The ‘Borcherds products everywhere’ construction [Gritsenko et al., ‘Borcherds products everywhere’, J. Number Theory148 (2015), 164–195] creates paramodular Borcherds products from certain theta blocks. We prove that the $q$ -order of every such Borcherds product lies in a sequence $\{C_{\unicode[STIX]{x1D708}}\}$ , depending only on the $q$ -order $\unicode[STIX]{x1D708}$ of the theta block. Similarly, the $q$ -order of the leading Fourier–Jacobi coefficient of every such Borcherds product lies in a sequence $\{A_{\unicode[STIX]{x1D708}}\}$ , and this is the sequence $\{a_{n}\}$ from work of Newman and Shanks in connection with a family of series for $\unicode[STIX]{x1D70B}$ . Our proofs use a combinatorial formula giving the Fourier expansion of any theta block in terms of its germ.

We identify the majority of Siegel modular eigenforms in degree four and weights up to 16 as being Duke–Imamoḡlu–Ikeda or Miyawaki–Ikeda lifts. We give two examples of eigenforms that are probably also lifts but of an undiscovered type.

We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler's lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Bergé and Martinet and find all dual-critical pairs.

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

We define the concept of a special positive matrix. We use the dyadic trace to prove the result that dim for odd k ≤ 13 and that dim ≤ 4.

We calculate the dimensions of using Erokhin's work on Niemeier lattices and geometric methods involving the hyperelliptic locus.