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These are some lecture notes takes during the Summer school on formulas of Siegel and Weil at Bielefeld. Feel free to let me know if there are some errors or typos.
In how many ways can one represenet \(n\) as the sum of \(2\) squares?
Define \[\begin{equation} r(n) = \# \{ (x,y) \in \mathbb{Z}^{2} \mid n = x^{2} + y^{2}\}. \end{equation}\]
(Jacobi 1829)
\[\begin{equation} r(n) = 4 ( \sum_{\substack{d \mid n \\ d = 1 \pmod{4}}}^{} 1 - \sum_{\substack{d \mid n \\ d \sim 3 \pmod{4}}}^{} 1 ) \end{equation}\]
Define \[\begin{equation} \Theta = \sum_{n \in \mathbb{Z}} q^{n^{2}}, \end{equation}\] where \(q \in \mathbb{C}, |q |\leq 1\).
Then \(\Theta^{2} = (\sum_{n \in \mathbb{Z}}q^{n^{2}})^{2} = \sum_{m=0}^{\infty}r(m) q^{m}\) . Use therefore the triple product identity to get \[\begin{equation} \Theta = 1 + 4( \frac{q }{1-q} - \frac{q^{3}}{1-q^{3}} + \frac{q^{5}}{1-q^{5}} - \frac{q^{7}}{1-q^{7}} + \dots ). \end{equation}\]
Complete the step.
Let \(\chi:(\mathbb{Z}/4 \mathbb{Z})^{\times} \to \{\pm 1\}\) be the unique non-trivial character. We define the Eisenstein series \[\begin{equation} G_{k}^{\chi}(\tau) = \sum_{\substack{(c,d) \in \mathbb{Z}^{2} \setminus \{0\} \\ 4 \mid c}} \frac{\chi(d)}{(c \tau + d)^{k}}, \end{equation}\] where \(\tau \in \mathbb{H}\) is a point on the upper half plane (i.e. \(\mathbb{H} = \{z \in \mathbb{C} \mid \Im(z)>0\}\).)
We also define \[\begin{equation} E_{1}^{\chi}(\tau) = 1 + 4 \sum_{d \mid n} \chi(d) q^{n} , \end{equation}\] for \(q = e^{\pi i \tau}\).
One then has that \(\Theta^{2} = E_{1}^{\chi}(\tau)^{2}\).
Let \(\Lambda\) be a positive definite quadratic lattice over \(\mathbb{Z}\) of rank \(m\). Let \(Q: \Lambda \rightarrow \mathbb{Z}\) be a quadratic form. Let \(B: \Lambda \times \Lambda \rightarrow \mathbb{Z}\) be given as \[\begin{equation} B (x,y) = Q(x+y) - Q(x) - Q(y). \end{equation}\] That is, \(B\) is the associated quadratic form of \(Q(z) = \tfrac{1}{2}B (x,y)\).
Let \(\Symm_{n}(\mathbb{Z})\) be the set of symmetric matrices given as \[\begin{equation} \begin{bmatrix} \mathbb{Z} & \tfrac{1}{2} \mathbb{Z} & \dots & \tfrac{1}{2} \mathbb{Z} \\ \tfrac{1}{2}\mathbb{Z} & \mathbb{Z} & \dots & \tfrac{1}{2} \mathbb{Z} \\ & \vdots & & \\ \tfrac{1}{2}\mathbb{Z}& \dots & \tfrac{1}{2}\mathbb{Z} & \mathbb{Z} \end{bmatrix}. \end{equation}\] Denote \(\Symm_{n}(\mathbb{Z})_{\geq 0}\) the subset of positive semi-definite matrices.For \(T \in \Symm_{n}(\mathbb{Z})_{\geq 0}\), the generalized representation number \[\begin{equation} r_{\Lambda}(T) = \#\{ (x_{1},\dots,x_{n})\in \Lambda^{n} \in \mid (\tfrac{1}{2} B (x_{i},x_{j})_{1 \leq i,j \leq n} = T \}. \end{equation}\]
We have Siegel’s half space \[\begin{equation} \mathbb{H}_{n} =\{ \tau = x+iy \mid x \in \Symm_{n}(\mathbb{R}), y \in \Symm_{n}(\mathbb{R})_{\geq 0} \}. \end{equation}\]
Siegel’s theta series associated to \(\Lambda\) is \[\begin{equation} \Theta_{\Lambda} ( \tau) = \sum_{\tau \in \Symm_{n}(\mathbb{Z})_{\geq 0}} r_{\Lambda}(\tau) e^{\pi i \Tr(T \cdot \tau)}. \end{equation}\]
Siegel proved using the Poisson summation formula that the Siegel theta function \(\Theta_{\Lambda}(\tau)\) is a Siegel modular form on \(\mathbb{H}_{n}\) of weight \(m/2\).
