grateful, if anyone could clarify it.

The detail in cases (c) and (d) appears to be similar, so I'll just write up the situation for one of them.

Let P(G) be an irreducible subgroup contained in a maximal subgroup of PSL$(3,l)$ that leaves invariant a triangle with coordinates in $\mathbb{F}_{l}$ and makes all six permutations of its vertices (see Mitchell's paper or Bloom's paper).

Then we have an exact sequence, $1 \to R \to P(G) \to U \to 1$, with $R$ diagonal and $U$ isomorphic to a subgroup of $S_3$. We want to show that the projective image of the inertia subgroup at $l$, $P(I)$, is contained in $R$. It is claimed that $P(I)$ is cyclic with order greater than or equal to $l - 1 > 3$, and so lies in $R$, as $S_3$ does not have cyclic groups of order greater than 3.

I can see from the description of $P(I)$ in section 5 that it has cyclic quotients of order greater than or equal to $l-1>3$, but not why it is cyclic. I don't see either why this must mean it lies in $R$. For example, we have the exact sequence, $1 \to \to Q_8 \to C_2 \to 1$, where $Q_8$ is the quaternion group of order 8. Yet there are cyclic subgroups of $Q_8$ of order greater than 2, which aren't contained in $$, for example $<j>$, and $<k>$.

1: MR2045504 11F80 Dieulefait, Luis; Vila, Núria On the images of modular and geometric three-dimensional Galois representations. Amer. J. Math. 126 (2004), no. 2, 335–361.