Timeline for The formula for (and computation of) the inverse p-adic mellin transform
Current License: CC BY-SA 4.0
18 events
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| Feb 10, 2020 at 0:03 | comment | added | wskrsk | Yes that is the inverse transform. The inversion formula in this setting apriori holds for functions that are locally constant and compactly supported in $\mathbb{Q}_p^{\times}$, i.e. on sets of the form $-n \leq \mathrm{val}(x) \leq n$. Their Mellin transforms correspond to Laurent polynomials in $p^{-s}$. The fact that $F$ has a pole is reflected in the fact that it's inverse is the characteristic function of $\mathbb{Z}_p$ which is not compact. This is in some sense the content of the so called "unramified calculation" in Tate's thesis. | |
| Feb 9, 2020 at 23:55 | comment | added | MCS | Also, for what $\mathfrak{z}\in\mathbb{Z}_{p}$ does this inversion formula hold? For almost every $\mathfrak{z}$? For every $\mathfrak{z}$? And what will the inversion formula output at values of $\mathfrak{z}$ where $f$ has a discontinuity? Edit: I meant $\mathbb{Z}_{p}\backslash\left\{ 0\right\}$ in the integral in the previous comment. $f$ is supported in $\mathbb{Z}_{p}$. | |
| Feb 9, 2020 at 23:53 | comment | added | MCS | So, the inverse transform is $$\mathscr{M}_{p}^{-1}\left\{ F\right\} \left(\mathfrak{z}\right)=\frac{\ln p}{2\pi i}\int_{-\frac{i\pi}{\ln p}}^{\frac{i\pi}{\ln p}}F\left(s\right)\left|\mathfrak{z}\right|_{p}^{-s}ds$$ where the integral is taken over the vertical line segment from $s=-\frac{i\pi}{\ln p}$ to $s=\frac{i\pi}{\ln p}$? So, if I evaluate this integral directly, it will give me the function $f$ such that $$F\left(s\right)=\int_{\mathbb{Z}_{p}}\left|\mathfrak{z}\right|_{p}^{s-1}f\left(\mathfrak{z}\right)d\mathfrak{z}$$ Is that correct? | |
| Feb 9, 2020 at 23:35 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 20:48 | comment | added | MCS | Finally, just as a piece of friendly advice, when someone asks a question about being unable to understand the notation in a document, using the notation of the question-asker (or something close to it) maximizes the chance of your answer being understood and appreciated by them. :) Note that all of the formulae I gave use p-adic absolute values and a complex variable $s$. I specifically avoided using the $\chi$ notation because I find it confusing. It's difficult for me to understand what you've written when you use the same ambiguous, abbreviated notation as the notes. | |
| Feb 9, 2020 at 20:43 | comment | added | MCS | Think of it like this: right now, I am like the student who asks "what is the derivative of sin(x)?" Your answer, to me, is like saying "Well, first you need to compute the difference quotient..." and giving a lengthy rambling explanation of how to derive the formula for the derivative of sin(x). Meanwhile, all I want to hear is "the derivative of sin(x) is cos(x)". If I know how the formulae work, I can eventually attain understanding of what they meaning and the reasoning behind them. But without knowing the purely formal rules for manipulating the symbols, I can't do anything. | |
| Feb 9, 2020 at 20:39 | comment | added | MCS | I don't understand your notation, you aren't parameterizing the integrals, you aren't parameterizing the characters, and you've said absolutely nothing about the heart of my question: giving me the correct formula for the inverse transform and explaining how (if at all) one can use the residue theorem to evaluate it. Although I appreciate you trying to explain the ideas behind the computations, I have no room for that in my head at the moment. Right now, I need formulae and algorithms to memorize. I can focus on understanding the reasoning behind them once I know how to do them. | |
| Feb 9, 2020 at 10:02 | comment | added | wskrsk | I tried to clarify my answer: the unramified characters are trivial on the unit group $\mathbb{Z}_p^{\times}$, and so are determined by their values at $p$. This introduces the complex variable. The inversion done by the contour integral, however you choose to parametrize it and keep in mind to use the Haar measure on the unit circle. | |
| Feb 9, 2020 at 9:54 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 5:45 | comment | added | MCS | I know the algorithm for computing $\int_{\mathbb{Z}_{p}}f\left(\mathfrak{z}\right)e^{-2\pi i\left\{ t\mathfrak{z}\right\} _{p}}d\mathfrak{z}$ when $f$ is a schwartz-bruhat function on $\mathbb{Z}_{p}$ and when $t\in\mathbb{Z}\left[\frac{1}{p}\right]/\mathbb{Z}$. I don't know how to do what you are describing. At this point, I just want to know the formula-rules for computing these p-adic mellin integral transforms and their inverses. | |
| Feb 9, 2020 at 5:08 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 5:02 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 4:47 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 4:27 | comment | added | MCS | You completely lose me by paragraph 3. I don't know the formula for a "character of $\mathbb{Q}_{p}^{\times}$ that is invariant under $\mathbb{Z}_{p}^{\times}". I also don't understand your notation from that point onward, and cannot follow any of it. | |
| Feb 9, 2020 at 4:21 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 4:15 | history | edited | wskrsk | CC BY-SA 4.0 |
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| Feb 9, 2020 at 4:13 | comment | added | MCS | I am fine with pontryagin duality. The moment anyone breathes a word about group representations, I run screaming for the hills. | |
| Feb 9, 2020 at 4:11 | history | answered | wskrsk | CC BY-SA 4.0 |