Kristian Holm · Lectures 3–4
Notes from Kristian Holm’s minicourse at RGNT in Lausanne, June 23, 2026. See the programme and lectures 1 and 2. Throughout, \(n\geq2\); the second-moment formula below requires \(n\geq3\).
We continue the derivation of Siegel’s and Rogers’ integration formulas using the Hecke averages \[ (T_pF)(g\mathbb{Z}^{n}) =\frac{1}{L_n(p)}\sum_{c=1}^{n}\sum_{x\in\mathbb{F}_p^{c-1}} F\bigl(p^{-1/n}gM_c(x)\mathbb{Z}^{n}\bigr). \] The intended limiting average, for the test functions under consideration, is \(\int_{X_n}F\,\mathrm{d}\mu\).
For a compactly supported continuous function \(f:\mathbb{R}^{n}\to\mathbb{R}_{\geq0}\), we want \[ \int_{X_n}\sum_{v\in\Lambda\setminus\{0\}}f(v)\,\mathrm{d}\mu(\Lambda) =\int_{\mathbb{R}^{n}}f(x)\,\mathrm{d}x. \] Take \[ F(g\mathbb{Z}^{n})=\sum_{m\in\mathbb{Z}^{n}\setminus\{0\}}f(gm). \] Since \(L_n(p)\sim p^{n-1}\), we can study the corresponding congruence sums with normalization \(p^{-(n-1)}\).
Recall the matrices \(M_c(x)\) from the previous lecture and set \[ \Lambda_c(x)=M_c(x)\mathbb{Z}^{n}. \] In block coordinates, \[ \Lambda_c(x)= \left\{ \begin{pmatrix}a_1\\xa_1+pa_2\\a_3\end{pmatrix} :a_1\in\mathbb{Z}^{c-1},\ a_2\in\mathbb{Z},\ a_3\in\mathbb{Z}^{n-c} \right\}. \] Here \(x\) is a row vector and \(xa_1\) is a scalar. Write \(m=(m_1,m_2,m_3)\in\mathbb{Z}^{c-1}\times\mathbb{Z}\times\mathbb{Z}^{n-c}\) and define \[ h_p(m)=f(p^{-1/n}gm),\qquad I_c(m_1,m_2)=\#\{x\in\mathbb{F}_p^{c-1}:xm_1\equiv m_2\pmod p\}. \] Then \[ (T_pF)(g\mathbb{Z}^{n}) =\frac{1}{L_n(p)}\sum_{c=1}^{n} \sum_{m\in\mathbb{Z}^{n}\setminus\{0\}}h_p(m)I_c(m_1,m_2). \] For \(c\geq2\), \[ I_c(m_1,m_2)= \begin{cases} p^{c-1},&m_1\equiv0\pmod p\text{ and }m_2\equiv0\pmod p,\\ 0,&m_1\equiv0\pmod p\text{ and }m_2\not\equiv0\pmod p,\\ p^{c-2},&m_1\not\equiv0\pmod p. \end{cases} \] For \(c=1\), the count is \(1\) if \(m_2\equiv0\pmod p\) and \(0\) otherwise.
For \(c=n\), the terms with \(m_1\not\equiv0\pmod p\) contribute asymptotically \[ \frac{1}{p} \sum_{\substack{(m_1,m_2)\in\mathbb{Z}^{n-1}\times\mathbb{Z}\\m_1\not\equiv0\pmod p}} f\bigl(p^{-1/n}g(m_1,m_2)\bigr). \] Without the congruence restriction, this is a Riemann sum converging to \(\int_{\mathbb{R}^{n}}f(x)\,\mathrm{d}x\). The removed non-zero terms are \[ \frac{1}{p} \sum_{\substack{(m_1,m_2)\in p\mathbb{Z}^{n-1}\times\mathbb{Z}\\(m_1,m_2)\ne0}} f\bigl(p^{-1/n}g(m_1,m_2)\bigr). \] Choose \(R\) such that \(g^{-1}\operatorname{supp}(f)\subseteq B_0(R)\). Then \(h_p(m)\ne0\) implies \(\|m\|\leq Rp^{1/n}\). If \(m_1\ne0\) and \(m_1\in p\mathbb{Z}^{n-1}\), then \(\|m_1\|\geq p\), so such points leave the support as \(p\to\infty\). The remaining points have \(m_1=0\) and lie on a one-dimensional subspace. There are \(O(p^{1/n})\) of them in the support, so their normalized contribution is \(O(p^{1/n-1})\to0\).
The terms with \(c<n\) are lower-order terms. This gives the pointwise limit of the Hecke averages in this calculation. Passing from that limit to the invariant integral requires control of the unbounded Siegel transform; the integrability discussion in lecture 4 addresses part of this issue.
