n-ary elliptic groups, rings, and primes in arithmetic progressions

Abstract

I introduced the notion of an elliptic group in [Elliptic groups and rings. Beitr\"age zur Algebra und Geometrie 66(2), 497-529]. It is a quasi-group based on the tangent-chord law of elliptic curves and thus, becomes an abelian group upon singling out an element. This close proximity to abelian groups is reflected in the theory, and among other things, we can define elliptic rings, which are monoidal objects in elliptic groups. An other way of expressing this is to say that they are commutative monoids with an elliptic group structure that distributes over them. In this paper, we generalise this theory from the binary elliptic group structure to the $n$-ary structure, which we call $n$-ary elliptic groups and $n$-ary elliptic rings. The latter are once again (binary) commutative monoids with an $n$-ary operation that distributes over the monoidal structure in an $n$-ary sense. The key interest of these objects for us is their arithmetic properties, which are surprisingly pleasant. The key result is that Dirichlet's famous theorem on arithmetic progressions becomes simply Euclid's theorem in these $n$-ary rings, at least for progressions of the form $an + 1$. Motivated by the hope to eventually prove this $n$-ary Euclidean theorem purely algebraically using the theory of $n$-ary rings (and thus give an alternative and purely algebraic proof of Dirichlet's theorem), we start by exploring the first arithmetic facts of these objects, including introducing the $n$-ary class group and showing that it indeed captures the unique $n$-ary factorisation. We also obtain a type of Dedekinds theorem for our main $n$-ary ring of interest: $\mathsf{nEl}(\mathbb{Z})$.