Genetics of reflective modular forms

In a recent article by V. Gritsenko and V.V. Nikulin (Proc., London Math., Soc., 116 (2018), no.3, 485-522), it was shown that reflective modular forms consist of families around one maternal form. For example, we built series of strictly reflective forms using the sublattices of the roots of the Niemeyer lattices and the Borcherds form $Phi_{12}$. However, the main reason for this phenomenon lies in the special purely arithmetic properties of the quadratic lattices, which will be described in the report. This allows us to obtain many new (not strictly) reflective modular forms, together with new applications in the theory of moduli spaces. Most of the talk will be designed for beginners. We give all the necessary definitions ans an overview of this subject. At the end we formulate several working problems.