$n$-Valued Algebraic Monoids, Cubics, and Discriminants

In this talk, we introduce the notions of algebraic $n$-valued monoids and groups, obtain structural results about them, and present important examples. We solve the problem of explicitly describing and classifying coset $n$-valued addition laws constructed from cubic curves. As a consequence, we show that all such addition laws are given by polynomials, whereas the addition laws of formal groups on general cubic curves are given by formal power series.

The discriminants of $n$-valued addition laws carry essential information about the structure of $n$-valued algebraic monoids and groups. For $n$-valued monoids we introduce the notion of iterating elements (for $n=2$ these are doubling elements). In the case $n=2$, the set of doubling elements forms a $2$-valued diagonal submonoid, which we call the doubling-point monoid. We give a description of iterating elements for all $n$-valued algebraic monoids and groups constructed from cubic curves.

For each natural number $n$ we introduce the notion of classes of symmetric $n$-algebraic $n$-valued groups. For $n=2$ and $n=3$ we describe the universal objects in these classes.

This talk is based on joint work with Victor M. Buchstaber, arXiv:2510.14010 and arXiv:2505.04296.