Timeline for Is Gauss sum a p-adic measure?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13, 2014 at 12:28 | comment | added | KConrad | What I wrote does not imply that different choices must change the $p$-adic valuation, but only that it might change it. You need to compute the sums to find out. In fact Odoni's paper, which I mentioned in a comment to David's answer, shows that the story of the $p$-adic valuation for Gauss sums mod $p^n$ is a lot simpler than for Gauss sums on finite fields, where the $p$-adic valuation is very sensitive to the choice of character, as described by Stickelberger's congruence (or the Gross-Koblitz formula). | |
| Sep 13, 2014 at 12:19 | comment | added | user57657 | @KConrad Thank you. I think I may get the point. Since at beginning, I have fixed an isomorphism $\mathbb{C}\simeq\mathbb{C}_p$, naively I view $G(\chi)$ as a $p$-adic one, and such isomorphism compositing with canonical complex additive character gives a $p$-adic additive character. What you say actually tells me that different choice of $\mathbb{C}\simeq\mathbb{C}_p$ changes the $p$-adic valuation of $G(\chi)$. | |
| Sep 13, 2014 at 11:19 | comment | added | KConrad | Your citation of Bump's book is not good. There he defines the complex Gauss sums, for which there is a canonical additive character. You are asking about $p$-adic valued Gauss sums, and there your notation in fact is incomplete. Such a sum relies on a choice of additive character as well, since there is not a canonical choice. Therefore I agree with Joël: you should have written out a formula for your Gauss sum instead of just saying it is well-known. | |
| Sep 13, 2014 at 6:06 | comment | added | user57657 | @GH from MO: For the definition of p-adic measure, please see chapter 3 of Hida "Elementary Theory of L-functions and Eisenstein Series" | |
| Sep 13, 2014 at 6:04 | comment | added | user57657 | @Joël We can view $\chi$ as a Dirichlet character, then the definition is Well-known, for instance, see page 4 of Bump "automorphic forms and representations". We note that the conductor of $\chi$ in my question is always a power of $p$. | |
| Sep 13, 2014 at 5:04 | answer | added | David Loeffler | timeline score: 4 | |
| Sep 12, 2014 at 23:31 | comment | added | GH from MO | Please explain what you mean by a $p$-adic measure on $\Gamma$. | |
| Sep 12, 2014 at 14:49 | comment | added | Joël | Dear user57657, can you please write down the formula defining $G(\chi)$ to ensure that everyone understands the same thing with this notation? | |
| Sep 12, 2014 at 11:59 | history | asked | user57657 | CC BY-SA 3.0 |