Timeline for Divisibility properties of Hurwitz numbers
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| May 25, 2015 at 23:30 | comment | added | Gerry Myerson | Numerators of Hurwitz numbers are tabulated at oeis.org/A002306 | |
| May 25, 2015 at 19:53 | answer | added | David Hansen | timeline score: 1 | |
| Sep 27, 2011 at 10:43 | comment | added | Igor Rivin | @David, thanks for the correction. Still, given Hurwitz' interests, it would make sense that your Hurwitz numbers would come from a similar place... | |
| Sep 26, 2011 at 23:25 | answer | added | Noam D. Elkies | timeline score: 13 | |
| Sep 26, 2011 at 22:40 | comment | added | David Hansen | @Francois: I messed up with $k=80$! Thanks for drawing my attention to that. | |
| Sep 26, 2011 at 22:39 | history | edited | David Hansen | CC BY-SA 3.0 |
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| Sep 26, 2011 at 22:30 | history | edited | David Hansen | CC BY-SA 3.0 |
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| Sep 26, 2011 at 22:21 | comment | added | David Hansen | @Igor: Unfortunately the term "Hurwitz number" is ambiguous; the objects in Ekedahl's article are an unrelated type of Hurwitz number. @Francois: The absence of 41 there is disturbing, especially since Katz's article indicates it should be there (if I calculated correctly, anyway). | |
| Sep 26, 2011 at 20:38 | comment | added | François Brunault | @David : maybe this article of Katz can be helpful : N. Katz, The congruences of Clausen-von Staudt and Kummer for Bernoulli-Hurwitz numbers, Math. Ann. 216. ams.u-strasbg.fr/mathscinet/search/… | |
| Sep 26, 2011 at 20:30 | comment | added | François Brunault | @David : Regarding the denominator of $H_k$, it seems that not all the primes come into play, for example in the case $k=80$ then $p=41$ is missing. | |
| Sep 26, 2011 at 20:29 | comment | added | Igor Rivin | It seems that Hurwitz came upon them through surface topology, see: arxiv.org/pdf/math/0004096v3 (of which Torsten Ekedahl is an author, so I expect he will shed more light). | |
| Sep 26, 2011 at 20:26 | comment | added | François Brunault | @darij : I'm not aware of a finite description of these numbes. The rationality comes from the theory of complex multiplication. The sum over $\mathbf{Z}[i]$ is the weight $k$ Eisenstein series evaluated at $z=i$. Since the algebra of modular forms with rational coeffs is generated by $E_4$ and $E_6$, it suffices to proves the result for $k=4$ and $k=6$. But the numbers $E_4(i)$ and $E_6(i)$ are linked with the coefficients of the elliptic curve $y^2=x^3-x$ which has CM by $\mathbf{Z}[i]$. The $\Gamma(1/4)^2$ arises from the real period of this elliptic curve. | |
| Sep 26, 2011 at 19:41 | comment | added | darij grinberg | To make life easier for us combinatorialists, could you perhaps give a finitary description of the Hurwitz numbers, or is there none known? (If none is known, how did Hurwitz prove their rationality?) | |
| Sep 26, 2011 at 19:36 | comment | added | Igor Rivin | That's a really interesting question, but could you give some pointer to where these numbers come from? | |
| Sep 26, 2011 at 17:25 | history | asked | David Hansen | CC BY-SA 3.0 |