Notes from Kristian Holm’s minicourse at RGNT in Lausanne, June 22, 2026. See the programme and lectures 3 and 4. Throughout, \(n \geq 2\).

Lecture 1

We say that \(\Lambda\) is unimodular if \(d(\Lambda)=1\). Let \[ X_n = \{\Lambda \subseteq \mathbb{R}^{n} : \Lambda \text{ is a lattice and } d(\Lambda)=1\}. \] We identify this space with the homogeneous space \[ X_n \simeq \mathrm{SL}_{n}(\mathbb{R})/\mathrm{SL}_{n}(\mathbb{Z}). \]

There are many interesting questions:

1. How many sublattices \(\Pi \subseteq \mathbb{Z}^{n}\) have index \([\mathbb{Z}^{n}:\Pi]=N\)?
2. How short can non-zero lattice vectors be? This is related to short closed geodesics on \(\mathbb{R}^{n}/\Lambda\) and to lattice-based cryptography.
3. Given a set \(S \subseteq \mathbb{R}^{n}\), estimate \(\#(\Lambda \cap S)\).

The Davenport–Heilbronn theorem counts cubic fields up to isomorphism. With both discriminant signs included, the normalization is \[ \#\{K/\mathbb{Q} : [K:\mathbb{Q}]=3,\ |\operatorname{disc}(K)|\leq T\} \sim \frac{T}{3\zeta(3)}. \] See Bhargava–Shankar–Tsimerman, Theorem 1 for the separate constants for positive and negative discriminants.

Sphere packings

If \(\Lambda \cap B_0(1)=\{0\}\), with \(B_0(1)\) the open unit ball, then the unit balls centred at \(2\Lambda\) form a lattice packing. Its density is \[ \frac{\operatorname{vol}(B_0(1))}{2^{n}d(\Lambda)}. \]

For \(f \in C_c^{\infty}(\mathbb{R}^{n})\), its Siegel transform is the function \(\widehat{f}:X_n\to\mathbb{C}\) defined by \[ \widehat{f}(\Lambda)=\sum_{v\in\Lambda\setminus\{0\}}f(v). \]

The formula also applies to \(f=\mathbf{1}_S\) for a measurable set \(S\) of finite volume, giving \[ \int_{X_n}\#\bigl((\Lambda\setminus\{0\})\cap S\bigr)\,\mathrm{d}\mu(\Lambda) =\operatorname{vol}(S). \] To obtain the extra factor of \(\zeta(n)\), one sieves for primitive points.

Primitive lattice points have asymptotic density \(\zeta(n)^{-1}\) among all lattice points.

Lecture 2

There is a natural measure on \(X_n\) with total mass \[ \widetilde{\mu}(X_n)=\frac{1}{n}\zeta(2)\zeta(3)\cdots\zeta(n). \] Thus \(\mu=\widetilde{\mu}(X_n)^{-1}\widetilde{\mu}\) is a probability measure.

Write \(G=\mathrm{SL}_{n}(\mathbb{R})\subseteq\operatorname{Mat}_{n}(\mathbb{R})\simeq\mathbb{R}^{n^2}\). For a measurable set \(A\subseteq G\), define \[ C_A(R)=\{tg:g\in A,\ 0<t\leq R\}. \] The cone normalization of Haar measure is \[ \widetilde{\mu}(A)=\operatorname{Leb}_{n^2}(C_A(1)). \] Let \(\mathcal{F}\subseteq G\) be a fundamental domain for the right action of \(\mathrm{SL}_{n}(\mathbb{Z})\). Scaling the cone gives \[ \widetilde{\mu}(X_n) =R^{-n^2}\operatorname{Leb}_{n^2}(C_{\mathcal{F}}(R)). \]

Define \(L_n(m)\) to be the number of sublattices \(\Pi\subseteq\mathbb{Z}^{n}\) with \([\mathbb{Z}^{n}:\Pi]=m\). Every full-rank integer matrix with positive determinant spans a sublattice, and choosing its normalized representative in \(\mathcal{F}\) selects one positively oriented basis for that sublattice. Consequently, \[ \#\bigl(\operatorname{Mat}_{n}(\mathbb{Z})\cap C_{\mathcal{F}}(R)\bigr) =\sum_{m\leq R^n}L_n(m). \] This suggests computing the cone volume through \[ R^{-n^2}\sum_{m\leq R^n}L_n(m). \] The passage from lattice-point counts to volume requires justification because the fundamental domain is unbounded; that step is not recorded in these notes.

Counting sublattices

Every sublattice \(\Pi\subseteq\mathbb{Z}^{n}\) has a basis in Hermite normal form: \[ \Pi= \begin{pmatrix} d_1 & 0 & \cdots & 0\\ * & d_2 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ * & * & \cdots & d_n \end{pmatrix}\mathbb{Z}^{n}. \] Here \(d_i>0\), the entries below the diagonal in row \(i\) lie in \(\{0,\ldots,d_i-1\}\), and \(d_1\cdots d_n=m\) for an index-\(m\) sublattice. The number of choices for the stars is \[ d_1^0d_2^1d_3^2\cdots d_n^{n-1}. \] Therefore \[ L_n(m)=\sum_{d_1\cdots d_n=m}d_1^0d_2^1\cdots d_n^{n-1}. \]

This is related to Dirichlet convolution: \[ \left(\sum_{r\geq1}\frac{a_r}{r^s}\right) \left(\sum_{t\geq1}\frac{b_t}{t^s}\right) =\sum_{m\geq1}\frac{\sum_{rt=m}a_rb_t}{m^s}, \] in a region of absolute convergence. In particular, \[ \sum_{m\geq1}\frac{L_n(m)}{m^s} =\zeta(s)\zeta(s-1)\cdots\zeta(s-n+1),\qquad \operatorname{Re}(s)>n. \] The rightmost pole is at \(s=n\). The Wiener–Ikehara Tauberian theorem gives \[ \sum_{m\leq X}L_n(m) \sim\frac{X^n}{n}\operatorname*{Res}_{s=n} \bigl(\zeta(s)\zeta(s-1)\cdots\zeta(s-n+1)\bigr) =\frac{X^n}{n}\prod_{j=2}^{n}\zeta(j). \]

Hecke operators

The matrices \(M_c(x)\) are exactly the Hermite normal forms of determinant \(p\). In particular, \[ L_n(p)=1+p+\cdots+p^{n-1}. \]

Two useful facts are:

1. The probability measure \(\mu\) is invariant under \(T_p\): \[ \int_{X_n}T_pF\,\mathrm{d}\mu=\int_{X_n}F\,\mathrm{d}\mu. \]
2. If \(F\in L^2(X_n)\) and \(T_pF\) converges almost everywhere to a constant as \(p\to\infty\) through the primes, then that constant is \(\int_{X_n}F\,\mathrm{d}\mu\). The \(L^2\) contraction property gives the uniform integrability needed to pass to the limit.

Conceptually, \(T_p\) averages over the unimodular rescalings of the index-\(p\) sublattices of a fixed lattice \(\Lambda\): \[ (T_pF)(\Lambda) =\frac{1}{L_n(p)} \sum_{\substack{\Pi\subseteq\Lambda\\{}[\Lambda:\Pi]=p}}F(p^{-1/n}\Pi). \]

Continue with lectures 3 and 4.