Timeline for On the pole of local L-function
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 23, 2014 at 9:32 | vote | accept | Monty | ||
| Aug 23, 2014 at 9:32 | comment | added | Monty | @GHfromMO, Thanks again. I will try to read the book you recommend. | |
| Aug 23, 2014 at 8:36 | comment | added | GH from MO | @classnumb: The statement in your second sentence is false. Indeed, for every nonzero complex number $s$, the function $\chi=|\cdot|^s$ on $\mathbb{A}^{\times}/F$ is a nontrivial Hecke character. On the other hand, the statement in Bump's book is true, it is a special case of Corollary 2 in Section VII.3 of Weil: Basic number theory. I recommend studying Sections VII.3 and VII.4 in Weil's book. Also, for new questions you might have, open a new question at MathOverflow (i.e. do not ask further questions as comments here). | |
| Aug 23, 2014 at 5:58 | comment | added | Monty | On the other hand, may I ask one another question? When $\chi$ is a nontrivial Hecke character of $\mathbb{A}^{\times}/F$ , then how do we know that there is no complex number $s$ such that $\chi \ne |\cdot|^s$ ? Because in the bump's book, it says that every quasi character $\chi$ of $\mathbb{A}^{\times}/F$ is a product of a unitary Hecke character $\chi_1$ and $|\cdot|^s$ for some complex $s$. | |
| Aug 23, 2014 at 5:57 | comment | added | Monty | Thanks for your clear answer. As you recommeded, I have read Tate's thesis contatined in Bump's book. And I learned that the above local L-function is the same form obtained from the meromorphic continuation of local zeta integral when the test function is the characteristic function. And so I think if $\chi$ is a Hecke character, the $L_v(\chi_v,x)$ would have no pole for $Re(s) >0$ and it may have pole only at $\Re(s)=0$. Your remark that the pole of global L-function do not pertains to that of local L-function is striking to me. I confused it for a long time. Thanks for your illumination. | |
| Aug 21, 2014 at 12:50 | comment | added | GH from MO | @paulgarrett: I agree, but of course the epsilon factors combine nicely! | |
| Aug 21, 2014 at 12:47 | comment | added | paul garrett | Another, similar, common misconception is that "local functional equations" should combine to give the "global functional equation", and this is not the case. | |
| Aug 21, 2014 at 8:32 | history | edited | GH from MO | CC BY-SA 3.0 |
added 336 characters in body
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| Aug 21, 2014 at 8:24 | history | answered | GH from MO | CC BY-SA 3.0 |