Timeline for Values of Artin L-functions at negative integers
Current License: CC BY-SA 3.0
12 events
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| Oct 19, 2017 at 16:07 | comment | added | reuns | @DavidLoeffler Some papers are unclear about that. For a general Artin representation $\rho :Gal(K/\mathbb{Q}) \to GL_n$ Brauer's theorem and Artin reciprocity gives $L(s,\rho) = \prod_{j=1}^l L(s,\psi_j)^{e_j}$ for some Hecke characters $\psi_j$ of fields $F_j \subset K$ and integers $e_j$. If some $e_j$ are negative then there could be infinitely many poles in $\Re(s) \in (0,1)$. With Artin conjecture we expect the product is entire (expect a pole at $s=1$). Do you have a picture of how quotients of Hecke L-functions are conjectured to "simplify" in general ? | |
| Oct 19, 2017 at 15:13 | comment | added | Asvin | Thanks for taking the time to explain everything, this was a very useful answer! | |
| Oct 19, 2017 at 15:04 | comment | added | David Loeffler | Let $M$ be a pure motive over a number field. Then $M$ has a weight $w \in \mathbf{Z}$ (which is 0 if $M$ comes from an Artin representation). If $M$ does not contain the Tate motive $\mathbf{Q}(w/2)$ as a summand, and $L(M, s)$ has analytic continuation of the expected form, then $L(M, s)$ vanishes at all integers $s \le w/2$ that are not critical. This is not a deep statement, it's just a routine consequence of the shape of the $\Gamma$-factors in the functional equation and the definition of "critical". | |
| Oct 19, 2017 at 14:56 | comment | added | Asvin | Very interesting! So if a point is not a critical point, the value is always 0 if it is algebraic? Or maybe just for negative integers...? | |
| Oct 19, 2017 at 14:52 | comment | added | David Loeffler | "It is even worse for a field like Z[i] where there seem to be no critical values despite the values at negative integers being known explicitly". Yes, they're explicitly known to be zero! (There are interpretations of the leading terms at these points in terms of regulator maps on higher K-groups, but now we are in very deep water indeed. If this interests you, read Beilinson's 1984 article "Higher regulators and values of $L$-functions".) | |
| Oct 19, 2017 at 12:42 | comment | added | Asvin | Does Deligne's article talk about values at non critical points or perhaps for an Elliptic curve, the only interesting points are the critical points? | |
| Oct 19, 2017 at 12:41 | comment | added | Asvin | Is that article the one where he defines the notion of a critical value for L-functions? If I understand correctly, this requires that the archimedean factor of the motive and it's dual not have a pole at the point. However, even in the case of the Dedekind zeta function, this seems to ignore the negative even integers even though the L function vanishes at that point. It is even worse for a field like Z[i] where there seem to be no critical values despite the values at negative integers being known explicitly. | |
| Oct 19, 2017 at 11:02 | comment | added | David Loeffler | Sorry, yes, you are right about the meromorphic continuation issue; the proof using Brauer's induction theorem does indeed give you that the meromorphic continuation has the "right" order of vanishing at integer points (and indeed everywhere outside the critical strip). For L-functions of more general arithmetic objects (motives), e.g. Hasse--Weil, you should read Deligne's article in the Corvallis proceedings. | |
| Oct 19, 2017 at 10:54 | vote | accept | Asvin | ||
| Oct 19, 2017 at 10:52 | comment | added | Asvin | Thanks a lot. I thought we knew that any poles of the meromorphic continuation would only have to occur at 1, if at all. If so, the values at negative integers should still be meaningful, right? Also, this might be too much to ask for but what is known in the case of Hasse-Weil zeta functions (say, for an elliptic curve)? Do you have a reference for the conjectures in the general higher dimensional representations case (and maybe even the Hasse-Weil case)? | |
| Oct 19, 2017 at 7:03 | history | edited | David Loeffler | CC BY-SA 3.0 |
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| Oct 19, 2017 at 5:51 | history | answered | David Loeffler | CC BY-SA 3.0 |