Timeline for Is there a notion of hyperbolicity for number rings?
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7 events
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| Mar 13 at 1:23 | comment | added | Will Sawin | @dummy The results I was mentioning are summarized at en.wikipedia.org/wiki/… For the $296.277$ statement, the claim is that $\mathbb Q(\sqrt{-3 \cdot 5 \cdot 7 \cdot 11 \cdot 19})$ has infinite fundamental group. In the other direction, the theorem is that there are only finitely many fields with discriminant less than $(22.3-\epsilon)^n$, which implies that all such fields have finite fundamental group. All of this is discussed in Tours le corps de classes et estimation de discriminants by Martinet. | |
| Mar 12 at 23:42 | comment | added | dummy | Is there a reference for the sentence "One property πΆ should have is that every number field of discriminant <πΆπ, being parabolic, should have finite etale fundamental group. One knows this is true for πΆ=22.3 and false for πΆ=296.277 but not a precise optimal value of πΆ."? Thanks! | |
| Jan 15, 2017 at 18:56 | vote | accept | Matthias Wendt | ||
| Dec 9, 2016 at 10:33 | comment | added | Will Sawin | @MatthiasWendt Cool. I am becoming increasingly convinced that one is supposed to investigate this via a Gauss-Bonnet type formula (or trace formula, or Kusnetsov formula). These should express the dimension of the cuspidal cohomology as a main term (the volume) plus secondary contributions - one from the cusps, one from each element of finite order in $SL_2(\mathcal O_{K})$, and one from the trivial representation. One expresses these all in terms of $\Delta_K$ and then tries to show that, for $\Delta_K$ sufficiently large, the main term dominates the error terms. | |
| Dec 9, 2016 at 9:45 | comment | added | Matthias Wendt | Thanks, that's an interesting suggestion. It is in fact true for imaginary quadratic fields that there are only finitely many where the rational cohomology of $SL_2(\mathcal{O}_{K,S})$ is trivial. | |
| Dec 8, 2016 at 11:16 | history | edited | Will Sawin | CC BY-SA 3.0 |
added 983 characters in body
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| Dec 8, 2016 at 11:11 | history | answered | Will Sawin | CC BY-SA 3.0 |