If we take such a theta function, can we expect that it coincides with some Eisenstein series?
Two quadratic lattices \(\Lambda,\Lambda'\) are in the same genus if they are isomotric over \(\mathbb{R}\) and over \(\mathbb{Z}_{p}\) for every prime \(p\).
We define \(\Gen(\Lambda)\) to be the set of isomorphism classes of quadratic lattices in the same genus as \(\Lambda\). We define \(\Aut(\Lambda)\) to be the automorphism group of \(\Lambda\) as a quadratic lattice.
(Siegel)
One has \[\begin{equation} \frac{1}{\sum_{\Lambda' \in \Gen(\Lambda)} (\# \Aut(\Lambda') )^{-1} } \sum_{\Lambda' \in \Gen(\Lambda)} \frac{\Theta_{\Lambda'}(\tau)}{\# \Aut(\Lambda') } = E_{\Lambda}(\tau) , \end{equation}\] where \(E_{\Lambda}(\tau)\) is certain normalized Siegel-Eisenstein series on \(\mathbb{H}_{n}\) of weight \(m/2\).
The lectures are based on the references [1], [2].
Let \(V\) be a quadratic space over \(\mathbb{Q}\) of dimension \(n\). Let \(V_{\mathbb{A}}\) be \(V \otimes _{\mathbb{Q}} \mathbb{A}\) and let \(\mathcal{S}(V_{\mathbb{A}}^{n})\) is the space of Shwartz Bruhat functions on \(V_{\mathbb{A}}^{n}\). Weil representation gives an action of a symplectic group (or its double cover) on \(\mathcal{S}(V_{\mathbb{A}}^{n})\). Let \((G,H) = (\Sp(2n),\mathcal{O}(V))\). We denote \(\omega\) to be the Weil representation of \(\Sp(2n)\) on \(\mathcal{S}(V(\mathbb{A})^{n})\).
Associated to \(\Phi \in \mathcal{S}(V(\mathbb{A})^{n})\) be the (two-variable) theta function \[\begin{equation} \Theta(h,g,\Phi) = \sum_{x\in V_{\mathbb{Q}}^{n}} \omega(h) \Phi(g^{-1} x). \end{equation}\]
For \(\tau \in \mathbb{H}_{n}\), and \(\Lambda\) a quadratic lattice, Define \(h_{\tau},g_{\tau}\) and \(\Theta_{\tau, \Lambda}\) set \(\Theta(h_{\tau},g_{\tau}, \Theta) = \Theta_{\Lambda}(\tau)\).
Associated to \(\Phi \in \mathcal{S}(V(\mathbb{A})^{n})\) define the Siegel-Eisenstein sries \[\begin{equation} E(h,\Phi) = \sum_{\gamma \in P(\mathbb{Q}) \backslash H(\mathbb{Q})} \omega( \gamma \cdot h) \Phi(0). \end{equation}\]
For \((H,G) = (\Sp(2n),O(V))\) and \(k=\mathbb{Q}\) one has \[\begin{equation} x \int_{G(\mathbb{Q}) \backslash G(\mathbb{A})} \Theta(h,g, \Theta) \diff g = E(h,\Phi), \end{equation}\] where \(x \in \{1,2\}\). There are also some conditions on \(m\) and \(n\) that guarantee convergence.
