Timeline for Local factors of Hasse-Weil zeta function - what do they have in common?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 7, 2014 at 11:22 | vote | accept | Andreas Holmstrom | ||
| Jun 7, 2014 at 10:41 | comment | added | Andreas Holmstrom | Sorry Felipe (and David), I realise my comment came across as overly disapproving. That was not my intention. | |
| Jun 7, 2014 at 2:58 | comment | added | user76758 | @AndreasHolmstrom: I agree 100% with your comment to Qiaochu. But note that in your elliptic curve example you are strictly speaking dropping a couple of Riemann-zeta factors that corresponding to degree-0 and degree-2 cohomology on fibers; i.e., you are really speaking about the degree-1 part, which is not the entire "zeta function". Curves are misleading in that a single cohomological part can still be expressed via point-counting; in general one cannot expect this to happen. My answer below discusses it in more detail. | |
| Jun 7, 2014 at 2:15 | comment | added | Felipe Voloch | @AndreasHolmstrom I, and believe others here, have no idea what it is that you are really asking. Instead of complaining that we don't understand, why don't you make an effort to ask a precise question. | |
| Jun 7, 2014 at 2:07 | answer | added | user76758 | timeline score: 23 | |
| Jun 7, 2014 at 0:18 | comment | added | Andreas Holmstrom | David, I get the feeling you didn't read the question, but maybe I didn't state it very clearly. Of course the Betti numbers govern the degrees of these polynomials, and of course the Weil conjectures give us even more information, in terms of the Riemann hypothesis which determines the absolute value of the numbers $\alpha_{ij}$. The question is whether one in general can get more precise information about how these polynomials vary in a flat family. | |
| Jun 6, 2014 at 23:46 | comment | added | David E Speyer | I agree with @Qiaochu's statement, at least as a first answer. The Weil conjectures state that, for all but finitely many $p$, the local factor should be of the form $\prod_{i=0}^{2 \dim X} \prod_{j=1}^{b_i} (1-\alpha_{ij}^{-s})^{(-1)^{i+1}}$ where $b_i$ is the $i$-th Betti number of $X(\mathbb{C})$. In particular, $b_i$ does not depend on $p$. This explains why the numerators in the elliptic curve case are quadratic: Because $b_1=2$. | |
| Jun 6, 2014 at 22:14 | history | edited | Andreas Holmstrom | CC BY-SA 3.0 |
Added reference to comments on point counts.
|
| Jun 6, 2014 at 22:08 | comment | added | KConrad | Are you seeking a definition of the local factors in terms of determinants on $\ell$-adic cohomology? See mathoverflow.net/questions/146081/good-factors-of-l-function and the paper by Serre that is mentioned in the question. | |
| Jun 6, 2014 at 21:54 | comment | added | Andreas Holmstrom | David, thanks for the comments! Sure, your points about Betti numbers and a_p being there by definition are valid. Let me try to reformulate the question slightly: What does Poincare duality say about the zeta coefficients of a genus 2 curve? Or a K3 surface, or a cubic threefold? In all cases, the degree of the polynomial factors will of course be governed by the Betti number, but I'm looking for a more concrete description of each coefficient, maybe in terms of point counts in the fibers. | |
| Jun 6, 2014 at 21:50 | answer | added | ACL | timeline score: 4 | |
| Jun 6, 2014 at 21:49 | comment | added | Andreas Holmstrom | Qiaochu, the Weil conjectures say a lot about the local factors, but they don't say anything about how they vary in a flat family. | |
| Jun 6, 2014 at 20:43 | comment | added | David Loeffler | I don't really see what you're trying to say by your statement "all these numerators are exactly the same, except of course that the prime $p$ varies". All these polynomials are quadratic because of the Betti numbers, as Qiaochu points out; the leading term is 1 by definition, the trailing term $p$ by Poincare duality, and the fact that the middle coefficient is $a_p$ is exactly the definition of $a_p$! | |
| Jun 6, 2014 at 20:19 | comment | added | Qiaochu Yuan | This is the question that the Weil conjectures answer, right? The shape is dictated by the Betti numbers of the complex points, at least in the smooth projective case. | |
| Jun 6, 2014 at 20:01 | history | asked | Andreas Holmstrom | CC BY-SA 3.0 |