For \(f\in C_c^{\infty}(\mathbb{R}^{n}\setminus\{0\})\), define \[ \Phi(f)=\int_{X_n}\sum_{v\in\Lambda\setminus\{0\}}f(v)\,\mathrm{d}\mu(\Lambda). \] The functional is positive, and local bounds follow from Schmidt’s estimate \[ |\widehat{f}(\Lambda)|\ll_{\operatorname{supp}(f)}\|f\|_{\infty}\alpha(\Lambda), \] where \[ \alpha(\Lambda)= \sup_{\substack{V\leq\mathbb{R}^{n},\ 1\leq\dim V\leq n\\ \operatorname{span}_{\mathbb{R}}(V\cap\Lambda)=V}} \operatorname{covol}_{V}(V\cap\Lambda)^{-1}. \] In two dimensions, this measures the size of an excursion into the cusp of \(\mathrm{SL}_2(\mathbb{Z})\backslash\mathbb{H}\).
The needed integrability statement is \[ \alpha\in L^p(X_n,\mu),\qquad p=1,\ldots,n-1. \] The original notes attribute this to Eskin–Margulis–Mozes, with the lemma number left uncertain.
By the Riesz representation theorem, there is a Radon measure \(\nu\) on \(\mathbb{R}^{n}\setminus\{0\}\) such that \[ \Phi(f)=\int_{\mathbb{R}^{n}\setminus\{0\}}f(x)\,\mathrm{d}\nu(x). \] For \(h\in\mathrm{SL}_{n}(\mathbb{R})\), invariance gives \(\Phi(f\circ h)=\Phi(f)\). Hence \(\nu\) is \(\mathrm{SL}_{n}(\mathbb{R})\)-invariant, so \[ \nu=c\,\mathrm{d}x \] for some constant \(c\geq0\). To recover Siegel’s formula, one still has to show \(c=1\). This normalization step is not recorded in the original proof sketch; integrability alone does not determine the constant.
For integers \(1\leq k\leq n-1\), Rogers’ formula computes moments of the Siegel transform: \[ \int_{X_n}\left(\sum_{v\in\Lambda\setminus\{0\}}f(v)\right)^k\,\mathrm{d}\mu(\Lambda). \]
For the second-moment formula, assume \(n\geq3\) and take a non-negative, continuous, compactly supported function \(f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}_{\geq0}\). Then \[ \begin{aligned} &\int_{X_n}\sum_{v,w\in\Lambda\setminus\{0\}}f(v,w)\,\mathrm{d}\mu(\Lambda)\\ &\quad=\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}f(x,y)\,\mathrm{d}x\,\mathrm{d}y\\ &\qquad\quad+\sum_{q=1}^{\infty}\frac{1}{q^n} \sum_{\substack{a\in\mathbb{Z}\setminus\{0\}\\\gcd(a,q)=1}} \int_{\mathbb{R}^{n}}f\left(x,\frac{a}{q}x\right)\,\mathrm{d}x. \end{aligned} \] The first term comes from linearly independent pairs, and the sum comes from linearly dependent pairs.
Proof sketch. Set \[ F(g\mathbb{Z}^{n})=\sum_{m,k\in\mathbb{Z}^{n}\setminus\{0\}}f(gm,gk), \qquad h_p(m,k)=f(p^{-1/n}gm,p^{-1/n}gk). \] Expanding the Hecke average gives \[ \begin{aligned} (T_pF)(g\mathbb{Z}^{n}) &=\frac{1}{L_n(p)}\sum_{c=1}^{n} \sum_{m,k\in\mathbb{Z}^{n}\setminus\{0\}}h_p(m,k)\\ &\quad\times\sum_{x\in\mathbb{F}_p^{c-1}} \mathbf{1}_{\{xm_1\equiv m_2\pmod p\}} \mathbf{1}_{\{xk_1\equiv k_2\pmod p\}}. \end{aligned} \] Here the block decomposition of \(m\) and \(k\) depends on \(c\), as in lecture 3.
For the \(c=n\) terms with \(\operatorname{rank}_{\mathbb{F}_p}(m_1,k_1)=1\), suppose \(k_1\not\equiv0\pmod p\) and write \(m_1\equiv\lambda k_1\pmod p\). Compatibility requires \(m_2\equiv\lambda k_2\pmod p\); when it holds, there are \(p^{n-2}\) admissible row vectors \(x\). The argument then separates linearly dependent integer pairs from the remaining congruence cases.
A linearly dependent non-zero integer pair can be written as \[ m=qs,\qquad k=as, \] with \(s\in\mathbb{Z}^{n}_{\mathrm{prim}}\), \(q\geq1\), and \(a\in\mathbb{Z}\setminus\{0\}\). The primitive-vector Riemann sum used here is, for a compactly supported continuous function \(\varphi\), \[ \frac{1}{p}\sum_{s\in\mathbb{Z}^{n}_{\mathrm{prim}}}\varphi(p^{-1/n}gs) \longrightarrow\frac{1}{\zeta(n)}\int_{\mathbb{R}^{n}}\varphi(x)\,\mathrm{d}x. \] Grouping the coefficients \(q,a\) by their common divisor gives the coprime sum in Rogers’ formula. This sketch does not record the estimates for the remaining congruence cases or the justification for passing the limit through the infinite sums.
Return to lectures 1 and 2.
This page was updated on October 19, 2026.
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