A group \(\mathcal{G}\) is said to be elementary when it is of the form \[\begin{equation} \mathcal{G} = \mathbb{R}^{n} \times \mathbb{Z}^{p} \times T^{q} \times F, \end{equation}\] where \(T = \{z \in \mathbb{C} \mid |z| = 1\}\) and \(F\) is a finite group.
If \(\mathcal{G}\) is elementary, the Schwartz space \(\mathcal{S}(\mathcal{G})\) is the space of \(C^{\infty}\)-functions \(\Phi\) such that for any polynomial \(P\) and transformation invariant differential operator \(D\), one has \(P \cdot D \Phi\) is bounded on \(\mathcal{G}\). The topology on \(\mathcal{S}(\mathcal{G})\) is the topology defined by the seminorms \(\sup_{x \in \mathcal{G}} |P \cdot D \Phi(x)|\).
Consider pairs of subgroup \((H,H')\) in \(\mathcal{G}\) with the following properties.
Consider the family \(\mathcal{S}(H,H')\) of continuous functions on \(\mathcal{G}\). With the following three properties:
Then, finally the Schwartz space \(\mathcal{S}(\mathcal{G})\) is the union of \(\mathcal{S}(H,H')\) and we give \(\mathcal{S}(\mathcal{G})\) the inductive limit topology, i.e. a convex set \(X\) is a neighbourhood of \(0\) in \(\mathcal{S}(\mathcal{G})\) if for any pair \(H,H'\) the image of \(X \cap \mathcal{S}(H,H')\) in \(\mathcal{S}(H/H')\) is a neighbourhood of zero.
In our case \(\mathcal{G}\) is \(X_{k_\mathbb{A}}\) for a finite-dimensional vector \(X\) space over a global field \(k\).Find a more practical definition for \(\mathcal{S}(k_{\mathbb{A}})\).
Let \(k\) be a number field. Let \(X\) be a vecor space over \(k\). Let \(\mathcal{G} = X_{\mathbb{A}}\).
One has \(\Phi = \Pi_{v} \Phi_{v}\) where \(\Phi_{v}\) is a function on \(X_{k_{v}}\).
If \(v\) is Archimedean , \(v = \infty \implies \Phi_{\infty}\) is a Schwartz function in usual sense.
If \(v\) is non-Arch. \(\implies \Phi_{v}\) is a compactly supported function and locally constant.
For almost all \(v\), \(\Phi_{v} = \mathbb{1}_{\mathcal{O}_{v}^{n}}\).
Let \(\mathcal{G}\) be locally compact group and let \(\mathcal{G}^{*}\) be a the Pontryagin dual of \(\mathcal{G}\).
This is the space of characters \[x^{*}: \mathcal{G} \rightarrow T =\{z \in \mathbb{C} \mid |z|=1 \} . \]
Let \(\diff x\) be a Haar measure on \(\mathcal{G}\). Let \(\Phi\) be a function on \(\mathcal{G}\), Let \(\Phi^{*}\) be a function on \(\mathcal{G}^{*}\) defined by \[\begin{equation} \Phi^{*}(x^{*}) = \int_{\mathcal{G}}^{} \Phi(x) \langle x,x^{*}\rangle\diff x, \end{equation}\] where \(\langle x, x^{*}\rangle\) is the evaluation of \(x^{*}\) on \(x\). This is defined formally for now, but we will talk about convergence later.
Let \(X\) be a vector space over \(k\). Let \(X^{*}\) be the dual vector space of \(X\). Let \(\Sp(X)\) be the group of automorphisms of \(X \oplus X^{*}\).
This can also be described as the set of \(G\) such that \[\begin{equation} \sigma = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}, \sigma^{\dagger} = \begin{bmatrix} \delta^{*} & - \beta^{*} \\ -\gamma^{*} & \alpha^{*} \end{bmatrix}, \end{equation}\] and \(\sigma \sigma ^{\dagger} = 1\). Here \(\alpha \in \Aut(X), \beta \in \Iso(X^{*},X), \gamma \in \Iso(X,X^{*}), \delta \in Aut(X^{*})\).
Show that this is how \(\Sp(X)\) is defined.
