Timeline for constants in Gamma factors in functional equation for zeta functions.
Current License: CC BY-SA 2.5
12 events
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| Mar 26, 2010 at 5:58 | answer | added | another anon | timeline score: 2 | |
| Feb 9, 2010 at 17:16 | comment | added | Kevin Buzzard | @Emerton: What they seem to do in J-L is, via the Whittaker model calculation, give a characterisation of the local factors for GL_1(R) and GL_1(C) (note: 1 not 2) which nails them up to a scalar, and then decree that the usual choices will work and that's what they'll stick with. And then for GL_2(R) and GL_2(C) they first classify all reps, and then define L-factors for all of them explicitly (e.g. L(PS(m1,m2))=L(m1)*L(m2)) and then prove the lines they span are canonical via those Whittaker/Kirillov model calculations and so say that they'll do. But they make choices for GL_1. | |
| Feb 9, 2010 at 17:09 | comment | added | Kevin Buzzard | @James: in the Jacquet-Langlands book they simply decree that at a non-arch place an Euler factor must be of the form P(q^{-s})^{-1} with P a polynomial with constant term 1. They give an "intrinsic" characterisation of the factor that gives it only up to a non-zero constant, and then they normalise it with this "constant term 1" trick. At the infinite places they basically give the same "intrinsic" characterisation, that only works up to a constant, and then they just decree what the answer is and note that this works fine. So there's a definite difference. | |
| Feb 5, 2010 at 23:14 | comment | added | Kevin Buzzard | At the finite Euler factors it's the same story: Tate would have a function f whose zeta function really did have an Euler factor 6(1-23^{-s})^{-1} at 23. All Tate would demand is that you're the usual Euler factor at all but finitely many places, basically. But the L-function of the motive really does have a well-defined Euler factor at all finite primes (and this is "the right one" in the sense that Deligne/Bloch-Kato etc predict its value). However at the infinite places all I've ever seen is "use this dictionary" rather than justification of the constants. | |
| Feb 5, 2010 at 22:36 | comment | added | JBorger | Actually, how does it work at the finite Euler factors? Does Tate's thesis give a nice explanation of why the usual normalization is the best one? What would go wrong if we multiplied the Euler factor at 23 by 6? | |
| Feb 5, 2010 at 22:26 | comment | added | JBorger | I don't have a serious understanding, but I can read his definitions and theorems (about 2 pages in the paper I mentioned above), and I don't see any choices of constants anywhere, although maybe someone with a deeper understanding would disagree. | |
| Feb 5, 2010 at 11:28 | comment | added | Kevin Buzzard | @James: do you understand what Deninger is doing well enough to know whether he's making a choice? I read Tate's thesis very carefully recently, and Tate proves a rather general functional equation that applies to any sufficiently rapidly decreasing function (on the ideles) with some properties. At the very end he shows how to deduce Hecke's result from his, but here he has to make a choice for his rapidly decreasing function, and what I am observing is that his choice for the component at the infinite places involves an "arbitrary" non-zero constant. | |
| Feb 5, 2010 at 3:48 | comment | added | Emerton | I was going to ask if Deninger's work suggested a particular normalization, but Jim seems to have answered that! Do I remember correctly that in Jacquet--Langlands they just choose the standard Gamma factor by convention (after having shown that the local factor has to lie in the one-dimensional space that it spans, by some kind of Whittaker model computation)? | |
| Feb 5, 2010 at 2:51 | comment | added | JBorger | Hi Kevin. This is probably orthogonal to your question, but according to my understanding of Deninger's point of view, Gamma factors have a preferred normalization---the one that makes his regularized products work. See "On the Gamma-factors attached to motives" (Inv. Math. 1991). From what I remember, he asserts elsewhere that this normalization is supported by the point of view in Tate's thesis. (NB This is all based on a cursory reading.) | |
| Feb 4, 2010 at 22:47 | comment | added | Kevin Buzzard | @Rob H: What I was trying to say about real places could be summarised like this: the zeta function has Gamma(s/2) in its Euler factor, and for an odd Dirichlet character it has Gamma((s+1)/2) in. And I'd like to think that I'd not made a choice of square root of -1...not yet at least. My Hecke characters are valued in C though, rather than R-bar (so I've made a choice in my coefficient field). | |
| Feb 4, 2010 at 21:28 | comment | added | Rob Harron | The gamma factor at a real place already considers the sign function (unless I'm misunderstanding what you mean), on the other hand, you've probably also made a choice of square root of -1. | |
| Feb 4, 2010 at 20:51 | history | asked | Kevin Buzzard | CC BY-SA 2.5 |