Tate's thesis
\(p\)-adic numbers
Let \(p\) be a prime and let \(N_p=\{0,\ldots,p-1\}\). Let \(\mathbb{N}\) be the nonnegative integers.
Define the \(p\)-adic rationals \(\mathbb{Q}_p\) to be the set of functions \(x:\mathbb{Z} \to N_p\) for which there exists some \(k_x\) such that \(x(k)=0\) for all \(k<k_x\). \(\mathbb{Q}_p\) is a field for which addition and multiplication are defined in 1 and 2
For \(x \in \mathbb{Q}_p\), define the \(p\)-adic valuation
\[v_p(x)=\inf \{k \in \mathbb{Z}: x(k) \neq 0\}.\]This is a normalized discrete valuation: (see 3)
\(v_p(\mathbb{Q}_p \setminus \{0\}) = \mathbb{Z}\),
If \(x(k)=0\) for all \(k \in \mathbb{Z}\), then \(v_p(x) = \infty\).
Define the \(p\)-adic integers
\[\mathbb{Z}_p = \{x \in \mathbb{Q}_p : v_p(x) \geq 0\}.\]Define the \(p\)-adic absolute value for \(x \in \mathbb{Q}_p\) by
\[\vert x \vert_p = p^{-v_p(x)}.\]If \(v_p(x)=\infty\), then \(\vert x \vert_p=0\).
The \(p\)-adic absolute value satisfies what is called an ultrametric/non-Archimedean property: for \(x,y \in \mathbb{Q}_p\),
\[|x+y|_p \leq \max\{|x|_p,|y|_p\}.\]For \(x \in \mathbb{Q}_p\) and for \(k \in \mathbb{Z}\), define \(p^k x \in \mathbb{Q}_p\) by
\[(p^k x)(j) = x(j-k). \qquad j \in \mathbb{Z}\]Then, for \(p\)-adic valuations,
\[v_p(p^k x) = -k + v_p(x)\]and \(p\)-adic absolute values,
\[|p^k x|_p = p^{-v_p(p^kx)} = p^{k-v_p(x)} = p^k |x|_p.\]Topology
Expressed using the \(p\)-adic absolute value,
\[\mathbb{Z}_p = \{x \in \mathbb{Q}_p: |x|_p \leq 1\}\]that is, the \(p\)-adic integers are a closed ball of radius 1 in the \(p\)-adic rationals.
Because \(v_p\) is a discrete valuation, it follows that there is some \(\epsilon>0\) such that
\[\{x \in \mathbb{Q}_p: |x|_p \leq 1\} = \{x \in \mathbb{Q}_p: |x|_p < 1 + \epsilon\},\]which shows that \(\mathbb{Z}_p\) is in fact an open set in \(\mathbb{Q}_p\).
Embedding \(\mathbb{Q} \hookrightarrow \mathbb{Q}_p\)
Additive Haar measure
The additive group \(\mathbb{Q}_p\) is a locally compact abelian group. \(\mathbb{Z}_p\) is a compact set in \(\mathbb{Q}_p\) so it has finite Haar measure, and it is an open set in \(\mathbb{Q}_p\) so it has positive Haar measure. Therefore there is a unique unique Haar measure \(\mu_p\) on \(\mathbb{Q}_p\) such that \(\mu_p(\mathbb{Z}_p)=1\).
\(\mathbb{Q}_p\) is a field and \(\mathbb{Z}_p\) is a ring. The multiplicative group of units of \(\mathbb{Z}_p\) is
\[\mathbb{Z}_p^\times = \{x \in \mathbb{Z}_p: |x|_p = 1\}.\]In particular,
\[p^k \mathbb{Z}_p = \{x \in \mathbb{Q}_p: |x|_p \leq p^{-k}\}\]and
\[p^k \mathbb{Z}_p^\times = \{x \in \mathbb{Q}_p: |x|_p = p^{-k}\},\]and one checks that
\[\mathbb{Q}_p = \{0\} \cup \bigcup_{k \in \mathbb{Z}} p^k \mathbb{Z}_p^\times\]and
\[\mathbb{Z}_p = \{0\} \cup \bigcup_{k \in \mathbb{N}} p^k \mathbb{Z}_p^\times.\]One checks
\[\mu_p(p^k \mathbb{Z}_p)=p^{-k}\]and
\[\mu_p(\mathbb{Z}_p^\times) = \frac{p-1}{p}.\]For a Borel set \(A\) in \(\mathbb{Q}_p\) and for \(x \in \mathbb{Q}_p\),
\[\mu_p(x \cdot A) = |x|_p \mu_p(A)\]For \(f \in L^1(\mathbb{Q}_p)\) and \(x \in \mathbb{Q}_p \setminus \{0\}\),
\[\int_{\mathbb{Q}_p} f(x^{-1}y)d\mu_p(y) = |x|_p \int_{\mathbb{Q}_p} f(y) d\mu_p(y).\]Multiplicative Haar measure
The multiplicative group
\[\mathbb{Q}_p^\times = \{x \in \mathbb{Q}_p : x \neq 0\} = \mathbb{Q}_p \setminus \{0\}\]is a locally compact abelian group. The restriction of \(\mu_p\) to the Borel \(\sigma\)-algebra of \(\mathbb{Q}_p^\times\) is a Borel measure on \(\mathbb{Q}_p^\times\).
One then proves (see 4) that
\[d\nu_p(x) = \frac{p}{p-1} \dfrac{1}{|x|_p} d\mu_p(x)\]is a Haar measure on \(\mathbb{Q}_p^\times\) with \(\nu_p(\mathbb{Z}_p^\times)=1\).
Then it is proved that for \(s \in \mathbb{C}\) with \(\mathrm{Re}(s)>-1\),
\[\int_{\mathbb{Z}_p^\times} |x|_p^s d\mu_p(x) = \dfrac{p-1}{p(1-p^{-1-s})}\]and that for \(\mathrm{Re}(s)>0\),
\[\int_{\mathbb{Z}_p^\times} |x|_p^s d\nu_p(x) = \dfrac{1}{1-p^{-s}}.\]\(p\)-adic fractional parts
For \(x \in \mathbb{Q}_p\), define \([x]_p \in \mathbb{Q}_p\) by
\[[x]_p(k) = \begin{cases} x(k)&k<0\\ 0&k \geq 0 \end{cases}, \quad k \in \mathbb{Z},\]called the \(p\)-adic fractional part of \(x\).
For \(x \in \mathbb{Q}\), 5
\[x - \sum_{p} [x]_p \in \mathbb{Z}.\]Additive characters on \(\mathbb{Q}_p\)
\(p\)-adic Fourier transform
\(p\)-adic Gaussians
Adeles
Chapter 2, “Preliminaries on p-adic and Adelic Technology” of 6.
Ideles
Haar measure on adeles
Tate’s thesis
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https://jordanbell.info/LaTeX/mathematics/padicharmonic/#S3 ↩
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Philipp Fleig, Henrik P. A. Gustafsson, Axel Kleinschmidt and Daniel Persson. Eisenstein Series and Automorphic Representations with Applications in String Theory. Cambridge Studies in Advanced Mathematics. Volume 176. Cambridge University Press. 2018. ↩ ↩2
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Dorian Goldfeld and Joseph Hundley. Automorphic Representations and L-Functions for the General Linear Group. Volume I. Cambridge Studies in Advanced Mathematics. Volume 129. Cambridge University Press. 2011. ↩