Elements \(\sigma \in \Sp(X)\) preserve the alternating \(k\)-form \([ \cdot , \cdot ] : (X \oplus X^{*}) \times (X \oplus X^{*})\) given by \[\begin{equation} [(x_{1},x_{1}^{*}) , (x_{2},x_{2}^{*}) ] = \{x_{1},x_{2}^{*}\} - \{x_{2}^{*},x_{1}\}. \end{equation}\]
Consider the pairs \(s = (\sigma,f)\) with \(\sigma \in \Sp(X)\), \(f\) is a quadratic form on \(X \oplus X^{*}\). For \(z_{1},z_{2} \in X \oplus X^{*}\), one has \[\begin{equation} f(z_{1}+z_{2}) - f(z_{1}) - f(z_{2}) = B(\sigma(z_{1}),\sigma(z_{2})) - B(z_{1},z_{2}). \end{equation}\] Here \(B\) is fixed bilinear symmetric form on \(X\) which fixes an isomorphism between \(X\) and \(X^{*}\). We use an isomorphism \(X \simeq X^{*}\) to extend \(B\) from \(X\) to \(X \oplus X^{*}\).
The set of pairs \((\sigma,f)\) forms a group. We denote this group by \(P_{S}(X)\). Here \(P_{S}\) stands for psuedo-symplectic.
Let \(\Phi \in L^{2}(X)\). For an automorphism \(\alpha \in \Aut X\), \(f \in Q(x)\) , \[\begin{equation} d(\alpha) \Phi (x) = |\alpha|^{\tfrac{1}{2}} \Phi(x) , \end{equation}\] \[\begin{equation} t(f)\Phi(x) = \Phi(x) \cdot \chi(f(x)) , \chi : (k,+) \rightarrow T$, \end{equation}\] \[\begin{equation} d'(\alpha) \Phi(x) = |\gamma|^{-1/2} \Phi^{*}(-x (\gamma^{*})^{-1}) \end{equation}\]
We consider \(Q(X/A)\) to be the space of \(A\)-quadratic forms for \(X\). \[\begin{equation} f(x) = \tau(F(x,x)), \end{equation}\] where \(F:X \times X \rightarrow A\) is a sesquilinear form. We get \[F(tx_{1},u x_{2}) = t F(x_{1},x_{2}) u^{i}\] where \(x_{1},x_{2} \in X\) and \(t,u \in A\).
We denote \(\Sp(X/A)\) to be the group of automorphisms \(\sigma\) of the left \(A\)-module \(X \oplus X^*\) that satisfy \(\sigma \sigma^{\dagger} = I\). \[\begin{equation} \sigma = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \end{bmatrix}, \sigma^{\dagger} = \begin{bmatrix} \delta^{*} & - \beta^{*} \\ -\gamma^{*} & \alpha^{*} \end{bmatrix}, \end{equation}\]
Elements \(\sigma \in \Sp(X/A)\) preserve the alternating \(k\)-linear form \[\begin{equation} [(x_{1},x_{1}^{*}), (x_{2},x_{2}^{*}) ] = \tau (\{x_{1},x_{2}^{*}\} - \{x_{2},x_{1}^{*}\}) \end{equation}\]
The group of automorphisms of \(X\) as an \(A\)-module is \(\Aut_{A}(X)\).
We denote \(P_{S_{k}}(X)\) to be the pseudosymplectic group attached to the \(k\)-vector space. We denote \(P_{S_{A}}(X)\) to be the subgroup of the pseudosymplectic group \(P_{S_{k}}(X)\) of those elements \((\sigma,f)\) for which \(\sigma \in \Aut_{A}(X \oplus X^{*}), f \in Q_{A}(X \oplus X^{*})\).
We recall and extend the maps
Let \(A\) be the an algebra over a field \(k\) (with an involution \(\iota\) and a trace function \(\tau\)). Suppose that either \(\char(k) \neq 2\) or \(\mathcal{A}\) is semisimple.
For \(X\) a left \(A\)-module and \(\Mp_{A}(X)\) being a metaplectic group, every element of \(\Mp_{A}(x)\) commutes with all elements of \(\Mp_{k}(X)\) of the form \(d(\chi_{a})\).
