Timeline for Good factors of L-function
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 28, 2013 at 18:05 | comment | added | LMN | Great! Here's a precise reference: Serre's Abelian L-adic representations book defines the L-function via arithmetic frobenius (p.I-16) and specifies the euler factor as the characteristic polynomial evaluated at the appropriate number. Thanks! | |
| Oct 28, 2013 at 4:29 | comment | added | Matt | For example, the place I tend to see this is with modularity of varieties, in particular Calabi-Yau threefolds, and Noriko Yui has notes here: arxiv.org/abs/1212.4308 that talks about this (maybe I just messed up the terms "arithmetic and geometric" in my previous post now that I look at this?) | |
| Oct 28, 2013 at 4:22 | comment | added | Matt | I'd have to re-look at places I've seen this, but I think when the local zeta functions are defined with geometric Frobenius they don't use the "characteristic polynomial" but rather $det(1-p^{-s}\rho(Frob_{\mathfrak{p}}))=P_{\mathfrak{p}}(p^{-s})$ or something? I was really worried about this as well at one point, and never really came to a definitive conclusion, but decided that there were enough minus signs and duals in papers by careful people that it seemed to work out as the same... | |
| Oct 28, 2013 at 2:06 | comment | added | LMN | This is pretty wierd. (1) Of course, fixed pts of arithmetic or geometric frobenius of vars/finite fields obviously agree, only their characteristic polynomials disagree. (2) The eigenvalues of geometric frobenius are algebraic integers, (hence not arithmetic frobenius). (3) Charpolys appearing in Weil conjectures (hence local zeta funcs) are of geometric frob. and I thought that the Hasse-Weil zeta function of vars/num fields are products of the local zeta functions (evaluated at various pts). If this last result is true, do we have a contradiction? | |
| Oct 27, 2013 at 21:27 | comment | added | Matt | I think it is standard to use arithmetic Frobenius on $H^i$ or equivalently take the contragredient representation of geometric Frobenius when thinking in terms of Galois representations on $H^i$ (even when this isn't explicitly mentioned). That way the definitions coincide for elliptic curves using the natural isomorphism $H^1_{et}(\overline{E}, \mathbf{Q}_\ell)^\vee \simeq T_\ell(E)\otimes \mathbf{Q}_\ell$. | |
| Oct 27, 2013 at 20:31 | history | asked | LMN | CC BY-SA 3.0 |