# How do we know if there are any better bounds than the Minkowski bound?

This question may be an exact replicate of some earlier question elsewhere.

I want to know if there are better bounds than the Minkowski bound for the norm of the smallest ideal in an ideal class of a number field, which is effective when $n$ is small but seems not very sharp when $n$ is large. I noticed the crucial part of Minkowski's bound comes from his theorem on the lattice points in $\mathbb{R}^{n}$, since the theorem is sharp (see wikipedia) there is no room for improvement in that direction. My friends told me there is the Bach bound which nevertheless assumes GRH and seems not so effective when $n$ is small.

So I want to know if there is any particular method that could have helped to bound the class number from above and below in statistic terms (so that for $\mathcal{A}\cap \mathbb{Q}[\sqrt{n}]$ with $n>m$, the probability that the class number of the number field to be less than $f(m)$ is about $g(m)$, etc).

I did have done some search for relevant papers in the past. I ask because I hope someone can help explain the current state of research in layman's terms.

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I would have found it helpful if you had stated what the bounds were for in the title (or at least the first sentence) instead of later on. –  marty cohen May 7 '12 at 4:35
I corrected "GBH" to "GRH", on the assumption that you didn't mean to threaten your number field with grievous bodily harm. I'm a bit sceptical that anything like your last-but-one paragraph is possible for $n > 0$; the class numbers of real quadratic fields are very mysterious, and at the moment I believe it's not even known if they're 1 infinitely often. –  David Loeffler May 7 '12 at 8:26
That is why I ask in here. –  Kerry May 7 '12 at 19:06
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## 1 Answer

There's a lot to say about generalizing Minkowski's result. For example, in my line of work it's more productive to think of Minkowski's result as a lower bound on the discriminant as a function of the degree, rather than an upper bound on norms of a minimal generating set of ideals. From this discriminant bound perspective, there are some notable substantial improvements coming from the analytic side, that go by the name "Odlyzko bounds," which are worth googling, though perhaps not directly applicable to your follow-up question.

More relevant in terms of your follow-up question however are probably the so-called Cohen-Lenstra heuristics. I don't have references in front of me to give exact formulas, but the basic idea is that they give a heuristic for the probability that any given prime $p$ divides a class number of a quadratic field, and more generally, that a prescribed p-rank of the class group is achieved. The probability that a number field has class number 1 is now just the probability that no prime divides the class number. Their prediction according to these heuristics is very close to 75%, which agrees closely with numerical results (there are some anomalies with $p=3$ coming from small $n$, which makes me think that your "for $n>m$" clause might actually improve the accuracy). By taking combinations of their computed probabilities, and making the universal assumption that, say, the 3-rank and 5-rank of the class group are independent variables, you could in principle write down pretty explicitly the formulas that you desire, i.e., the probability that a real quadratic field has class number at most a given $h$.

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I see the state of research now. I ask this because I was wondering if there is any better ways to access the size of the class group instead of relying on Cohen-Lenstra heuristics, because in real data there are substantial deviations even for small class groups. Thanks for the answer. –  Kerry May 8 '12 at 1:52
It sounds like you're asking for something that probably doesn't exist. While there are tiny refinements to Cohen-Lenstra (and significant progress in the function field setting), there's certainly nothing more accurate that's non based on something conjectural. –  Cam McLeman May 9 '12 at 15:10
If I know the question is about something exist, then I would not have raised it in the first place. I think there is a lot of subtlety in the data. Thanks for your comment. –  Kerry May 9 '12 at 20:47
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