Hecke actions on certain strongly modular genera of lattices (Q1764943)

In a previous article the first author and \textit{B. Venkov} [J. Reine Angew. Math. 531, 49--60 (2001; Zbl 0997.11039)] studied Siegel theta series of lattices in the genus of the Leech lattice \(\Gamma_1\) in dimension 24. Its automorphism group contains the Mathieu group \(M_{23}\) as a subgroup. Now the authors aim at a generalization in the following sense. Consider a cyclic subgroup of square-free order \(l\) in \(M_{23}\). It exists if and only if the sum \(\sigma_1(l)\) of positive divisors of \(l\) divides 24. Let \(\Gamma_l\) be the fixed lattice of this subgroup. It is an extremal strongly \(l\)-modular lattice of even dimension \(2k_l= 24\sigma_0(l)/\sigma_1(l)\). (We will not explain these concepts here.) Let \(V\) be the complex vector space of formal linear combinations of isomorphy classes of lattices in the genus of \(\Gamma_l\), endowed with a certain Hermitian scalar product and with a commutative and associative multiplication. Taking Siegel theta series (of any degree \(m\)) of lattices defines a linear map from \(V\) into spaces of modular forms of weight \(k_l\) (and degree \(m\)). For primes \(p\nmid l\), Hecke operators \(T(p)\) and Kneser neighbouring operators \(K_p\) generate a commutative algebra \({\mathcal H}\) of self-adjoint endomorphisms of \(V\) and of spaces of modular forms. Common eigenspaces are of dimension 1: an orthogonal basis of eigenvectors can be constructed explicitly. An important role plays a cusp form in the span of the Siegel theta series which is constructed following a method of \textit{R. E. Borcherds}, \textit{E. Freitag} and \textit{R. Weissauer} [J. Reine Angew. Math. 494, 141--153 (1998; Zbl 0885.11034)]. The program is performed for all possible values of \(l\). Among them are the cases \(l= 2\), where \(\Gamma_2\) is the Barnes-Wall lattice in dimension 16, and \(l= 3\), where \(\Gamma_3\) is the Coxeter-Todd lattice in dimension 12.