Here \(\xi_{a}\) is the homothety \(x \mapsto a x\) and \(a \in A\) satisfies \(a \cdot a^{\iota} = 1\).
We can put all these in one group and denote \[\begin{equation} G = \{a \in A \mid a \cdot a^{\iota} = 1\}. \end{equation}\]
When \(\char(k) \neq 2\), the group \(\Mp_{A}(X)\) and \(P_{S_{A}}(X)\) are the same (and resp. for \(\Mp_{k}(X)\) and \(P_{S_{k}}(X)\)).
What should be \((A,\iota,\tau)\) for this to correspond to Jan’s lectures?
Let \(\mathbb{A}_{k}\) be the ring of adeles attached to a number field or a function field \(k\). Let \(A\) be a \(k\)-algebra over \(k\) with the inclusion \(i\) and trace function \(\tau\). # Suppose that \(\char(k) \neq 2\) or \(A\) is semisimple. Let \(G_{k}\) and \(G_{\mathbb{A}}\) be as before. Let \(X_{k}\) be a left \(A\)-module and let \(X_{\mathbb{A}} = X_{k} \otimes \mathbb{A}_{k}\).
Then for all \(S \in \Mp(X/A)_{\mathbb{A}}\) and all \(a \in G_{\mathbb{A}}\) the operator \(\Phi \mapsto S \Phi\) and \(\Phi(x) \mapsto \Phi(ax)\) in \(L^{2}(X_{\mathbb{A}})\) commute.
We have now developed enough machinery to be able to express the Eisenstein series. Let \(\Phi \in \mathcal{S}(X_{\mathbb{A}})\). Let \[\begin{equation} I(\Phi) = \int_{G_{\mathbb{A}}/G_{k}}^{} \Phi(u \xi)\diff \mu(u), \end{equation}\] where \(\mu\) is a Haar measure such that \(G_{\mathbb{A}}/G_{k}=1\).
One has \(P_{k}(X) \subseteq P_{S_{k}}(X)\) and \(P_{A}(X)\) to be the subgroup of those elements \[\begin{equation} s = \Big( \Big( \begin{smallmatrix} \alpha & \beta \\ \gamma & \delta \end{smallmatrix} \Big), f \Big) \text{ such that } \gamma=0. \end{equation}\]
We define \[\begin{equation} E(\Phi) = \sum_{s \in P_{S}(X_{k}/A_{k}) / P_{k}(X_{k}/A_{k})}^{} \gamma_{k}(S) \Phi(0) \end{equation}\]
The series \(E(\Phi)\) is absolutely convergent is \(A_{k}\) and \(X_{k}\) satisfy the convergence criterion that (assuming that \(A_{k}\) is of Type (I)) \[\begin{equation} m > 2n + 4 \epsilon - 2, \end{equation}\] where \(m = \dim(A/k)\) and \(n = \rank(X/A)\). Here \(\epsilon\) is an invariant of the algebra \(A\) which Maryna did not define.
We want to now show that the theta integral is the Eisenstein series.
Define \[\begin{equation} X \otimes_{\iota} X = (X \otimes_{k} X ) / \{ t^{\iota}x \otimes y - x \otimes t y = 0, \text{ for each } x,y \in X, t \in A\}. \end{equation}\]
Take the swap map $s : X {} X X {i} X $ to be \(s(x \otimes y) = y \otimes x\).
We write \[\begin{equation} I(X) = \{u \in X \otimes_{i} X \mid s u = s\}, \end{equation}\] and \[\begin{equation} i_{X} : X \rightarrow X \otimes_{i X }, \quad x \mapsto x \otimes_{\iota} x. \end{equation}\] Denote \[\begin{equation} Q(X/A) \simeq I(X)^{*}, f(x) = [\iota_{X}(x),f]. \end{equation}\]
For any \(i \in I(X)\), let \(U(i)\) be the set of elements of \(\iota_{X}^{-1}(\{i\})\) which of of maximal rank in \(X\) (i.e. \(x\) has maximal rank if \(A x = X\)) ; then if \(U(i)\) is not empty, it is an orbit of \(G\) in \(X\).
