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The subject line pretty much says it all. To expand just a little bit:

1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known to occur regularly over $\mathbb{Q}$; not known to occur regularly over every Hilbertian field.)

2) Is there a convenient list of the simple groups of order, say, at most $10,000$ together with information about which of them are known to be Galois groups over $\mathbb{Q}$?

3) Is there perhaps some nice website keeping track of information like this? (There should be!)

Added: The table in $\S 8.3.4$ of Serre's Topics in Galois Theory (the 1992 edition; I don't know if this matters) lists $\operatorname{SL}_2(\mathbb{F}_{16})$ -- which has order $4080$ -- as the smallest simple group which is not known to occur regularly over $\mathbb{Q}$. On the other hand this 2007 paper of Johan Bosman shows that $\operatorname{SL}_2(\mathbb{F}_{16})$ occurs as a Galois group over $\mathbb{Q}$, but mentions that the problem of realizing it regularly is still open. Thus the answer to the "regular" variant of the question seems to be $\operatorname{SL}_2(\mathbb{F}_{16})$. But to be clear, I am really looking for as much data as possible, not just "records".

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    $\begingroup$ Realizing groups as Galois groups over Q is known as the "inverse Galois problem". en.wikipedia.org/wiki/Inverse_Galois_problem has some results / references. Moreover there is a nice book from Serre about the subject: math.mcgill.ca/~darmon/pub/Articles/Serre/c.pdf. But probably I'm telling you nothing new. $\endgroup$
    – Ralph
    Commented Nov 8, 2011 at 5:28
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    $\begingroup$ @Ralph: Thanks. But, yes, I am aware of IGP and Serre's book: see e.g. math.uga.edu/~pete/atkinlehnerfinal.pdf. $\endgroup$
    – Pete L. Clark
    Commented Nov 8, 2011 at 5:32
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    $\begingroup$ Not sure if this helps, but in Jennifer Andreotti's master's thesis, she says that $SL(2,16)$, order 4080, is the only one of the 10 smallest non-abelian simple groups not yet known to be a Galois group over $\mathbb Q$. $\endgroup$
    – Max Flander
    Commented Nov 8, 2011 at 5:33
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    $\begingroup$ I think Jürgen Klüners was collecting a lot of data concerning inverse Galois problem for simple/almost simple groups. Perhaps he has some sort of database? $\endgroup$
    – Tim Dokchitser
    Commented Nov 8, 2011 at 8:43
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    $\begingroup$ Isn't this kind of a duplicate of mathoverflow.net/questions/70155/on-the-inverse-galois-problem ? $\endgroup$
    – Dror Speiser
    Commented Nov 8, 2011 at 21:53

1 Answer 1

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I decompose the question into four parts, all of which Pete already knows a lot about, and/or are mentioned in the comments:

(1) The books by Malle-Matzat and Völklein thoroughly explain the workhorses of the field such as the rigidity method of John Thompson.

(2) Later results: Pete himself is responsible for a significant further result in the $A_1(2,p)$ case. It seems well-known that the case $A_1(2,q)$ (and $G(q)$ for other many other algebraic groups $G$) is difficult when $q$ is a prime power rather than a prime. (But that was the wisdom 10 years ago --- see the update below.) Pete already gave an interesting recent reference for the case $A_1(2,16)$.

(3) People tabulating results. It seems that the main activity is by Klüners with help from Malle. The on-line database of Klüners is organized by the permutation degree of the group, which is of course not the same as the cardinality. Also Klüners looks at all finite groups, not just finite simple ones.

(4) Sizes of finite simple groups. The Wikipedia page, list of finite simple groups, is excellent. It lists all 16 non-abelian finite simple groups of order less than 10,000.

So it could be best to correspond directly with Klüners concerning specific inverse Galois results.

If you're interested in organizing finite simple groups by cardinality --- this is a question that comes up from time to time --- it could be very useful to convert the Wikipedia page to a Python/SAGE program.

Update 2: David Madore had the same idea 8 years ago and wrote a code not in Python but in Scheme. So if you're interested in the smallest simple group not proven to be a Galois group, you only need to go down Madore's list. Note the Shih-Malle theorem that Pete summarizes in his paper: $A_1(2,p)$ is $\mathbb{Q}$-Galois when $p$ is prime and not a square mod 210. The first prime $p$ that is a square mod 210 is 311. (Pete's paper settles some other such $p$, but not that one.) Madore's table shows that the open case $A_1(2,311)$ shows up just a few entries after the only open sporadic case, $M_{23}$. So what does that leave that's smaller than $M_{23}$?

Update 1: Some Googling found an interesting paper by Dieulefait and Wiese and a more recent paper by Bosman. It seems that a lot more is known about the $A_1(2,q)$ case in recent years using the number of theory of curves Galois representations attached to modular forms, rather than the rigidity method of Thompson. In particular, no open cases remain in the Wikipedia list of finite simple groups of order less than 10,000, other than possibly $A_1(27)$ and ${}^2A_2(9)$, which is the unitary group $\text{PSU}(3,\mathbb{F}_9)$. Actually it looks like ${}^2A_2(9)$ doesn't survive either because (like $M_{11}$) it is handled by older methods, according to Malle and Matzat. The smallest finite simple group which is not known to be a Galois group over $\mathbb{Q}$ could be an interesting trivia question for the moment.

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    $\begingroup$ @Greg: Thanks for this. I did not know about Klüners's online database, so that seems suspiciously close to being an answer to my question. $\endgroup$
    – Pete L. Clark
    Commented Nov 8, 2011 at 18:39
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    $\begingroup$ @Pete I think that it could be quite useful to create a living document, presumably a web page, summarizing the finite simple case. $\endgroup$
    – Greg Kuperberg
    Commented Nov 8, 2011 at 19:22
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    $\begingroup$ @Greg: I whole-heartedly agree....Wait, you're not expecting me to do it, are you? Well, maybe I could enlist a graduate student... $\endgroup$
    – Pete L. Clark
    Commented Nov 8, 2011 at 22:05
  • $\begingroup$ @Pete Over the years I have seen many mathematical web page and web site projects get started, and I have been involved with some of them. For this scale of project, the idea of enlisting a graduate student is both too much and too little. It wouldn't take more an afternoon or at the most a few days for you to mock up something useful in a wiki environment. You don't need a grad student for that, and if you did merely brush it off onto a student, it might or might not get started. (continued in next comment) $\endgroup$
    – Greg Kuperberg
    Commented Nov 8, 2011 at 23:23
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    $\begingroup$ Dear Greg, Thanks for this interesting answer. One minor comment: I wouldn't describe Dieulefait and Wiese's result as using "the number theory of curves", but rather "the theory of Galois representations attached to modular forms"; both Dieulefait and Wiese are experts in this field, and they construct the relevant extensions of $\mathbb Q$ by constructing certain modular forms. Best wishes, Matt $\endgroup$
    – Emerton
    Commented Nov 9, 2011 at 0:09

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