Let \(A_{k}\) be an algebra of Type (I). Let \(X_{k}\) be a left \(A\)-module satisfying the covergence condition (\(m>2n + 4 \epsilon - 2\)). Let \(v\) be a place of \(k\) such that \(U(0)_{v}\) is not empty. Let \(G_{v}'\) be a subgroup of \(G_{v}\) acting transitively on \(U(i)_{\sigma}\) for all \(i \in I(X)_{k}\). Let \(E'\) be a positive temepered measure on \(X_{\mathbb{A}}\) invariant by \(P_{S}(X_{k}/A_{k})\) and under \(G'_{v}\).
Assume that \(E-E'\) is a sum of measures carried by the \(U(i)_{\mathbb{A}}\) for \(i \in I(X)_{k}\). Then \(E' = E\).
Recall \(I(X)\) and the map \(\iota_{X} :X \rightarrow I(X)\) given by \(x \mapsto x \otimes_{\iota} x\). We have \[\begin{equation} U(i) =\{x \in \iota_{X}^{-1}({i}) \mid x \text{ has minimal rank } \}. \end{equation}\] In fact, \(x\) has maximal rank \(\Leftrightarrow A x = X\) . (Why?)
Let \(A_{k}\) be an algebra of type (I). Let \(X_{k}\) be a left \(A_{k}\)-module + convergence condition (\(m > 2n + 4e - 2\)). Let \(v\) be a place of \(k\) such that \(U(0)_{v}\) is non-empty. Let \(G_{v}'\) a subgroup of \(G_{v}\) acting trasnsitively on \(U(i)_{v}\) for any \(i \in I(X)_{k}\).
Let \(E'\) be a positive tempered measure on \(X_{\mathbb{A}}\) invariant by \(P_{S}(X_{k},A_{k})\) and invariant under \(G'_{v}\). If \(E-E'\) is a sum of measures supported on \(U(i)_{\mathbb{A}}\) for \(i \in I(X)_{k}\), then we have \(E'=E\).
We will only be able to give an idea of the proof. Define \[\begin{equation} E'' = E'-E. \end{equation}\] We know that \(E''\) is invariant under \(P_{S}(X_{k}/A_{k})\).
The function \(S \mapsto E''(S \Phi)\) is bounded \(\Mp_{\mathbb{A}}\) for any \(\Phi \in \mathcal{S}(X_{\mathbb{A}})\). So for any \(\Phi \in S(X_{\mathbb{A}})\), we will denote by \[\begin{equation} M(\Phi) := \sup_{S \in \Mp(\mathbb{A})} |E''(S \Phi)|. \end{equation}\]
Then \(E''(\Phi) = \sum_{i \in I(X)_{k}}^{} \int_{X_{\mathbb{A}}} \Phi \diff \mu_{i}''\) where \(\mu_{i}\) is a measure supported on \(U(i)_{\mathbb{A}}\).
Next step: Show that \[\begin{equation} \Big|\int S \Phi \diff \mu''_{} \Big| \leq M(\Phi) , \end{equation}\] Let \(i^{*} \in I(X)^{*}_{\mathbb{A}}\). Let \(q \in Q(X,A)_{\mathbb{A}}\) corresponds to \(i^{*}\) by \(q(x) = [i{X}(x),i^{*}]\), \(x \in X_{\mathbb{A}}\).
For \(\Phi \in \mathcal{S}(X_{\mathbb{A}})\) we have \[\begin{equation} t(q) \Phi(x) = \Phi(x) \chi (q(x)), \end{equation}\] given by \[\begin{equation} \Phi(x) \chi( [i_{X}(x),i^{*}]). \end{equation}\] This is now given by \[\begin{equation} E''(t(q) \Phi ) = \sum_{i \in I(X)_{k}} \chi([i,i^{*}]) \int_{}^{} \Phi \diff \mu_{i}^{''}. \end{equation}\] Fourier series of the function is \(i^{*} \mapsto E''(t(q) \Phi)\) on a compact abelian group \(I(X)_{\mathbb{A}}^{*}/ I(X)_{k}^{*}\). By Fourier formula, we get \[\begin{equation} |\Phi \diff \mu_{i}''| \leq M(\Phi), \end{equation}\] replace \(\Phi\) by \(S \Phi\) given by \(\Big| \int_{}^{} S \Phi \diff \mu_{i}^{''}\Big| < M(\Phi)\).
At this point there was a question if the \(\mu_{i}''\) are normalized so that \(U(i)\) have probability measure. Maryna said, it does not matter for the conclusion.
We know that \(\mu_{i}''\) are invariant under \(G_{v}'\). Take \(X_{\mathbb{A}} = X_{v} \cdot X'\). Then \(\Phi = \Phi_{v} \cdot \Phi'\) , for \(\Phi_{v} \in \mathcal{S}(X_{v})\), \(\Phi' \in \mathcal{S}(X')\).
\[\begin{equation} \int_{}^{} \Phi \diff \mu_{i}'' = c_{i}(\Phi') \cdot \int_{U(i)_{v}}^{} \Phi_{v} |\theta_{i}|_{v} , \end{equation}\] where \(\theta_{i} = ( \tfrac{\diff x}{ \diff \iota_{X}(X)})_{i}\), \(|\theta_{i}|_{v}\) is a measure on \(\overline{U(i)}\). \(c_{i}\) is a measure on \(X'\). In this formula, replace \(\Phi\) by \(d \lambda_{t} \Phi,\) \(t \in T_{v}\)
Maryna said that this is some kind of integration in radial coordinates.
In this formula, replace \(\Phi\) by \(d(\lambda_{t} \Phi), t \in T_{v}\). \[\begin{equation} \Phi_{v} \cdot \Phi' \mapsto ( d(\lambda_{t}) \Phi_{v} ) \cdot \Phi'. \end{equation}\]\[\begin{equation} \theta_{i}(x \lambda_{t}^{-1}) = (t_{1}\dots t_{n})^{\delta(-m + n _ 2 \epsilon -1 )} \cdot \theta_{i'}(x). \end{equation}\] Then \(i' = i \cdot \overline{\lambda_{t}}\), and \(\lambda_{t}\) acts on \(X_{k}\) and \(\overline{\lambda}_{t}\) acts on \(I(X)\).
Let \(t=(t_{1},\dots,t_{n}) \rightarrow 0\) . Let \(\int_{}^{} d(\lambda_{t}) \Phi \diff \mu_{i}''\) is bounded as \(t \in T_{v}\). The \(c_{i}(\Phi')\) is constant and \(|t_{1} \dots t_{n}|_{v}^{ \delta (-m + 2n + 4 \epsilon -2)} \rightarrow \infty, t \rightarrow 0\). Finally $${U(i’){v}}^{} {v} |{i’}|{v} {U(0){v}}^{} {v |{0}|{v}} , t . Then \(U(0)_{v}\) is non-empty \(\implies\) we can choose \(\Phi_{v}\) so that \(\int_{U(0)_{v}}^{} \Phi_{v}|\theta_{0}|_{v} \neq 0\) \[\begin{equation} \implies \mu_{i}''=0. \end{equation}\]
Let \(A_{k}\) (type I) and \(X_{k}\) be as before. Let \(v\) be a place of \(k\) as before. Let \(G_{v}'\) be a subgroup of \(G_{v}\) as in the uniqueness theorem (the place \(v\) has non-empty condition). Let \(\nu\) be a positive measure on \(G_{\mathbb{A}}/G_{k}\) is invariant under \(G_{v}'\), such that \(\nu(G_{\mathbb{A}}/G_{k}) =1\) .
We define \[\begin{equation} I_{\nu}(\Phi) = \int_{G(\mathbb{A}/G_{k})}^{} \sum_{\xi \in X_{k}}^{} \Phi(u \xi) \diff \mu(u), \end{equation}\] here the integration is with respect to the other measure.
Then we have \(I_{\nu}(\Phi) = E(\Phi)\).
Proof is by induction over \(n\). We must show that the difference \(I_{v}(\Phi) - E(\Phi)\) satisfies the condition of the uniqueness theorem. For this we need to show that the difference of measures need to be supported on \(U(i)\), and everything of smaller rank can be worked out somehow by induction.
This page was updated on October 3, 2025.
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